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‎projects/sudoku/docs/codeforces_algorithm_toolbox.md‎

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# The Competitive Programming Algorithm Toolbox
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## **1. Basics & Foundations**
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* **Prefix sums / Difference arrays**
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* **Two pointers / Sliding window**
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* **Binary search (on array, on answer, parametric search)**
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* **Sorting tricks (coordinate compression, custom comparators)**
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* **Hashing (rolling hash, string hashing, double hashing)**
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* **Modular arithmetic (mod inverse, fast exponentiation, CRT basics)**
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---
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## **2. Greedy Techniques**
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* Interval scheduling (earliest finishing time)
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* Activity selection / Sweep line methods
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* Huffman coding (optimal merge patterns)
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* Greedy with sorting + priority queue (classic scheduling, intervals, tasks)
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* Exchange argument proofs
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---
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## **3. Divide & Conquer / Binary Search Variants**
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* Divide & conquer DP optimization
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* Convex hull trick (for DP optimization)
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* Ternary search (on unimodal functions)
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* Meet-in-the-middle (subset sum, exponential problems)
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---
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## **4. Graph Theory**
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* BFS / DFS (basic traversals)
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* Topological sort
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* Connected components (Union-Find / DSU)
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* Dijkstra's algorithm (with heap)
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* Bellman-Ford (for negative edges)
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* Floyd-Warshall (all-pairs shortest paths)
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* Minimum spanning tree (Kruskal, Prim)
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* Bridges & articulation points (Tarjan's algorithm)
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* Strongly connected components (Kosaraju, Tarjan)
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* Eulerian path / circuit (Hierholzer's algorithm)
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* Hamiltonian path (bitmask DP or backtracking)
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* Max flow (Ford-Fulkerson, Edmonds-Karp, Dinic)
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* Min-cost max flow
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* Bipartite matching (Hungarian, Hopcroft-Karp)
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* Network flow applications (assignment problems, circulation)
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---
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## **5. Trees**
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* Binary lifting (LCA, ancestor queries)
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* Heavy-light decomposition (path queries)
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* Euler tour + RMQ trick for LCA
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* Centroid decomposition
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* Tree DP (classic rooted DP problems)
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* Rerooting techniques (DP on trees with re-root transitions)
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---
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## **6. Dynamic Programming**
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* Classic 1D/2D DP (knapsack, LIS, edit distance, coin change)
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* Digit DP
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* Subset DP (bitmask DP)
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* SOS DP (sum over subsets)
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* Interval DP (matrix chain multiplication, optimal binary search trees)
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* DP on graphs (DAG longest path, etc.)
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* DP + bitmasks (traveling salesman)
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* DP with convex hull / divide & conquer optimization
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* Tree DP (independent set, diameter, etc.)
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---
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## **7. Advanced Data Structures**
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* Segment tree (point update, range query)
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* Fenwick tree / BIT
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* Persistent segment tree (k-th order statistics)
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* Lazy propagation (range updates, range queries)
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* Treaps, Splay trees
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* Sparse table (RMQ, idempotent functions)
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* Disjoint set union (with path compression, union by rank)
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* Link-cut trees (dynamic trees)
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* Mo's algorithm (offline queries)
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* Wavelet trees (frequency/rank queries)
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---
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## **8. String Algorithms**
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* KMP (pattern matching)
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* Z-function
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* Prefix function (pi array)
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* Rabin-Karp hashing
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* Suffix array (with LCP)
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* Suffix automaton
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* Aho-Corasick automaton (multiple pattern matching)
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* Manacher's algorithm (longest palindrome)
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* Trie / compressed trie
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* Rolling hash tricks (substring equality, palindrome checking)
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---
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## **9. Geometry & Math**
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* Convex hull (Graham scan, Andrew monotone chain)
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* Rotating calipers (diameter, closest pair, width)
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* Line sweep algorithms (events, intersections)
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* Closest pair of points (divide & conquer)
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* Circle intersection, polygon area (shoelace formula)
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* Pick's theorem
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* Fast Fourier Transform (FFT, NTT for polynomial multiplication)
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* Matrix exponentiation
