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Algorithms
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name: algorithms
compatibility: opencode
completeness: 95
content-types:
- code
- guidance
- examples
- diagrams
description: '''''Provides Comprehensive algorithm selection guide u2014 choose, implement,
and'' optimize algorithms based on time/space trade-offs, input characteristics,
and problem constraints'''''
domain: programming
license: MIT
maturity: stable
metadata:
domain: programming
output-format: code
role: reference
scope: implementation
triggers: algorithms, comprehensive, algorithm, selection
archetypes:
- educational
anti_triggers:
- brainstorming
- vague ideation
response_profile:
verbosity: medium
directive_strength: medium
abstraction_level: tactical
version: "1.0.0"
output-format: code
role: reference
scope: implementation
triggers: algorithm, algorithms, big-o, complexity, data structure, searching, sorting
version: "1.0.0"
---
related-skills: abl-v10-learning, abl-v12-learning
# Skill: programming-algorithms
# Programming Algorithms: A Comprehensive Guide for Algorithm Selection
**Role:** Senior Algorithm Engineer — select, implement, and optimize algorithms for problem-solving across domains.
**Philosophy:** Algorithmic Precision — choose the right tool for the job based on time/space trade-offs, input characteristics, and problem constraints. No fabrication — leverage established, proven algorithms.
## Purpose: Why Standard Algorithms
**The Problem with Fabrication:**
- Reinventing algorithms introduces bugs and suboptimal solutions
- Standard algorithms have decades of academic analysis and optimization
- Edge cases are already understood and handled
- Performance characteristics are well-documented
**When to Use Standard Algorithms:**
1. Problem matches a known pattern (sorting, shortest path, etc.)
2. Input size suggests complexity constraints
3. Resource limits (time/space) are known
4. Industry standards exist for the domain
**Key Principles:**
- **Trade-offs are inevitable:** Time vs space, simplicity vs performance
- **Context matters:** Input size, distribution, and constraints dictate choices
- **Proof first, optimize later:** Ensure correctness before micro-optimizations
---
related-skills: abl-v10-learning, abl-v12-learning
## Table of Contents
1. [Sorting Algorithms](#sorting-algorithms)
2. [Searching Algorithms](#searching-algorithms)
3. [Graph Algorithms](#graph-algorithms)
4. [Dynamic Programming](#dynamic-programming)
5. [Greedy Algorithms](#greedy-algorithms)
6. [String Algorithms](#string-algorithms)
7. [Mathematical Algorithms](#mathematical-algorithms)
8. [Geometric Algorithms](#geometric-algorithms)
9. [Backtracking Algorithms](#backtracking-algorithms)
10. [Numerical Algorithms](#numerical-algorithms)
11. [Probabilistic Algorithms](#probabilistic-algorithms)
12. [Streaming Algorithms](#streaming-algorithms)
13. [Algorithm Selection Guide](#algorithm-selection-guide)
---
related-skills: abl-v10-learning, abl-v12-learning
## Sorting Algorithms
### Quick Sort
- **Alternative Names:** Partition-exchange sort
- **Time Complexity:**
- Best: O(n log n) (balanced partitions)
- Average: O(n log n)
- Worst: O(n²) (poor pivot selection)
- **Space Complexity:** O(log n) (recursion stack)
- **Key Use Cases:**
- General-purpose sorting when cache locality matters
- In-place sorting with minimal memory overhead
- Average-case optimal for random data
- **When to Choose:**
- Data is randomly distributed
- Memory is constrained (in-place)
- Average performance matters more than worst-case guarantee
- Not suitable for linked lists or nearly-sorted data
- **Optimizations:**
- Use median-of-three pivot selection
- Switch to insertion sort for small partitions (n < 10-20)
- Iterative implementation to avoid stack overflow
### Merge Sort
- **Alternative Names:** Mergesort
- **Time Complexity:**
- Best: O(n log n)
- Average: O(n log n)
- Worst: O(n log n)
- **Space Complexity:** O(n) (temporary arrays)
- **Key Use Cases:**
- Linked list sorting
- External sorting (files too large for memory)
- Stable sorting required
- Parallel processing (tasks are independent)
- **When to Choose:**
- Stability is required (equal elements maintain order)
- Sorting linked lists (O(1) extra space)
- External sorting with disk I/O
- Predictable performance needed
- **Optimizations:**
- Bottom-up (iterative) implementation
- Use insertion sort for small subarrays
- Parallel merge sort for multi-core systems
### Heap Sort
- **Alternative Names:** Heap sorting
- **Time Complexity:**
- Best: O(n log n)
- Average: O(n log n)
- Worst: O(n log n)
- **Space Complexity:** O(1) (in-place)
- **Key Use Cases:**
- In-place sorting with guaranteed O(n log n) performance
- Priority queue implementation
- Finding top-k elements (use min-heap of size k)
- Memory-constrained environments
- **When to Choose:**
- Worst-case performance guarantee needed
- Memory is extremely constrained
- Building priority queues
- Not suitable for nearly-sorted data (no early exit)
- **Optimizations:**
- Build heap in O(n) time (Floyd's algorithm)
- Use binary heap for arrays, binomial heap for decreases
### Radix Sort
- **Alternative Names:** Bucket sort (for integers), LSD radix sort
- **Time Complexity:**
- Best: O(nk)
- Average: O(nk)
- Worst: O(nk)
- Where k = number of digits/characters
- **Space Complexity:** O(n + k) (buckets)
- **Key Use Cases:**
- Sorting integers with fixed width
- Sorting strings by characters
- Stable sorting for digit-by-digit processing
- When k is small relative to n
- **When to Choose:**
- Integers with bounded range
- Strings with fixed maximum length
- Need stable sorting
- k is O(1) or very small
- Not suitable for floating-point or large k values
- **Optimizations:**
- MSD (Most Significant Digit) for string sorting
- LSD (Least Significant Digit) for fixed-width integers
- Use counting sort as stable subroutine
### Bucket Sort
- **Alternative Names:** Bin sort
- **Time Complexity:**
- Best: O(n + k) (uniform distribution)
- Average: O(n + k)
- Worst: O(n²) (all elements in one bucket)
- **Space Complexity:** O(nk) (buckets)
- **Key Use Cases:**
- Uniformly distributed floating-point numbers
- Sorting data that can be partitioned into ranges
- Database partitioning
- Parallel processing (independent buckets)
- **When to Choose:**
- Input is uniformly distributed over a range
- Can create appropriate number of buckets
- Parallel sorting allowed
- Not suitable for skewed distributions
### Insertion Sort
- **Time Complexity:**
- Best: O(n) (already sorted)
- Average: O(n²)
- Worst: O(n²)
- **Space Complexity:** O(1) (in-place)
- **Key Use Cases:**
- Nearly-sorted data
- Small datasets (n < 20)
- Online algorithms (inserting elements one at a time)
- Subroutine for hybrid algorithms
- **When to Choose:**
- Small input sizes (often used as base case)
- Data is already partially sorted
- Online sorting (streaming input)
- Minimal memory overhead required
### Bubble Sort
- **Time Complexity:**
- Best: O(n) (optimized with swapped flag)
- Average: O(n²)
- Worst: O(n²)
- **Space Complexity:** O(1) (in-place)
- **Key Use Cases:**
- Educational demonstrations
- Detecting nearly-sorted data
- Small datasets with few swaps needed
- **When to Choose:**
- Almost never for production (except teaching)
- Detect if data is already sorted (O(n))
- Very small datasets where simplicity matters
### Selection Sort
- **Time Complexity:**
- Best: O(n²)
- Average: O(n²)
- Worst: O(n²)
- **Space Complexity:** O(1) (in-place)
- **Key Use Cases:**
- Minimizing number of swaps
- Small datasets where swaps are expensive
- Educational demonstrations
- **When to Choose:**
- Memory writes are costly (minimize to n swaps)
- Small datasets
- Never for large datasets
### Shell Sort
- **Alternative Names:** Shell's method
- **Time Complexity:**
- Best: O(n log² n)
- Average: O(n log² n) to O(n^(3/2))
- Worst: O(n²)
- **Space Complexity:** O(1) (in-place)
- **Key Use Cases:**
- Middle ground between O(n²) and O(n log n)
- When quick sort/merge sort are too complex
- Embedded systems with limited resources
- **When to Choose:**
- Need better than O(n²) without complexity of O(n log n)
- Memory-constrained but need better performance
---
related-skills: abl-v10-learning, abl-v12-learning
## Searching Algorithms
### Binary Search
- **Alternative Names:** Half-interval search, logarithmic search
- **Time Complexity:**
- Best: O(1)
- Average: O(log n)
- Worst: O(log n)
- **Space Complexity:** O(1) iterative, O(log n) recursive
- **Key Use Cases:**
- Finding elements in sorted arrays
- Finding first/last occurrence
- Finding minimum/maximum in bitonic/unimodal functions
- Floating-point binary search (precision search)
- **When to Choose:**
- Data is sorted or can be sorted
- Need O(log n) lookup time
- Static data (not frequently updated)
- Not suitable for unsorted or frequently changing data
- **Variants:**
- Lower bound / Upper bound (first ≥ / first >)
- Rotated array search
- 2D matrix search (row-wise and column-wise sorted)
- Real number binary search (for precision)
### Interpolation Search
- **Time Complexity:**
- Best: O(log log n) (uniform distribution)
- Average: O(log log n)
- Worst: O(n) (non-uniform distribution)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Uniformly distributed sorted data
- Large datasets with known distribution
- Numeric data with continuous values
- **When to Choose:**
- Data is uniformly distributed
- Data is sorted and large
- Not suitable for sparse or non-uniform data
### Exponential Search
- **Alternative Names:** Galloping search, doubling search
- **Time Complexity:**
- Best: O(1)
- Average: O(log i)
- Worst: O(log i)
- Where i is the position of the element
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Unbounded/infinite sorted arrays
- Finding element position for binary search
- When element might be near the beginning
- **When to Choose:**
- Sorted array but size unknown
- Element likely near start
- As preprocessing for binary search
### Linear Search
- **Time Complexity:**
- Best: O(1)
- Average: O(n)
- Worst: O(n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Unsorted data
- Small datasets
- Single search (sorting not worth it)
- Linked lists
- **When to Choose:**
- Data is unsorted
- Small n where O(n) is acceptable
- Single search on large dataset
### Ternary Search
- **Time Complexity:**
- Best: O(1)
- Average: O(log n)
- Worst: O(log n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Finding minimum/maximum of unimodal function
- Convex/concave functions
- Golden section search alternative
- **When to Choose:**
- Optimization of unimodal functions
- When binary search doesn't apply
- Compare with golden section search
### Jump Search
- **Alternative Names:** Block search
- **Time Complexity:**
- Best: O(1)
- Average: O(√n)
- Worst: O(√n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Sorted data where jumping back is expensive
- Large datasets on disk
- Intermediate between linear and binary search
- **When to Choose:**
- Jumping back is costly (disk seeks)
- O(√n) is acceptable
---
related-skills: abl-v10-learning, abl-v12-learning
## Graph Algorithms
### Breadth-First Search (BFS)
- **Time Complexity:** O(V + E)
- **Space Complexity:** O(V) (queue)
- **Key Use Cases:**
- Shortest path in unweighted graphs
- Level-order traversal
- Connected components
- Bipartite checking
- Web crawling
- **When to Choose:**
- Unweighted shortest path
- Need all nodes at distance k
- Flow networks (Ford-Fulkerson)
- Social network analysis
### Depth-First Search (DFS)
- **Time Complexity:** O(V + E)
- **Space Complexity:** O(V) (recursion stack)
- **Key Use Cases:**
- Topological sorting
- Cycle detection
- Strongly connected components
- Maze solving
- Path finding
- **When to Choose:**
- Need to explore all paths
- Stack-based iteration
- Topological sort
- Tarjan's SCC algorithm
- **Variants:**
- Iterative DFS (explicit stack)
- DFS with parent tracking
- DFS forest (multiple components)
### Dijkstra's Algorithm
- **Alternative Names:** Dijkstra's shortest path
- **Time Complexity:**
- O(V²) (naive)
- O((V + E) log V) (with priority queue)
- O(V log V + E) (Fibonacci heap)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Single-source shortest path (non-negative weights)
- Routing protocols
- GPS navigation
- Network optimization
- **When to Choose:**
- Non-negative edge weights
- Single source to all destinations
- Need exact shortest path
- Not suitable for negative weights
- **Optimizations:**
- Use Fibonacci heap for O(V log V + E)
- Early termination when target reached
- Bidirectional Dijkstra for source-target
### Bellman-Ford Algorithm
- **Time Complexity:** O(VE)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Single-source shortest path with negative weights
- Negative cycle detection
- Distributed routing
- Linear programming
- **When to Choose:**
- Graph may have negative edge weights
- Need negative cycle detection
- Distributed systems
- Not suitable for dense graphs (too slow)
### Floyd-Warshall Algorithm
- **Alternative Names:** Floyd's algorithm, Roy-Warshall
- **Time Complexity:** O(V³)
- **Space Complexity:** O(V²)
- **Key Use Cases:**
- All-pairs shortest path
- Transitive closure
- Negative cycle detection
- Density graphs
- **When to Choose:**
- Need all-pairs shortest paths
- Graph is dense (V³ acceptable)
- Small V (V < 200-500)
- Transitive closure needed
- **Optimizations:**
- Use only when V is small
- Can detect negative cycles
### Kruskal's Algorithm
- **Alternative Names:** Minimum spanning tree (Kruskal)
- **Time Complexity:** O(E log E) or O(E log V)
- **Space Complexity:** O(V) (disjoint set)
- **Key Use Cases:**
- Minimum spanning tree
- Network design
- Approximation algorithms
- Clustering
- **When to Choose:**
- Sparse graphs
- Need MST
- Edge-based processing
- Disjoint set data structure available
- **Optimizations:**
- Union by rank + path compression
- Pre-sort edges
### Prim's Algorithm
- **Time Complexity:**
- O(V²) (naive)
- O((V + E) log V) (priority queue)
- O(E + V log V) (Fibonacci heap)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Minimum spanning tree
- Dense graphs
- Network design
- Image segmentation
- **When to Choose:**
- Dense graphs (more edges)
- Need MST
- Vertex-based processing
- Adjacency matrix available
- **Comparison with Kruskal:**
- Kruskal better for sparse
- Prim better for dense
### Topological Sort
- **Time Complexity:** O(V + E)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Dependency resolution
- Course scheduling
- Build systems
- Job scheduling
- **When to Choose:**
- Directed acyclic graph (DAG)
- Need linear ordering
- Dependency ordering required
- Cycle detection (if not DAG)
- **Methods:**
- DFS-based (post-order)
- Kahn's algorithm (BFS with in-degrees)
### A* Search Algorithm
- **Time Complexity:** O(b^d) worst case (where b = branching, d = depth)
- **Space Complexity:** O(b^d)
- **Key Use Cases:**
- Heuristic path finding
- Game AI
- Robotics
- Puzzle solving
- **When to Choose:**
- Need shortest path with heuristic
- Admissible heuristic available
- Want to reduce search space
- Not suitable without good heuristic
- **Heuristic Requirements:**
- Admissible (never overestimates)
- Consistent (triangle inequality)
- **Variants:**
-IDA* (Iterative Deepening A*)
- SMA* (Simplified Memory-Bounded A*)
### Tarjan's SCC Algorithm
- **Time Complexity:** O(V + E)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Strongly connected components
- Graph condensation
- Dependency analysis
- Circuit simulation
- **When to Choose:**
- Find SCCs in directed graph
- Graph condensation needed
- Cycle analysis
- Topological sort on SCCs
### Johnson's Algorithm
- **Time Complexity:** O(V² log V + VE)
- **Space Complexity:** O(V²)
- **Key Use Cases:**
- All-pairs shortest path (sparse graphs)
- Graphs with negative weights
- **When to Choose:**
- Sparse graphs, all-pairs shortest path
- Negative weights allowed
- Better than Floyd-Warshall for sparse
### Chinese Postman Problem
- **Time Complexity:** O(V² log V + E) for undirected
- **Space Complexity:** O(V²)
- **Key Use Cases:**
- Route optimization (mail carrier)
- Circuit board inspection
- Street cleaning
- **When to Choose:**
- Need to traverse all edges
- Minimize total distance
- Graph may have odd-degree vertices
### Traveling Salesman Problem (Approximations)
- **Time Complexity:** Varies by heuristic
- **Space Complexity:** O(V²)
- **Key Use Cases:**
- Route optimization
- Logistics
- Manufacturing (drill positioning)
- **Heuristics:**
- Nearest neighbor: O(V²)
- Christofides: O(V³) (3/2 approximation)
- Simulated annealing
- Genetic algorithms
- **When to Choose:**
- NP-hard problem, need approximation
- Real-world constraints
- Exact solution not required
### Minimum Cut (Stoer-Wagner)
- **Time Complexity:** O(V³) or O(VE + V² log V)
- **Space Complexity:** O(V²)
- **Key Use Cases:**
- Network reliability
- Image segmentation
- Clustering
- **When to Choose:**
- Find minimum edge cut
- Graph partitioning
- No source-sink constraint
### Maximum Flow (Ford-Fulkerson)
- **Time Complexity:** O(E * max_flow) (integer capacities)
- **Space Complexity:** O(V + E)
- **Key Use Cases:**
- Network flow
- Bipartite matching
- Image segmentation
- transportation problems
- **When to Choose:**
- Flow network optimization
- Matching problems
- Integer capacities
- **Variants:**
- Edmonds-Karp: O(VE²) (BFS)
- Dinic's: O(V²E) (level graph)
- Push-relabel: O(V²E) (more efficient in practice)
### Hopcroft-Karp Algorithm
- **Time Complexity:** O(E * √V)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Maximum bipartite matching
- Assignment problems
- Job scheduling
- **When to Choose:**
- Bipartite graph matching
- Better than Ford-Fulkerson for bipartite
- Sparse graphs
### Max-Flow Min-Cut Theorem Applications
- **Time Complexity:** Same as underlying max-flow algorithm
- **Key Use Cases:**
- Image segmentation (graph cuts)
- Computer vision
- Parallel computing
- VLSI design
---
related-skills: abl-v10-learning, abl-v12-learning
## Dynamic Programming
### 0/1 Knapsack Problem
- **Time Complexity:** O(nW) where W = capacity
- **Space Complexity:** O(nW) or O(W) (optimized)
- **Key Use Cases:**
- Resource allocation
- Investment portfolio
- Container packing
- Knapsack variations
- **When to Choose:**
- Items can only be taken once
- Capacity constraint
- Optimal substructure exists
- **Variants:**
- Unbounded knapsack (unlimited items)
- Bounded knapsack (limited quantities)
- Multiple knapsack
- Fractional knapsack (greedy, not DP)
- **Optimizations:**
- Space optimization (1D array)
- Pruning based on bounds
- Meet-in-the-middle for large n
### Longest Common Subsequence (LCS)
- **Time Complexity:** O(mn) where m, n = string lengths
- **Space Complexity:** O(mn) or O(min(m, n))
- **Key Use Cases:**
- Diff utilities
- Bioinformatics (DNA matching)
- Version control
- Plagiarism detection
- **When to Choose:**
- Two sequences common subsequence
- Order matters, continuity not required
- Not suitable for substring (use KMP/Rabin-Karp)
- **Reconstruction:**
- Track decisions during DP
- Backtrack to build actual LCS
- **Optimizations:**
- Hirschberg's algorithm: O(min(m,n)) space
- Early termination if no match
### Longest Increasing Subsequence (LIS)
- **Time Complexity:** O(n²) (DP) or O(n log n) (patience sorting)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Pattern recognition
- Stock market analysis
- Bioinformatics
- Data smoothing
- **When to Choose:**
- Strictly increasing (or non-decreasing)
- Need longest monotonic subsequence
- **O(n log n) Method:**
- Maintain active lists
- Binary search for insertion
- Track predecessors for reconstruction
- **Variants:**
- Longest decreasing subsequence
- Bitonic subsequence
- Circular variant
### Matrix Chain Multiplication
- **Time Complexity:** O(n³)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Optimal parenthesization
- Compiler optimization
- Dynamic programming example
- **When to Choose:**
- Matrix multiplication order
- Minimize scalar multiplications
- All matrices compatible
- **Optimizations:**
- Store optimal split points
- Reconstruction for actual multiplication order
### Edit Distance (Levenshtein)
- **Time Complexity:** O(mn)
- **Space Complexity:** O(mn) or O(min(m,n))
- **Key Use Cases:**
- Spell checking
- DNA sequence alignment
- Fuzzy string matching
- Version control
- **When to Choose:**
- Minimum edits to transform string A to B
- Insert, delete, replace operations
- **Variants:**
- Hamming distance (same length, replace only)
- Damerau-Levenshtein (adjacent swap)
- Wagner-Fischer (generalization)
- **Optimizations:**
- Space optimization
- Early termination for small distances
### Coin Change Problem
- **Time Complexity:** O(n * amount) where n = coin types
- **Space Complexity:** O(amount)
- **Key Use Cases:**
- Making change (min coins)
- Combinatorial counting
- Resource allocation
- **When to Choose:**
- Minimize number of coins
- Count ways to make amount
- DP applies (optimal substructure)
- **Variants:**
- Minimum coins (0/1 or unlimited)
- Count combinations
- With limited coins
- Greedy doesn't always work
### Subset Sum Problem
- **Time Complexity:** O(n * sum) or O(n * 2^(n/2)) (meet-in-middle)
- **Space Complexity:** O(n * sum)
- **Key Use Cases:**
- Scheduling
- Resource allocation
- Cryptography
- NP-complete problems
- **When to Choose:**
- Find subset with given sum
- Decision problem (existential)
- Optimization variant exists
- **Optimizations:**
- Meet-in-the-middle for large n
- Bitset optimization
- Pseudo-polynomial DP
### Traveling Salesman Problem (Dynamic Programming)
- **Time Complexity:** O(n² * 2^n)
- **Space Complexity:** O(n * 2^n)
- **Key Use Cases:**
- Exact TSP for small n
- Algorithm comparison
- Benchmarking
- **When to Choose:**
- n < 20-25
- Need exact solution
- Not suitable for large n
- **Held-Karp Algorithm:**
- DP with bitmask
- Track visited set and last city
### Partition Problem
- **Time Complexity:** O(n * sum)
- **Space Complexity:** O(sum)
- **Key Use Cases:**
- Fair division
- Load balancing
- NP-complete problems
- **When to Choose:**
- Split into equal-sum subsets
- Decision variant
- Optimization (minimize difference)
### Longest Palindromic Subsequence
- **Time Complexity:** O(n²)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Palindrome analysis
- Bioinformatics
- String algorithms
- **When to Choose:**
- Find longest palindromic subsequence
- Not substring (LPS can skip chars)
- **Variants:**
- Longest palindromic substring (manacher's O(n))
- Minimum deletions to make palindrome
### Word Break Problem
- **Time Complexity:** O(n²) with dictionary lookup
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Text segmentation
- Dictionary matching
- Natural language processing
- **When to Choose:**
- Can string be segmented into dictionary words?
- Count all possible segmentations
- **Optimizations:**
- Trie for dictionary lookup
- Memoization
- Early termination
### Wildcard Pattern Matching
- **Time Complexity:** O(mn)
- **Space Complexity:** O(mn) or O(min(m,n))
- **Key Use Cases:**
- Regex matching
- File pattern matching
- Text processing
- **When to Choose:**
- Pattern with ? and * wildcards
- Match against text
- **Variants:**
- Regex with character classes
- Case sensitivity
- Multiline support
### Unique Paths
- **Time Complexity:** O(mn)
- **Space Complexity:** O(mn) or O(min(m,n))
- **Key Use Cases:**
- Grid path counting
- Combinatorics
- Robot motion planning
- **When to Choose:**
- Grid with obstacles
- Count paths from top-left to bottom-right
- Only right/down moves allowed
- **Variants:**
- With obstacles (grid[i][j] = 1 blocked)
- With costs (minimum cost path)
- With forbidden cells
### Egg Dropping Puzzle
- **Time Complexity:** O(n * k²) or O(n * log k)
- **Space Complexity:** O(nk)
- **Key Use Cases:**
- Testing/quality assurance
- Optimization under uncertainty
- Decision theory
- **When to Choose:**
- Minimize trials to find critical floor
- k eggs, n floors
- Binary search when 2 eggs
### Catalan Numbers Applications
- **Time Complexity:** O(n²) for DP, O(n) for formula
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Parentheses matching
- Binary tree counting
- Polygon triangulation
- Dyck paths
- **When to Choose:**
- Problems with Catalan structure
- Combinatorial counting
- Recursive structure
- **Applications:**
- n pairs of valid parentheses
- n+1 leaves in full binary tree
- n×n grid monotonic paths
- Convex polygon triangulation
### Optimal Binary Search Tree
- **Time Complexity:** O(n³)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Compiler design
- Database indexing
- Optimal search structure
- **When to Choose:**
- Given probabilities, build optimal BST
- Minimize search cost
- Static search set
- **Optimizations:**
- Knuth's optimization (if quadrangle inequality)
- O(n²) with Knuth optimization
### Bitmask DP Applications
- **Time Complexity:** O(n * 2^n) or O(m * 3^(n/2))
- **Space Complexity:** O(2^n)
- **Key Use Cases:**
- Subset problems
- Graph problems (TSP, Hamiltonian)
- Set cover
- **When to Choose:**
- n < 20-25
- Subsets or states can be encoded as bitmask
- State space is 2^n
### DP on Trees
- **Time Complexity:** O(V) for simple, O(V * k²) for k-state
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Tree diameter
- Tree center
- Tree coloring
- Tree independence
- **When to Choose:**
- Tree structure
- Root the tree arbitrarily
- Combine children's results
- **Common Patterns:**
- Tree diameter (two DFS)
- Tree center (eccentricity)
- Tree isomorphism
- Tree knapsack
### DP with Bitwise Operations
- **Time Complexity:** Varies
- **Space Complexity:** Varies
- **Key Use Cases:**
- Subset XOR sums
- Bit manipulation problems
- State compression
- **When to Choose:**
- Bitwise operations on subsets
- XOR-based problems
- Bitmask DP
---
related-skills: abl-v10-learning, abl-v12-learning
## Greedy Algorithms
### Activity Selection Problem
- **Time Complexity:** O(n log n) (sorting) or O(n) (if sorted)
- **Space Complexity:** O(1) extra
- **Key Use Cases:**
- Scheduling resources
- Meeting room allocation
- Single-resource scheduling
- **When to Choose:**
- Select maximum non-overlapping activities
- Greedy choice works (earliest finish time)
- Not for weighted activities (need DP)
### Huffman Coding
- **Time Complexity:** O(n log n) (priority queue)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Data compression
- Prefix codes
- Optimal binary encoding
- **When to Choose:**
- Character frequencies known
- Minimal expected codeword length
- Prefix-free encoding needed
- **Algorithm:**
- Build frequency table
- Create min-heap of nodes
- Combine two smallest frequencies
- Build tree and assign codes
### Kruskal's MST (Greedy)
- **Time Complexity:** O(E log E)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Minimum spanning tree
- Network design
- Clustering
- **When to Choose:**
- Sparse graphs
- Edge-based processing
- Union-find available
### Prim's MST (Greedy)
- **Time Complexity:** O(V²) or O(E log V)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Minimum spanning tree
- Dense graphs
- Vertex-based processing
- **When to Choose:**
- Dense graphs
- Adjacency matrix
- Vertex expansion
### Dijkstra's Algorithm (Greedy)
- **Time Complexity:** O((V + E) log V)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Shortest path (non-negative)
- Routing
- Network optimization
- **When to Choose:**
- Non-negative edge weights
- Single source
- Greedy choice (shortest known distance)
### Fractional Knapsack
- **Time Complexity:** O(n log n) (sorting)
- **Space Complexity:** O(1) extra
- **Key Use Cases:**
- Resource allocation
- Maximizing value with weight limit
- Continuous items
- **When to Choose:**
- Items can be split
- Value/weight ratio matters
- Not 0/1 knapsack (needs DP)
### Job Sequencing with Deadlines
- **Time Complexity:** O(n²) or O(n log n) with union-find
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Job scheduling
- Profit maximization
- Deadline constraints
- **When to Choose:**
- Jobs with deadlines and profits
- One unit time per job
- Maximize total profit
### Coin Change (Greedy)
- **Time Complexity:** O(n) where n = number of coins
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Standard currency systems
- USD, EUR coin systems
- Greedy-valid denominations
- **When to Choose:**
- Greedy-valid currency (US, EUR)
- Not for arbitrary denominations
- Check if greedy works first
### Graph Coloring (Greedy)
- **Time Complexity:** O(V + E)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Register allocation
- Scheduling
- Map coloring
- **When to Choose:**
- Approximation needed
- Order matters
- Not optimal but fast
### Stable Marriage Problem (Gale-Shapley)
- **Time Complexity:** O(n²)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Hospital-resident matching
- School choice
- Two-sided matching
- **When to Choose:**
- Two sets with preferences
- Stable matching required
- Men-optimal/women-optimal
### Minimum Spanning Tree (General)
- **Time Complexity:** O(E log V)
- **Space Complexity:** O(V)
- **Key Use Cases:**
- Network design
- Approximation algorithms
- Clustering
- **When to Choose:**
- Connected, undirected graph
- Minimum total edge weight
- Greedy algorithms work
### Job Scheduler (Shortest Job First)
- **Time Complexity:** O(n log n) (priority queue)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Process scheduling
- Batch processing
- Minimize average wait time
- **When to Choose:**
- Process burst times known
- Minimize average waiting time
- Non-preemptive or preemptive
### Interval Scheduling (Weighted)
- **Time Complexity:** O(n log n) with binary search
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Resource allocation with weights
- Profit maximization
- Job selection
- **When to Choose:**
- Weighted activities
- Non-overlapping subset
- Greedy doesn't work, use DP
---
related-skills: abl-v10-learning, abl-v12-learning
## String Algorithms
### KMP (Knuth-Morris-Pratt)
- **Time Complexity:** O(n + m) where n = text, m = pattern
- **Space Complexity:** O(m) (LPS array)
- **Key Use Cases:**
- Pattern matching
- DNA sequence search
- Text editors
- Security scanning
- **When to Choose:**
- Multiple pattern occurrences
- Pattern has repetitions
- Need linear time guarantee
- Preprocessing pattern allowed
- **LPS Array:**
- Longest proper prefix which is also suffix
- Avoids re-comparing characters
### Rabin-Karp
- **Time Complexity:** O(n + m) average, O(nm) worst
- **Space Complexity:** O(1) (constant operations)
- **Key Use Cases:**
- Plagiarism detection
- Multi-pattern matching
- String hashing
- Duplicate detection
- **When to Choose:**
- Multiple patterns to search
- Rolling hash useful
- Average case acceptable
- Hash collisions manageable
### Boyer-Moore
- **Time Complexity:** O(n/m * m!) worst, O(n/m) average
- **Space Complexity:** O(σ) where σ = alphabet size
- **Key Use Cases:**
- Large alphabet (ASCII, Unicode)
- Large text, small pattern
- Text editors (grep)
- Bioinformatics
- **When to Choose:**
- Large alphabet (letters, not just ACGT)
- Pattern near end of text
- Good heuristic behavior
- Not for small alphabet
### Z-Algorithm
- **Time Complexity:** O(n + m)
- **Space Complexity:** O(n + m)
- **Key Use Cases:**
- Pattern matching
- String prefix matching
- String repetition detection
- Concatenation problems
- **When to Choose:**
- Z-array computation
- Prefix matching
- Alternative to KMP
- Suffix matching with sentinel
### Manacher's Algorithm
- **Time Complexity:** O(n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Longest palindromic substring
- All palindromes in string
- Palindrome density
- **When to Choose:**
- Linear time palindrome
-Substring (not subsequence)
- All palindromes needed
- **Key Insight:**
- Uses symmetry to avoid re-computation
- Expands around centers with memoization
### Suffix Array
- **Time Complexity:** O(n log n) (sort) or O(n) (SAIS)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Pattern matching (with binary search)
- Burrows-Wheeler transform
- Data compression
- Genomics
- **When to Choose:**
- Multiple queries on same text
- Memory efficiency
- LCP array for additional queries
- Alternative to suffix tree
- **Construction:**
- Sorting all suffixes
- Radix sort for O(n)
- DC3 algorithm for O(n)
### Suffix Tree (Ukkonen's)
- **Time Complexity:** O(n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Fast pattern matching
- Longest repeated substring
- Substring queries
- Bioinformatics
- **When to Choose:**
- Single query, fast lookup
- Multiple pattern queries
- Space permits
- Complex query support
- **Applications:**
- Longest repeated substring
- Longest common substring
- Palindrome detection
### Rolling Hash (Rabin-Karp)
- **Time Complexity:** O(1) per shift
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Rabin-Karp
- String matching
- Duplicate detection
- Streaming
- **When to Choose:**
- Multiple substring hashes needed
- Window sliding
- Hash collision handling
### Longest Prefix Suffix (LPS) / Failure Function
- **Time Complexity:** O(m)
- **Space Complexity:** O(m)
- **Key Use Cases:**
- KMP algorithm
- String border
- Periodicity detection
- **When to Choose:**
- KMP preprocessing
- String borders
- Pattern analysis
### Boyer-Moore-Horspool
- **Time Complexity:** O(n) average
- **Space Complexity:** O(σ)
- **Key Use Cases:**
- Simplified Boyer-Moore
- Text search
- Binary files
- **When to Choose:**
- Simpler than Boyer-Moore
- Good average performance
- Fixed alphabet
### Apostolico-Giancarlo
- **Time Complexity:** O(n) average
- **Space Complexity:** O(m)
- **Key Use Cases:**
- Speeding up KMP
- Avoiding re-comparisons
- Pattern matching
- **When to Choose:**
- KMP with early termination
- Memory bandwidth limited
- Large pattern
### Multiple Pattern Matching (Aho-Corasick)
- **Time Complexity:** O(n + m + z) where z = matches
- **Space Complexity:** O(m * σ)
- **Key Use Cases:**
- Multiple keywords
- Security scanning
- Text processing
- intrusion detection
- **When to Choose:**
- Multiple patterns (3+)
- All occurrences needed
- Linear time in text length
- Dictionary matching
- **Structure:**
- Trie with failure links
- Output function
### Suffix Trie
- **Time Complexity:** O(m) for query
- **Space Complexity:** O(m * σ^m) (exponential)
- **Key Use Cases:**
- Educational
- Small strings only
- Pattern matching
- **When to Choose:**
- Never for production
- Only for small m
- Understand suffix trees
### Longest Repeated Substring
- **Time Complexity:** O(n) (suffix tree) or O(n log n) (suffix array)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Plagiarism detection
- DNA analysis
- Code duplication
- **When to Choose:**
- Repeat detection
- Suffix tree/array available
- Overlapping allowed or not
### Longest Common Substring
- **Time Complexity:** O(n + m) (suffix tree) or O(nm) (DP)
- **Space Complexity:** O(n + m)
- **Key Use Cases:**
- DNA comparison
- File diff
- Code similarity
- **When to Choose:**
- Contiguous match
- Suffix tree/array for linear
- DP for simplicity
- **DP Approach:**
- Table[i][j] = length of common substring ending at i, j
- Maximum value is answer
### Palindromic Tree (Eertree)
- **Time Complexity:** O(n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- All palindromes in string
- Palindrome counting
- Palindromic density
- **When to Choose:**
- All palindromes
- Online algorithm
- Memory efficient
- **Structure:**
- Two roots (even/odd length)
- Suffix links
- Palindromic nodes
### String Matching with Wildcards
- **Time Complexity:** O(mn)
- **Space Complexity:** O(mn)
- **Key Use Cases:**
- Shell globbing
- File matching
- Query patterns
- **When to Choose:**
- Pattern with ? and *
- DP approach
- Memoization for optimization
---
related-skills: abl-v10-learning, abl-v12-learning
## Mathematical Algorithms
### Euclidean GCD
- **Time Complexity:** O(log min(a, b))
- **Space Complexity:** O(1) iterative, O(log n) recursive
- **Key Use Cases:**
- Simplifying fractions
- LCM calculation
- Cryptography
- Number theory
- **When to Choose:**
- Greatest common divisor
- Euclidean algorithm
- Binary GCD alternative for bit operations
- **Extensions:**
- Extended GCD (Bezout coefficients)
- Multiple number GCD
- LCM = (a * b) / GCD(a, b)
### Binary Exponentiation (Fast Power)
- **Time Complexity:** O(log n)
- **Space Complexity:** O(log n) recursive, O(1) iterative
- **Key Use Cases:**
- Power computation
- Matrix exponentiation
- Modular exponentiation
- Fibonacci numbers
- **When to Choose:**
- Large exponents
- Modular arithmetic
- Matrix powers
- Exponentiation by squaring
- **Variants:**
- Iterative implementation
- Modular exponentiation (a^b mod m)
- Matrix exponentiation
- Fast Fibonacci (O(log n))
### Sieve of Eratosthenes
- **Time Complexity:** O(n log log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Prime generation
- Primality testing
- Number theory
- Cryptography preprocessing
- **When to Choose:**
- Generate primes up to n
- Multiple primality tests
- Sieve is better for batch
- **Optimizations:**
- Sieve of Atkin (O(n / log log n))
- Segmented sieve (for large n)
- Only odd numbers
- Bitset compression
### Extended Euclidean Algorithm
- **Time Complexity:** O(log min(a, b))
- **Space Complexity:** O(log n)
- **Key Use Cases:**
- Modular inverse
- Bezout coefficients
- Chinese Remainder Theorem
- RSA cryptography
- **When to Choose:**
- Find x, y such that ax + by = GCD(a, b)
- Modular inverse exists
- Linear Diophantine equations
### Miller-Rabin Primality Test
- **Time Complexity:** O(k * log³ n) where k = iterations
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Large number primality
- Cryptography
- Probabilistic testing
- BigInteger libraries
- **When to Choose:**
- Large numbers (100+ bits)
- Probabilistic acceptable
- Deterministic for 64-bit (specific bases)
- **Deterministic:**
- For n < 2^64, specific bases guarantee correctness
- Common bases: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
### Lucas-Lehmer Primality Test
- **Time Complexity:** O(log² p) for Mersenne number M_p
- **Space Complexity:** O(log p)
- **Key Use Cases:**
- Mersenne primes
- Large prime discovery
- GIMPS
- **When to Choose:**
- Mersenne numbers (2^p - 1)
- specifically for Mersenne primes
- Deterministic for Mersenne
### Modular Arithmetic
- **Key Operations:**
- Addition: (a + b) mod m
- Multiplication: (a * b) mod m
- Division: a * mod_inverse(b, m) mod m
- Subtraction: (a - b + m) mod m
- **When to Choose:**
- Avoid overflow
- Cryptography
- Large number arithmetic
### Fast Fourier Transform (FFT)
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Polynomial multiplication
- Signal processing
- Large integer multiplication
- Convolution
- **When to Choose:**
- Polynomial multiplication
- Convolution theorem
- Signal analysis
- Circular convolution
### Karatsuba Multiplication
- **Time Complexity:** O(n^log₂3) ≈ O(n^1.585)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Large integer multiplication
- divide and conquer
- Algorithm comparison
- **When to Choose:**
- Large integers (beyond native type)
- Recursive approach
- Better than O(n²) for big n
### Strassen's Matrix Multiplication
- **Time Complexity:** O(n^log₂7) ≈ O(n^2.807)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Large matrix multiplication
- Divide and conquer
- Algorithm comparison
- **When to Choose:**
- Large matrices
- Beyond naive O(n³)
- Space for recursion
- Crossover point exists (n > 100-1000)
### Chinese Remainder Theorem
- **Time Complexity:** O(k * log² n) where k = congruences
- **Space Complexity:** O(k)
- **Key Use Cases:**
- Modular arithmetic
- Cryptography (RSA)
- Large number computation
- Number theory
- **When to Choose:**
- System of congruences
- Pairwise coprime moduli
- Reconstruct from remainders
- CRT optimization
### Discrete Logarithm
- **Algorithms:**
- Brute force: O(n)
- Baby-step Giant-step: O(√n) time/space
- Pollard's rho: O(√n) time, O(1) space
- Pohlig-Hellman: for smooth order
- **When to Choose:**
- Solve a^x ≡ b (mod n)
- Cryptography (Diffie-Hellman, ECC)
- Group theory
- **Baby-step Giant-step:**
- Time: O(√n)
- Space: O(√n)
- Precompute baby steps
### BigInteger Arithmetic
- **Algorithms:**
- Addition: O(n)
- Multiplication: O(n²) naive, O(n^1.585) Karatsuba
- Division: O(n²) long division
- GCD: O(log n) Euclidean
- **When to Choose:**
- Arbitrary precision
- Libraries available (GMP, BigInteger)
- Algorithm selection based on size
### Polygon Area (Shoelace Formula)
- **Time Complexity:** O(n) where n = vertices
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Geometry
- Computer graphics
- Geographic information
- **When to Choose:**
- Simple polygon area
- Vertices in order
- Positive for CCW, negative for CW
### Point in Polygon (Ray Casting)
- **Time Complexity:** O(n) where n = polygon vertices
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Graphics
- GIS
- Collision detection
- **When to Choose:**
- Point containment
- Simple polygons
-射线投射法
### Convex Hull (Graham Scan)
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Computational geometry
- Collision detection
- Pattern recognition
- Image processing
- **When to Choose:**
- Convex hull needed
- Points in plane
- Graham scan or Andrew's monotone chain
- **Andrew's Chain:**
- Sort by x, then y
- Upper and lower hulls
- O(n log n), numerically stable
### Convex Hull (Jarvis March / Gift Wrapping)
- **Time Complexity:** O(nh) where h = hull vertices
- **Space Complexity:** O(h)
- **Key Use Cases:**
- Few hull vertices
- Adaptive algorithms
- Incremental
- **When to Choose:**
- h << n (few hull points)
- Output-sensitive
- Simple implementation
### Line Segment Intersection
- **Algorithms:**
- Brute force: O(n²)
- Sweep line (Bentley-Ottmann): O((n + k) log n)
- **Key Use Cases:**
- GIS
- CAD
- Collision detection
- **When to Choose:**
- Line segment intersection
- Sweep line for many intersections
### Closest Pair of Points
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Computational geometry
- Graphics
- Clustering
- **When to Choose:**
- 2D closest pair
- Divide and conquer
- Not for high dimensions (curse of dimensionality)
### Farthest Pair of Points (Diameter)
- **Time Complexity:** O(n log n) (rotating calipers on convex hull)
- **Space Complexity:** O(n)
- **Key Use Cases:**
-Bounding box
- Clustering
- Shape analysis
- **When to Choose:**
- Convex hull first
- Rotating calipers
- Diameter of point set
### Triangulation (Ear Clipping)
- **Time Complexity:** O(n²) for simple polygon
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Computer graphics
- Finite element analysis
- Game development
- **When to Choose:**
- Simple polygon triangulation
- O(n²) acceptable
- Simple implementation
- **Alternative:**
- Monotone partitioning: O(n log n)
- Triangulation by diagonal
### Delaunay Triangulation
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Mesh generation
- Interpolation
- Terrain modeling
- **When to Choose:**
- Maximum minimum angle
- Circumcircle property
- Voronoi dual
### Voronoi Diagram
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Nearest neighbor
- Facility location
- GIS
- Biology (cell modeling)
- **When to Choose:**
- Nearest site queries
- Partitioning plane
- Delaunay dual
### Line Clipping (Cohen-Sutherland)
- **Time Complexity:** O(n) where n = line segments
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Graphics
- Rendering
- Clipping windows
- **When to Choose:**
- Rectangle clipping
- Code-based
- Simple implementation
- **Alternative:**
- Liang-Barsky: O(n) with fewer divisions
- Cyrus-Beck: general convex clip
### Sutherland-Hodgman Clipping
- **Time Complexity:** O(n * m) where n = polygon, m = clip window
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Polygon clipping
- Graphics
- GIS
- **When to Choose:**
- Convex clip window
- Polygon clipping
- Simple implementation
### Point in Convex Polygon
- **Time Complexity:** O(log n) with preprocessing, O(n) without
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Graphics
- Collision detection
- GIS
- **When to Choose:**
- Convex polygon
- Multiple queries
- Binary search on angles
### Rotating Calipers
- **Time Complexity:** O(n) after convex hull
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Convex hull diameter
- Minimum width
- Minimum area rectangle
- All antipodal pairs
- **When to Choose:**
- Convex polygon properties
- Antipodal pairs
- After convex hull
### Cross Product Applications
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Collinearity
- Polygon orientation
- Point line position
- Convexity testing
- **When to Choose:**
- 2D geometry
- Orientation test
- Signed area
### Dot Product Applications
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Angle calculation
- Projection
- Perpendicular
- Work calculation
- **When to Choose:**
- Angle between vectors
- Projection length
- Orthogonality
### Polygon Classification
- **Time Complexity:** O(n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Geometry validation
- Graphics
- GIS
- **When to Choose:**
- Convex/concave
- Simple/complex
- Orientation
### Centroid Calculation
- **Time Complexity:** O(n) where n = vertices
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Center of mass
- Graphics
- Physics simulation
- **When to Choose:**
- Polygon centroid
- Weighted average
- Triangle centroid
### Distance from Point to Line/Segment
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Collision detection
- Graphics
- GIS
- **When to Choose:**
- Shortest distance
- Line or segment
- Perpendicular projection
### Angle Between Three Points
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Geometry analysis
- Graphics
- Robot kinematics
- **When to Choose:**
- Turn angle
- Interior angle
- Vector angle
### Circle Operations
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Circle intersection
- Point in circle
- Tangent lines
- **When to Choose:**
- Circle geometry
- Distance comparison
- Quadratic equations
### Line-Circle Intersection
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Collision detection
- Graphics
- Physics
- **When to Choose:**
- Find intersection points
- 0, 1, or 2 intersections
- Quadratic formula
### Circle-Circle Intersection
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Venn diagrams
- Collision detection
- Geometry
- **When to Choose:**
- Two circles intersection
- 0, 1, 2, or infinite points
- Distance between centers
### Triangle Centers
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Geometry
- Graphics
- Engineering
- **Centers:**
- Centroid (medians)
- Circumcenter (perpendicular bisectors)
- Incenter (angle bisectors)
- Orthocenter (altitudes)
### Polygon Triangulation (Monotone Partition)
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Efficient triangulation
- Graphics
- Finite elements
- **When to Choose:**
- Better than ear clipping
- O(n log n) guarantee
- Monotone polygons easy
### Convex Polygon Union
- **Time Complexity:** O(n + m)
- **Space Complexity:** O(n + m)
- **Key Use Cases:**
- Computational geometry
- Graphics
- GIS
- **When to Choose:**
- Two convex polygons
- Merge hulls
- Linear time after convex hull
### Minkowski Sum
- **Time Complexity:** O(nm) or O(n log n + m log m)
- **Space Complexity:** O(n + m)
- **Key Use Cases:**
- Robotics (configuration space)
- Collision detection
- Morphological operations
- **When to Choose:**
- Sum of two polygons
- Configuration space obstacle
- Motion planning
### Line Intersection (General)
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- CAD
- Graphics
- Geometry
- **When to Choose:**
- Two lines intersection
- Parametric form
- Determinant method
### Segment-Segment Intersection
- **Time Complexity:** O(1)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Collision detection
- Graphics
- GIS
- **When to Choose:**
- Finite segments
- Parametric with bounds
- Orientation tests
### Polygon Boolean Operations
- **Algorithms:**
- CGAL library
- Greiner-Hormann
- Sutherland-Hodgman (limited)
- **Time Complexity:** O(n log n)
- **Space Complexity:** O(n)
- **Key Use Cases:**
- CAD
- GIS
- Graphics
- **When to Choose:**
- Union, intersection, difference
- Complex polygons
- Library recommended
---
related-skills: abl-v10-learning, abl-v12-learning
## Backtracking Algorithms
### N-Queens Problem
- **Time Complexity:** O(N!) worst, O(N^N) with pruning
- **Space Complexity:** O(N²) or O(N) with 1D array
- **Key Use Cases:**
- Constraint satisfaction
- Permutation with constraints
- Algorithm demonstration
- **When to Choose:**
- Place N queens on NxN board
- No two queens attack
- Backtracking with pruning
- **Optimizations:**
- 1D array (column positions)
- Symmetry breaking
- Bitmask for O(1) check
- Warnsdorff's heuristic (for knights tour)
### Sudoku Solver
- **Time Complexity:** O(9^81) worst, much better with pruning
- **Space Complexity:** O(81) = O(1)
- **Key Use Cases:**
- Constraint satisfaction
- Puzzle solving
- Logic puzzles
- **When to Choose:**
- Fill 9x9 grid
- Row, column, 3x3 box constraints
- Backtracking with constraint propagation
- **Optimizations:**
- Least constraining value
- MRV (minimum remaining values)
- Forward checking
- Bitmask for quick check
### Rat in Maze
- **Time Complexity:** O(2^(n+m)) worst
- **Space Complexity:** O(nm)
- **Key Use Cases:**
- Path finding
- Maze generation
- Educational
- **When to Choose:**
- Find path from start to end
- Obstacles in grid
- All paths or just one
- **Variants:**
- Find all paths
- Shortest path (BFS)
- Multiple rats
### Knight's Tour
- **Time Complexity:** O(8^n) for n moves
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Hamiltonian path on chessboard
- Backtracking example
- Performance benchmarking
- **When to Choose:**
- Visit every square once
- Knight moves only
- Open or closed tour
- **Optimizations:**
- Warnsdorff's heuristic (fewest onward moves)
- Dividing board
- Parallel search
### Subset Sum
- **Time Complexity:** O(2^n) worst, O(n * sum) DP
- **Space Complexity:** O(n) or O(n * sum)
- **Key Use Cases:**
- Resource allocation
- Cryptography
- NP-complete problems
- **When to Choose:**
- Find subset with given sum
- Decision or optimization
- Backtracking for small n
- **Alternatives:**
- DP for pseudo-polynomial
- Meet-in-the-middle for large n
### Permutation Generation
- **Time Complexity:** O(n * n!)
- **Space Complexity:** O(n) for recursion
- **Key Use Cases:**
- Combinatorics
- Cryptography
- Algorithm testing
- **When to Choose:**
- All permutations of n elements
- Heap's algorithm (minimal changes)
- Lexicographic order
- **Methods:**
- Heap's algorithm
- Lexicographic generation
- Steinhaus-Johnson-Trotter
### Combination Generation
- **Time Complexity:** O(C(n, k) * k)
- **Space Complexity:** O(k)
- **Key Use Cases:**
- Combinatorics
- Probability
- Statistics
- **When to Choose:**
- Choose k from n
- All combinations
- Lexicographic or Gray code
- **Methods:**
- Recursive generation
- Iterative with indices
- Bit manipulation
### Graph Coloring
- **Time Complexity:** O(m^n) where m = colors, n = vertices
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Scheduling
- Register allocation
- Map coloring
- **When to Choose:**
- m-coloring problem
- Find valid coloring
- Backtrack with constraint checking
- **Optimizations:**
- Degree ordering
- Forward checking
- Heuristic ordering
### Hamiltonian Path/Cycle
- **Time Complexity:** O(n!) worst
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Route planning
- Network design
- Algorithm comparison
- **When to Choose:**
- Visit each vertex exactly once
- Path or cycle
- Backtracking with pruning
- **Optimizations:**
- Degree 1 vertices first
- Backtracking with adjacency check
### Maze Generation (Recursive Backtracking)
- **Time Complexity:** O(width * height)
- **Space Complexity:** O(width * height)
- **Key Use Cases:**
- Game development
- Puzzle generation
- Procedural content
- **When to Choose:**
- Generate perfect maze
- All cells reachable
- One solution
- **Algorithm:**
- Start with grid of walls
- Carve path recursively
- Backtrack when stuck
### Partition Problem
- **Time Complexity:** O(2^n) or O(n * sum)
- **Space Complexity:** O(n) or O(sum)
- **Key Use Cases:**
- Fair division
- Load balancing
- NP-complete
- **When to Choose:**
- Split into equal sum subsets
- Decision problem
- Backtracking or DP
### Crossword Puzzle Solver
- **Time Complexity:** Exponential
- **Space Complexity:** O(word count * puzzle size)
- **Key Use Cases:**
- Puzzle solving
- AI application
- Natural language
- **When to Choose:**
- Place words in grid
- Cross constraints
- Backtracking with constraint satisfaction
- **Optimizations:**
- Most constrained variable
- Least constraining value
- Forward checking
### Bin Packing (Backtracking)
- **Time Complexity:** Exponential
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Resource allocation
- Container loading
- Optimization
- **When to Choose:**
- Pack items into bins
- Minimize number of bins
- Exact solution for small n
- **Alternatives:**
- First-fit, best-fit heuristics
- DP for special cases
### Traveling Salesman (Backtracking)
- **Time Complexity:** O(n!) or O(n² * 2^n) DP
- **Space Complexity:** O(n) or O(n * 2^n)
- **Key Use Cases:**
- Route optimization
- Logistics
- Exact solution for small n
- **When to Choose:**
- Visit all cities once
- Return to start
- Exact solution for n < 20-25
- **Optimizations:**
- Branch and bound
- Lower bound pruning
- Symmetry breaking
---
related-skills: abl-v10-learning, abl-v12-learning
## Numerical Algorithms
### Newton-Raphson Method
- **Time Complexity:** O(log precision) iterations, O(1) per iteration
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Root finding
- Optimization (derivative = 0)
- Nonlinear equations
- Square roots
- **When to Choose:**
- Function differentiable
- Good initial guess
- Quadratic convergence near root
- Not for discontinuous functions
- **Optimizations:**
- Secant method (no derivative)
- Hybrid with bisection
- Damped Newton
- **Applications:**
- sqrt(x): f(y) = y² - x
- Reciprocal: f(y) = 1/y - x
- Optimization: find f'(x) = 0
### Bisection Method
- **Time Complexity:** O(log((b-a)/tolerance))
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Root finding (continuous functions)
- Guaranteed convergence
- Interval methods
- **When to Choose:**
- Continuous function
- Sign change in interval
- Slow but reliable
- Not for multiple roots in interval
- **Comparison with Newton:**
- Bisection: guaranteed, linear convergence
- Newton: fast but may not converge
### Gaussian Elimination
- **Time Complexity:** O(n³)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Solving linear systems
- Matrix inverse
- Determinant
- Linear algebra
- **When to Choose:**
- Ax = b for square matrix
- Direct solver
- Not for sparse large systems
- **Optimizations:**
- Partial pivoting (avoid division by small)
- Full pivoting (best accuracy)
- LU decomposition reuse
### LU Decomposition
- **Time Complexity:** O(n³) factorization, O(n²) solve
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Multiple right-hand sides
- Matrix inverse
- Determinant
- Linear systems
- **When to Choose:**
- Solve Ax = b multiple times
- Same matrix, different b
- Matrix invertibility check
### Cholesky Decomposition
- **Time Complexity:** O(n³/3)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Positive definite matrices
- Monte Carlo simulation
- Optimization
- Numerical PDEs
- **When to Choose:**
- Symmetric positive definite
- Faster than LU
- No pivoting needed
### Gradient Descent
- **Time Complexity:** O(iterations * n) where n = parameters
- **Space Complexity:** O(n)
- **Key Use Cases:**
- Optimization
- Machine learning
- Function minimization
- Neural networks
- **When to Choose:**
- Differentiable function
- Large number of parameters
- Stochastic for big data
- **Variants:**
- Batch gradient descent
- Stochastic (SGD)
- Mini-batch
- Momentum
- Adam, RMSProp
### Binary Search (Numerical)
- **Time Complexity:** O(log((high-low)/tolerance))
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Root finding (monotonic functions)
- Inverse functions
- Optimization (unimodal)
- **When to Choose:**
- Monotonic function
- Sign change or monotonic
- Guaranteed convergence
- Slower than Newton but reliable
### Simpson's Rule (Integration)
- **Time Complexity:** O(n) where n = intervals
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Numerical integration
- Area calculation
- Probability
- **When to Choose:**
- Smooth functions
- Higher accuracy than trapezoidal
- Even number of intervals
- **Composite Simpson:**
- Divide into subintervals
- O(1/n⁴) error
### Trapezoidal Rule
- **Time Complexity:** O(n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Numerical integration
- Approximate area
- Simple implementation
- **When to Choose:**
- Quick integration
- Smooth functions
- Simpler than Simpson
- **Comparison:**
- Trapezoidal: O(1/n²) error
- Simpson: O(1/n⁴) error
### Monte Carlo Integration
- **Time Complexity:** O(n) where n = samples
- **Space Complexity:** O(1)
- **Key Use Cases:**
- High-dimensional integration
- Complex domains
- Statistical physics
- **When to Choose:**
- High dimensions (d > 3)
- Complex boundaries
- Probabilistic error estimate
- **Advantages:**
- Dimension doesn't affect complexity
- Easy to parallelize
- Error decreases as 1/√n
### Runge-Kutta Methods (RK4)
- **Time Complexity:** O(n) where n = steps
- **Space Complexity:** O(1) per step
- **Key Use Cases:**
- ODE solving
- Physics simulation
- Engineering
- Dynamics
- **When to Choose:**
- Initial value problems
- Fourth-order accuracy
- Non-stiff equations
- **Standard RK4:**
- Four slope estimates
- Weighted average
- O(h⁴) local error
### Jacobi Iteration
- **Time Complexity:** O(n² * iterations)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Linear systems
- Sparse matrices
- Parallel computing
- **When to Choose:**
- Diagonally dominant
- Sparse systems
- Parallel implementation
- **Alternative:**
- Gauss-Seidel (faster, but sequential)
- SOR (Successive Over-Relaxation)
### Gauss-Seidel Iteration
- **Time Complexity:** O(n² * iterations)
- **Space Complexity:** O(n²)
- **Key Use Cases:**
- Linear systems
- PDE discretization
- Iterative solver
- **When to Choose:**
- Convergent systems
- Better than Jacobi
- Sequential updates
### Fixed-Point Iteration
- **Time Complexity:** O(iterations)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Equation solving
- Functional equations
- Proofs
- **When to Choose:**
- g(x) = x formulation
- Contraction mapping
- Simple implementation
- **Convergence:**
- |g'(x)| < 1 near fixed point
- Linear convergence
### Secant Method
- **Time Complexity:** O(log precision) iterations
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Root finding (no derivative)
- Numerical analysis
- Engineering applications
- **When to Choose:**
- Derivative unavailable or expensive
- Good alternative to Newton
- Superlinear convergence
---
related-skills: abl-v10-learning, abl-v12-learning
## Probabilistic Algorithms
### Bloom Filter
- **Time Complexity:** O(k) where k = hash functions
- **Space Complexity:** O(m) where m = bit array size
- **Key Use Cases:**
- Membership testing
- Spell checking
- Network routers
- Database query optimization
- **When to Choose:**
- Allow false positives
- No false negatives
- Memory efficient
- Large dataset
- **Parameters:**
- m = bit array size
- k = number of hash functions
- n = expected items
- Optimal k = (m/n) * ln(2)
- **Operations:**
- add(x): set k bits
- query(x): check k bits
- No delete (counting Bloom filter allows)
### HyperLogLog
- **Time Complexity:** O(1) per update/query
- **Space Complexity:** O(log log N) where N = universe size
- **Key Use Cases:**
- Cardinality estimation
- Unique visitor count
- Network traffic analysis
- Database statistics
- **When to Choose:**
- Estimate distinct elements
- Massive scale (billions)
- Small memory footprint
- Allow ~2% error
- **Algorithm:**
- Hash elements
- Track maximum leading zeros
- Harmonic mean of estimates
- Bias correction
### Count-Min Sketch
- **Time Complexity:** O(k) per operation
- **Space Complexity:** O(k * w) where k = hash functions, w = width
- **Key Use Cases:**
- Frequency estimation
- Heavy hitters
- Network monitoring
- Text analysis
- **When to Choose:**
- Approximate frequency
- Overestimated (never underestimate)
- Stream processing
- Heavy hitters problem
- **Parameters:**
- w = width (accuracy)
- k = depth (confidence)
- Estimate = minimum of k rows
### Reservoir Sampling
- **Time Complexity:** O(n)
- **Space Complexity:** O(k) where k = sample size
- **Key Use Cases:**
- Random sampling from stream
- Unknown stream length
- Memory constraints
- Online algorithms
- **When to Choose:**
- Sample k items from n (n unknown)
- Equal probability for each item
- Single pass
- Each item has k/n probability
- **Algorithm (Algorithm R):**
- Fill reservoir with first k items
- For item i > k, replace with probability k/i
### MinHash
- **Time Complexity:** O(nk) where n = elements, k = hash functions
- **Space Complexity:** O(k)
- **Key Use Cases:**
- Jaccard similarity estimation
- Duplicate detection
- News clustering
- NLP
- **When to Choose:**
- Similarity between sets
- Large-scale data
- Allow approximation
- Jaccard similarity = P(min hash same)
- **Jaccard Similarity:**
- |A ∩ B| / |A ∪ B|
- MinHash estimates this
### Skip List
- **Time Complexity:** O(log n) average, O(n) worst
- **Space Complexity:** O(n) (expected)
- **Key Use Cases:**
- Ordered dictionary
- Concurrent data structure
-替代 balanced trees
- **When to Choose:**
- Sorted data structure
- Concurrent access
- Simpler than balanced trees
- Probabilistic balancing
- **Operations:**
- Search, insert, delete: O(log n) average
- Space: O(n) expected
- No rebalancing needed
### Randomized Quicksort
- **Time Complexity:** O(n log n) expected, O(n²) worst
- **Space Complexity:** O(log n) expected
- **Key Use Cases:**
- Average-case optimal
- Avoiding worst-case
- Input order unknown
- **When to Choose:**
- Input adversarial or unknown
- Expected performance matters
- Random pivot selection
- Same as deterministic but better practical performance
### Monte Carlo Methods
- **Time Complexity:** O(n) where n = samples
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Numerical integration
- Optimization
- Risk analysis
- Physics simulations
- **When to Choose:**
- Deterministic hard
- Probabilistic acceptable
- Parallelization easy
- Error decreases as 1/√n
### Las Vegas Algorithms
- **Time Complexity:** Varies (randomized)
- **Space Complexity:** Varies
- **Key Use Cases:**
- Always correct
- Randomized time
- Quicksort (expected)
- Randomized algorithms
- **When to Choose:**
- Must be correct
- Randomized running time
- Contrasts with Monte Carlo
### Karp-Rabin (Hash-based)
- **Time Complexity:** O(n + m) average
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Pattern matching
- String hashing
- Duplicate detection
- **When to Choose:**
- Substring search
- Hash collisions handled
- Rolling hash application
---
related-skills: abl-v10-learning, abl-v12-learning
## Streaming Algorithms
### Majority Element (Boyer-Moore)
- **Time Complexity:** O(n)
- **Space Complexity:** O(1)
- **Key Use Cases:**
- Finding majority element
- Stream processing
- Memory-constrained
- **When to Choose:**
- Element appears > n/2 times
- Single pass
- O(1) space
- Verify pass if not guaranteed
- **Algorithm:**
- Candidate with count
- Increment on match, decrement otherwise
- When count = 0, new candidate
### FM-Sketch (Flajolet-Martin)
- **Time Complexity:** O(k) per update
- **Space Complexity:** O(k) where k = hash functions
- **Key Use Cases:**
- Distinct element estimation
- Cardinallity estimation
- Predecessor to HyperLogLog
- **When to Choose:**
- Distinct elements
- Large scale
- Allow approximation
- Basic probabilistic counting
### Greenwald-Khanna
- **Time Complexity:** O(log(εn)) per update
- **Space Complexity:** O(ε⁻¹ log(εn))
- **Key Use Cases:**
- Quantile estimation
- Percentiles
- Streaming data
- **When to Choose:**
- Compute approximate quantiles
- Stream with unknown length
- Guaranteed error bound
- ε-error with O(1/ε log(εn)) space
### t-Sketch
- **Time Complexity:** O(log n) per update
- **Space Complexity:** O(ε⁻² log² n)
- **Key Use Cases:**
- Frequency estimation
- Heavy hitters
- Streaming
- **When to Choose:**
- Frequent items
- High accuracy
- Larger space for better accuracy
### Count-Sketch
- **Time Complexity:** O(k) per operation
- **Space Complexity:** O(k * w) where w = width, k = depth
- **Key Use Cases:**
- Frequency estimation
- Sparse recovery
- Machine learning
- **When to Choose:**
- L2 frequency estimation
- Allow sign errors
- Better than Count-Min for L2
### Lossy Counting
- **Time Complexity:** O(n/m) where m = bucket size
- **Space Complexity:** O(m) where m = O(ε⁻¹)
- **Key Use Cases:**
- Frequent items
- Market basket analysis
- Streaming
- **When to Choose:**
- Frequent itemsets
- Space O(1/ε)
- Error bound ε
- Simple implementation
### Space-Saving
- **Time Complexity:** O(1) per operation
- **Space Complexity:** O(ε⁻¹)
- **Key Use Cases:**
- Frequent items
- Top-k elements
- Stream summary
- **When to Choose:**
- Top-k elements
- Space O(k)
- Count-min with update tracking
- Better than Lossy Counting
### Stable Sampling
- **Time Complexity:** O(1) per element
- **Space Complexity:** O(k) where k = sample size
- **Key Use Cases:**
- Streaming sampling
- Sliding window
- Approximate queries
- **When to Choose:**
- Stream with update frequency
- Sample representative items
- Approximate aggregate queries
### Sliding Window Algorithms
- **Key Techniques:**
- Sliding window histogram
- Smooth histograms
- Exponential histograms
- **Key Use Cases:**
- Time-based window
- Decay functions
- Real-time analytics
- **When to Choose:**
- Recent data only
- Time-sensitive analytics
- Window size fixed or decaying
### DGIM (Dwork et al.)
- **Time Complexity:** O(log N) per update
- **Space Complexity:** O(log² N)
- **Key Use Cases:**
- Windowed sum estimation
- Text processing
- Sliding windows
- **When to Choose:**
- Binary stream sum
- Approximate count
- Logarithmic space
---
related-skills: abl-v10-learning, abl-v12-learning
## Algorithm Selection Guide
### Problem Type: Sorting
| Input Size | Distribution | Memory | Recommended Algorithm | Why |
|------------|--------------|--------|----------------------|-----|
| Small (n < 20) | Any | Any | Insertion Sort | Low overhead, fast for small n |
| Small (n < 20) | Nearly sorted | Any | Insertion Sort | O(n) best case |
| Medium (n < 10K) | Random | Constrained | Quick Sort | Average O(n log n), in-place |
| Medium (n < 10K) | Any | Not constrained | Merge Sort | Stable, consistent O(n log n) |
| Medium (n < 10K) | Any | Tight constraint | Heap Sort | Guaranteed O(n log n), O(1) space |
| Large (n > 10K) | Random | Any | Quick Sort (optimized) | Cache-friendly, average optimal |
| Large (n > 10K) | Nearly sorted | Any | Shell Sort or Merge Sort | Avoid O(n²) behavior |
| Large (n > 10K) | Fixed width integers | Any | Radix Sort | O(n) for k = O(1) |
| Linked List | Any | Any | Merge Sort | O(1) extra space, stable |
| External (disk) | Any | Limited RAM | Merge Sort | Sequential I/O, external |
### Problem Type: Searching
| Data Structure | Sorted? | Query Count | Recommended Algorithm | Why |
|----------------|---------|-------------|----------------------|-----|
| Array | Yes | 1 | Binary Search | O(log n) |
| Array | Yes | Many | Binary Search or TBL | Preprocessing amortized |
| Array | No | 1 | Linear Search | No preprocessing |
| Array | No | Many | Sort + Binary Search | O(n log n) preprocessing |
| Linked List | Yes | Any | Linear Search | No random access |
| BST | Yes | Any | BST Search | O(h) = O(log n) balanced |
| Unbalanced BST | Yes | Any | May need balancing | O(n) worst case |
| Hash Table | No | Many | Hash Lookup | O(1) average |
| Array | Yes, uniform | Any | Interpolation Search | O(log log n) for uniform |
| Infinite Array | Yes | Any | Exponential + Binary | Find bound first |
### Problem Type: Graph - Shortest Path
| Graph Type | Edge Weights | Source | Destination | Recommended Algorithm | Why |
|------------|--------------|--------|-------------|----------------------|-----|
| Unweighted | All 1 | Single | Single | BFS | O(V + E), queue-based |
| Unweighted | All 1 | Single | All | BFS | Single run gives all |
| Weighted | Non-negative | Single | Single | Dijkstra | O((V+E) log V) |
| Weighted | Non-negative | Single | All | Dijkstra or Johnson | Johnson for all-pairs |
| Weighted | May be negative | Single | All | Bellman-Ford | O(VE), detects cycles |
| Dense | Any | Single | Single | Dijkstra (V²) | Dense makes heap overhead high |
| Sparse | Non-negative | All | All | Johnson | V × Dijkstra = O(VE log V) |
| All pairs | Any | All | All | Floyd-Warshall | O(V³), simple, negative allowed |
| All pairs | Non-negative | All | All | Dijkstra V times | Better than Floyd for sparse |
### Problem Type: Graph - MST
| Graph Type | Density | Recommended Algorithm | Why |
|------------|---------|----------------------|-----|
| Sparse | E = O(V) | Kruskal | O(E log V), simple |
| Dense | E = O(V²) | Prim | O(V²) or O(E log V) with heap |
| Connected | Any | Both work | Choose based on implementation |
| Disconnected | Any | N/A | No spanning tree exists |
### Problem Type: Graph - Connectivity
| Problem | Recommended Algorithm | Time | Space |
|---------|----------------------|------|-------|
| Connected Components | BFS/DFS | O(V + E) | O(V) |
| Strongly Connected Components | Kosaraju/Tarjan | O(V + E) | O(V) |
| Articulation Points | Tarjan | O(V + E) | O(V) |
| Bridges | Tarjan | O(V + E) | O(V) |
| Bipartite Check | BFS 2-coloring | O(V + E) | O(V) |
| Cycle Detection | DFS | O(V + E) | O(V) |
### Problem Type: Dynamic Programming
| Problem | Input Size | Recommended Approach | Why |
|---------|------------|---------------------|-----|
| 0/1 Knapsack | n < 100, W < 10K | DP table | O(nW) time/space |
| 0/1 Knapsack | n < 40, W large | Meet-in-the-middle | O(n × 2^(n/2)) |
| 0/1 Knapsack | n large, W large | Greedy approximation | FPTAS available |
| LCS | n, m < 1000 | DP table | O(nm) |
| LCS | One string small | DP with O(min) space | Space optimization |
| LCS | Very long strings | Hirschberg | O(min(n,m)) space |
| LIS | n < 10K | DP O(n²) or patience O(n log n) | Patience for large n |
| LIS | Need sequence | DP with parent pointers | Track reconstruction |
| TSP | n < 20 | DP (Held-Karp) | O(n² × 2^n) exact |
| TSP | n > 20 | Heuristics (2-opt, 3-opt) | Exact infeasible |
| Matrix Chain | n < 500 | DP | O(n³) time/space |
| Matrix Chain | n large | Greedy (not optimal) | Heuristic |
| Coin Change | Amount < 10K | DP | O(n × amount) |
| Coin Change | Greedy-valid | Greedy | O(n) |
| Partition | sum < 10K | DP | O(n × sum) |
| Partition | sum large | Karmarkar-Karp | Heuristic |
| Subset Sum | sum < 10K | DP | O(n × sum) |
| Subset Sum | n < 40 | Meet-in-the-middle | O(n × 2^(n/2)) |
### Problem Type: String Matching
| Pattern Size | Text Size | Pattern Characteristics | Recommended Algorithm | Why |
|--------------|-----------|------------------------|----------------------|-----|
| Small (m < 10) | Any | Random | Naive O(mn) | Low overhead |
| Small | Small | Any | Naive | Simple, fast for small |
| Any | Single query | Any | Naive/KMP | KMP for pattern reuse |
| Any | Multiple queries | Any | Build index (suffix array/tree) | Preprocessing amortized |
| Single pattern | Large | Repetitive | KMP | O(n+m) guaranteed |
| Single pattern | Large | Large alphabet | Boyer-Moore | O(n/m) average |
| Multiple patterns | Large | Any | Aho-Corasick | O(n + m + z) |
| With wildcards | Any | ? and * | DP | O(mn) |
| Palindrome | Any | Any | Manacher | O(n) linear |
| Anagram | Any | Any | Sliding window + hash | O(n) |
### Problem Type: Mathematical
| Problem | Number Size | Precision | Recommended Algorithm | Why |
|---------|-------------|-----------|----------------------|-----|
| GCD | Any | Exact | Euclidean | O(log min(a,b)) |
| LCM | Any | Exact | GCD-based | (a × b) / GCD |
| Prime test | Small (< 10^12) | Exact | Trial division | Simple |
| Prime test | Large | Exact | Miller-Rabin | O(k log³ n) |
| Prime test | 64-bit | Exact | Deterministic Miller-Rabin | Specific bases |
| Prime generation | < 10^8 | All | Sieve of Eratosthenes | O(n log log n) |
| Prime generation | Large | All | Segmented sieve | O(n) time, O(√n) space |
| Factorization | Small (< 10^12) | Exact | Trial division | Simple |
| Factorization | Medium | Exact | Pollard's Rho | Fast for small factors |
| Factorization | Large | Exact | Quadratic sieve | Sub-exponential |
| Modular inverse | Prime mod | Exact | Fermat's little theorem | a^(p-2) mod p |
| Modular inverse | Composite | Exact | Extended Euclidean | ax + by = 1 |
| Modular exponentiation | Large | Exact | Binary exponentiation | O(log exp) |
| FFT multiplication | Very large | Exact | FFT | O(n log n) |
| Matrix inverse | Small | Exact | Gaussian elimination | O(n³) |
| Matrix inverse | Large sparse | Approx | Iterative methods | Conjugate gradient |
| Root finding | Smooth function | Approx | Newton-Raphson | Quadratic convergence |
| Root finding | Any continuous | Approx | Bisection | Guaranteed convergence |
| Integration | Smooth | Approx | Simpson's rule | O(1/n⁴) error |
| Integration | High-dim | Approx | Monte Carlo | Dimension-independent |
| ODE solving | Non-stiff | Approx | RK4 | O(h⁴) local error |
### Problem Type: Backtracking
| Problem | Size | Recommended Approach | Why |
|---------|------|---------------------|-----|
| N-Queens | N < 15 | Backtracking with pruning | O(N!) worst case |
| Sudoku | 9×9 | Backtracking + MRV | 81 cells, constraints |
| Sudoku | Larger | CP solver or heuristic | NP-complete |
| Subset Sum | n < 25 | Backtracking | O(2^n) |
| Subset Sum | n large | DP or approximation | Pseudo-polynomial |
| Graph Coloring | Small | Backtracking | NP-complete |
| Hamiltonian Path | Small | Backtracking | NP-complete |
| Maze Generation | Standard | Recursive backtracking | O(V) for perfect maze |
### Problem Type: Numerical
| Problem | Function Properties | Precision | Recommended Algorithm | Why |
|---------|--------------------|-----------|----------------------|-----|
| Root finding | Differentiable | High | Newton-Raphson | Quadratic convergence |
| Root finding | Continuous | Medium | Bisection | Guaranteed, linear |
| Root finding | No derivative | Medium | Secant | Superlinear |
| System Ax=b | Dense | High | Gaussian elimination | Direct, accurate |
| System Ax=b | Sparse | Medium | Iterative (CG, GMRES) | Memory efficient |
| System Ax=b | Symmetric posdef | High | Cholesky | Faster than LU |
| Optimization | Differentiable | Medium | Gradient descent | Scalable |
| Optimization | No gradient | Low | Nelder-Mead | Derivative-free |
| Integration | Smooth 1D | High | Simpson's rule | Accurate |
| Integration | High-dim | Low/Medium | Monte Carlo | Dimension-independent |
| ODE y'=f(t,y) | Non-stiff | Medium | RK4 | Good accuracy |
### Problem Type: Probabilistic
| Problem | Scale | Error Tolerance | Recommended Algorithm | Why |
|---------|-------|----------------|----------------------|-----|
| Membership test | Large | 1% false positive | Bloom filter | O(1) query, space-efficient |
| Cardinality | Billions | 2% error | HyperLogLog | log log N space |
| Frequency estimation | Large | 5% error | Count-Min Sketch | L1 approximation |
| Sampling | Stream | Exact | Reservoir sampling | O(k) space, uniform |
| Similarity | Large | 5% error | MinHash | Jaccard estimate |
| Sorted structure | Concurrent | Probabilistic | Skip list | O(log n) operations |
### Problem Type: Streaming
| Problem | Window | Error | Recommended Algorithm | Why |
|---------|--------|-------|----------------------|-----|
| Majority element | Full stream | Exact | Boyer-Moore | O(1) space |
| Count distinct | Full stream | 2% | HyperLogLog | log log N space |
| Frequency | Full stream | 5% | Count-Min | L1 error bound |
| Quantiles | Full stream | ε | Greenwald-Khanna | Guaranteed error |
| Top-k | Full stream | ε | Space-Saving | O(k/ε) space |
| Windowed sum | Sliding | ε | DGIM | Logarithmic space |
| Heavy hitters | Full stream | ε | Lossy Counting | O(1/ε) space |
---
related-skills: abl-v10-learning, abl-v12-learning
## Code Patterns for Implementation
### Dynamic Programming Template
```python
def dp_template(problem_state):
# Initialize DP table
dp = initialize_dp_table()
# Base cases
dp[base_case] = base_value
# State transition
for state in problem_space:
for choice in valid_choices(state):
dp[state] = optimal(dp[state], transition(dp[previous_state], choice))
return dp[target_state]
```
### Backtracking Template
```python
def backtrack(state, constraints):
if is_solution(state):
record_solution(state)
return
for candidate in generate_candidates(state):
if is_valid(candidate, constraints):
make_move(candidate, state)
backtrack(state, constraints)
undo_move(candidate, state) # Backtrack
```
### Graph Traversal Template
```python
def graph_traversal(graph, start):
visited = set()
queue = [start] # BFS queue or DFS stack
while queue:
node = queue.pop(0) if BFS else queue.pop() # DFS
if node not in visited:
visited.add(node)
process(node)
for neighbor in graph[node]:
if neighbor not in visited:
queue.append(neighbor)
```
### Greedy Algorithm Template
```python
def greedy_selection(items):
result = []
while not_done(items):
candidate = best_candidate(items)
if is_valid(candidate, result):
result.append(candidate)
items.remove(candidate)
return result
```
### Binary Search Template
```python
def binary_search(sorted_array, target):
left, right = 0, len(sorted_array) - 1
while left <= right:
mid = left + (right - left) // 2
if sorted_array[mid] == target:
return mid
elif sorted_array[mid] < target:
left = mid + 1
else:
right = mid - 1
return -1 # Not found
```
### Memoization Template
```python
def memoized_recursive(state, memo={}):
if state in memo:
return memo[state]
if is_base_case(state):
return base_value
result = compute(memoized_recursive(substate1),
memoized_recursive(substate2))
memo[state] = result
return result
```
### Sliding Window Template
```python
def sliding_window(arr, k):
n = len(arr)
if n < k:
return []
# Initialize window
window = initialize_window(arr[0:k])
result = [process(window)]
# Slide window
for i in range(k, n):
remove_old(arr[i - k], window)
add_new(arr[i], window)
result.append(process(window))
return result
```
### Union-Find Template
```python
class UnionFind:
def __init__(self, n):
self.parent = list(range(n))
self.rank = [0] * n
def find(self, x):
if self.parent[x] != x:
self.parent[x] = self.find(self.parent[x]) # Path compression
return self.parent[x]
def union(self, x, y):
px, py = self.find(x), self.find(y)
if px == py:
return False
# Union by rank
if self.rank[px] < self.rank[py]:
px, py = py, px
self.parent[py] = px
if self.rank[px] == self.rank[py]:
self.rank[px] += 1
return True
```
### Monotonic Stack Template
```python
def monotonic_stack(arr):
stack = []
result = [None] * len(arr)
for i, num in enumerate(arr):
# Maintain decreasing/increasing property
while stack and compare(stack[-1], num):
idx = stack.pop()
result[idx] = num # or process
stack.append(i)
return result
```
### Two-Pointer Template
```python
def two_pointers(arr1, arr2):
i, j = 0, 0
result = []
while i < len(arr1) and j < len(arr2):
if condition(arr1[i], arr2[j]):
process(arr1[i], arr2[j])
i += 1
else:
j += 1
# Process remaining
while i < len(arr1):
process_remaining(arr1[i])
i += 1
while j < len(arr2):
process_remaining(arr2[j])
j += 1
return result
```
---
related-skills: abl-v10-learning, abl-v12-learning
## Common Pitfalls
### Dynamic Programming Pitfalls
1. **Wrong State Definition**: Define state precisely. `dp[i][j]` should have clear meaning.
2. **Incorrect Base Cases**: Test with small inputs. Edge cases often fail.
3. **Off-by-One Errors**: Arrays vs indices. `dp[0]` vs `dp[1]`.
4. **Missing Transitions**: Ensure all valid states are reachable.
5. **Space Optimization Bugs**: When reducing to 1D, ensure no used values are overwritten.
6. **Order Matters**: Top-down needs correct recursion order. Bottom-up needs correct iteration order.
### Backtracking Pitfalls
1. **Missing Pruning**: Without pruning, exponential time. Prune early when possible.
2. **State Not Restored**: Always undo moves after recursive call.
3. **Duplicate Solutions**: Use set or careful ordering to avoid duplicates.
4. **Incorrect Termination**: Base case should check complete solution.
5. **Memory Leaks**: Large state objects not cleaned up.
### Graph Algorithm Pitfalls
1. **Indexing Confusion**: 0-indexed vs 1-indexed. Consistency matters.
2. **Undirected vs Directed**: Edge direction affects algorithm choice.
3. **Disconnected Graphs**: Handle multiple components.
4. **Negative Cycles**: Bellman-Ford detects, Dijkstra doesn't.
5. **Memory Limits**: Adjacency matrix O(V²) vs list O(V + E).
### String Algorithm Pitfalls
1. **Index Bounds**: String indexing can be tricky with encoding.
2. **Off-by-One in LPS**: LPS array construction often has off-by-one.
3. **Hash Collisions**: Rabin-Karp needs good hash function.
4. **Pattern Empty**: Handle edge case of empty pattern.
5. **Unicode Issues**: Consider character vs byte length.
### Mathematical Algorithm Pitfalls
1. **Overflow**: Intermediate calculations may overflow. Use mod carefully.
2. **Division in Modular**: Multiply by modular inverse, not divide.
3. **Floating Point Precision**: Newton's method may not converge perfectly.
4. **Matrix Singularity**: Check for zero pivot in Gaussian elimination.
5. **Edge Cases**: GCD(0, x) = x, power(x, 0) = 1.
### Greedy Algorithm Pitfalls
1. **Greedy Doesn't Always Work**: Prove greedy choice property first.
2. **Local vs Global Optimum**: Greedy finds local, may not be global.
3. **Sorting Overhead**: Sometimes sorting is the answer, not just a step.
4. **Weighted Cases**: Standard greedy fails; need DP or other approach.
### Numerical Algorithm Pitfalls
1. **Division by Zero**: Newton's method can hit zero derivative.
2. **Convergence**: Not all methods converge for all functions.
3. **Stiff Equations**: Standard ODE solvers may fail; use stiff solvers.
4. **Round-off Error**: Accumulated in iterative methods.
5. **Initial Guess**: Newton's method is sensitive to starting point.
### Probabilistic Algorithm Pitfalls
1. **False Positives/Negatives**: Understand and handle them.
2. **Seed Quality**: Randomness quality affects results.
3. **Error Bounds**: Understand and communicate error tolerance.
4. **Space-Time Tradeoff**: Higher accuracy uses more space.
5. **Deterministic Alternative**: Sometimes deterministic is faster.
### Implementation Pitfalls
1. **Integer Division**: Use // in Python, be careful with casting.
2. **Off-by-One in Loops**: `i < n` vs `i <= n`.
3. **Mutable Defaults**: Don't use mutable objects as default arguments.
4. **Pass by Reference**: Ensure copying when needed.
5. **Edge Cases**: Always test with empty input, single element, max size.
---
related-skills: abl-v10-learning, abl-v12-learning
## References
### Primary References
1. **CLRS - Introduction to Algorithms** (3rd Edition) by Cormen, Leiserson, Rivest, Stein
- Comprehensive coverage of all algorithms
- Clear proofs and analysis
- Standard reference for algorithm courses
2. **The Art of Computer Programming** by Donald Knuth
- Volume 1: Fundamental Algorithms
- Volume 2: Seminumerical Algorithms
- Volume 3: Sorting and Searching
- Volume 4: Combinatorial Algorithms
3. **Algorithm Design** by Kleinberg and Tardos
- Great for algorithm design techniques
- Real-world applications
- Clear explanations
4. **Concrete Mathematics** by Graham, Knuth, Patashnik
- Mathematical foundations
- Recurrences, summations, asymptotics
- Essential for algorithm analysis
5. **Programming Challenges** by Skiena and Revilla
- ACM-style problems
- Algorithm categorization
- Practical implementation tips
### Online Resources
1. **GeeksforGeeks** - Great for code examples and explanations
2. **Wikipedia** - Good reference for algorithm details
3. **Algorithm Wiki** - Comprehensive algorithm catalog
4. **CP-algorithms** - Competitive programming algorithms
5. **Visualgo** - Visual algorithm demonstrations
### Specialized Topics
1. **Computational Geometry** by de Berg et al.
2. **Approximation Algorithms** by Vazirani
3. **Randomized Algorithms** by Motwani and Raghavan
4. **Streaming Algorithms** by McGregor
5. **Numerical Recipes** for numerical methods
### Data Structures
1. **Dynamic Arrays** - Amortized O(1) append
2. **Hash Tables** - O(1) average lookup
3. **Balanced BSTs** - O(log n) operations
4. **Heaps** - Priority queues
5. **Tries** - String indexing
6. **Disjoint Sets** - Union-Find
7. **Segment Trees** - Range queries
8. **Fenwick Trees** - Prefix sums
---
related-skills: abl-v10-learning, abl-v12-learning
## Conclusion
Algorithm selection is a critical skill in computer science. This guide provides:
1. **Comprehensive Coverage**: 15+ algorithm categories with detailed entries
2. **Selection Criteria**: Time/space complexity, use cases, and when to choose
3. **Decision Guide**: Quick lookup for common problem types
4. **Code Patterns**: Implementation templates for common paradigms
5. **Pitfall Avoidance**: Common mistakes to watch for
**Remember:**
- Standard algorithms beat fabrication every time
- Trade-offs are inevitable (time vs space, simplicity vs performance)
- Context matters (input size, distribution, constraints)
- Test edge cases thoroughly
- Document your algorithm choice reasoning
**Pro Tip:** When in doubt, start with the simplest algorithm that meets your requirements. Optimize only after profiling confirms it's necessary.
---
*This skill document is designed for OpenCode algorithm selection. For specific implementation questions, refer to the code-philosophy skill for implementation guidelines.*
---
## When to Use
Use this skill when:
- **Implementing or analyzing algorithms** — You need to understand, implement, or optimize algorithms for a specific problem domain
- **Comparing algorithmic approaches** — You are evaluating different algorithms for correctness, performance, or trade-offs
- **Studying computational patterns** — You need reference implementations and complexity analysis for standard algorithmic patterns
---
## Core Workflow
1. **Understand the Problem** — Identify inputs, outputs, constraints, and edge cases. **Checkpoint:** Clearly define the problem statement and expected behavior.
2. **Select Algorithm** — Choose the most appropriate algorithm based on constraints (time/space complexity, data size, ordering requirements). **Checkpoint:** Justify the algorithm choice with complexity analysis.
3. **Implement & Test** — Write clean, readable code with type hints, docstrings, and edge case handling. **Checkpoint:** Verify correctness with multiple test cases including edge cases.
4. **Analyze & Optimize** — Review time/space complexity, identify optimization opportunities, and validate against benchmarks. **Checkpoint:** Confirm complexity bounds match analysis and all test cases pass.
---
## Constraints
### MUST DO
- Include time and space complexity analysis for all algorithms
- Show at least one working example with input/output
- Include diagrams for non-trivial algorithm flows
### MUST NOT DO
- Present an algorithm without its complexity bounds
- Use recursive solutions without discussing tail-call or memoization alternatives
- Omit edge case handling in examples
---
## Live References
> Authoritative documentation links for this skill's domain. The model follows markdown links to resolve external references and inline content.
- [Wikipedia — Algorithm](https://en.wikipedia.org/wiki/Algorithm)
- [Big-O Cheat Sheet](https://bigocheatsheet.com/)
- [Introduction to Algorithms (CLRS) — MIT Press](https://mitpress.mit.edu/9780262033848/introduction-to-algorithms/)
- [GeeksforGeeks — Algorithm Design Techniques](https://www.geeksforgeeks.org/algorithm-design-techniques/)
- [Stanford CS 161 — Algorithm Design and Analysis](https://web.stanford.edu/class/cs161/)
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