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* Linear algebra (Gaussian elimination, matrix rank)
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* Number theory (sieve of Eratosthenes, segmented sieve)
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* GCD/LCM, extended Euclidean algorithm
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* Modular inverse, Chinese remainder theorem
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* Euler's totient, Mobius function
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* Miller-Rabin primality test, Pollard rho factorization
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---
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## **10. Probability / Randomization**
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* Monte Carlo algorithms (random primality testing)
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* Randomized hashing
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* Reservoir sampling
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* Randomized quicksort / treaps
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---
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## **11. Specialized Techniques**
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* Bit tricks (lowest set bit, bit masks for subsets)
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* Meet-in-the-middle (for exponential search space)
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* Inclusion-exclusion principle
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* Fast Walsh-Hadamard transform (XOR convolutions)
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* Dynamic connectivity (DSU rollback, divide & conquer on queries)
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* Heavy-light + segment tree for dynamic paths
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* Sprague-Grundy theorem (Nim games, impartial games)
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---
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## **Russian-Style Problem-Solving Map**
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### **1. Spotting Hidden Invariants**
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| Technique | Example Problems | Difficulty | Note |
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| -------------------------- | ----------------------------------------------------------------- | ---------- | --------------------------------------------------- |
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| Track invariant quantities | [Game of Stones](https://codeforces.com/problemset/problem/359/B) | Medium | Find what never changes to simplify decision-making |
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| Parity / Modulus tricks | [Even-Odd Game](https://codeforces.com/problemset/problem/451/B) | Easy | Often reduces complex operations to parity checks |
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---
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### **2. Reduce to Canonical Form**
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| Technique | Example Problems | Difficulty | Note |
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| --------------------------- | ---------------------------------------------------------------------------- | ---------- | -------------------------------------- |
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| Sorting / Normalizing input | [Remove Duplicate Intervals](https://leetcode.com/problems/merge-intervals/) | Medium | Makes greedy solutions apparent |
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| Coordinate compression | [Compressed Array Queries](https://codeforces.com/problemset/problem/276/C) | Hard | Reduce large ranges to smaller indices |
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---
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### **3. Backward Reasoning**
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| Technique | Example Problems | Difficulty | Note |
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| -------------------------- | ------------------------------------------------------------------------ | ---------- | ----------------------------------- |
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| Reverse simulation | [Last Person Standing](https://codeforces.com/problemset/problem/602/B) | Medium | Think from end state to start |
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| Goal-oriented construction | [Constructing Sequence](https://codeforces.com/problemset/problem/977/C) | Medium | Build answer by reasoning backwards |
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---
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### **4. Extremal Principle**
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| Technique | Example Problems | Difficulty | Note |
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| ---------------------------- | ----------------------------------------------------------------------------- | ---------- | --------------------------------------- |
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| Consider largest/smallest | [Maximum Pair Sum](https://codeforces.com/problemset/problem/489/C) | Medium | Anchor reasoning on extremal element |
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| First/last element as anchor | [Minimum Swaps to Sort](https://leetcode.com/problems/minimum-swaps-to-sort/) | Medium | Helps find forced moves and constraints |
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---
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### **5. Greedy Guess + Proof**
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| Technique | Example Problems | Difficulty | Note |
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| ---------------------- | --------------------------------------------------------------------------------------- | ---------- | ------------------------------------------------------- |
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| Natural greedy choices | [Activity Selection](https://leetcode.com/problems/non-overlapping-intervals/) | Easy | Pick earliest end or largest gain and prove correctness |
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| Check if greedy fails | [Weighted Interval Scheduling](https://leetcode.com/problems/maximum-intervals-weight/) | Medium | Test greedy on small cases |
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---
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### **6. Simplify to 1D / Projection**
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| Technique | Example Problems | Difficulty | Note |
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| ------------------------- | --------------------------------------------------------------------- | ---------- | ------------------------------------- |
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| Project 2D -> 1D | [Points on a Line](https://codeforces.com/problemset/problem/103/B) | Medium | Reduces complexity |
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| Treat features separately | [Skyline Problem](https://leetcode.com/problems/the-skyline-problem/) | Hard | Separate dimensions / simplify object |
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---
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### **7. Symmetry & Grouping**
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| Technique | Example Problems | Difficulty | Note |
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| ------------------------------ | --------------------------------------------------------------------------- | ---------- | --------------------------- |
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| Treat equivalent objects alike | [Graph Coloring](https://codeforces.com/problemset/problem/1139/B) | Medium | Reduce state space |
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| Exploit symmetry | [Mirror Array Operations](https://codeforces.com/problemset/problem/1669/C) | Medium | Avoid redundant computation |
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---
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### **8. Small Examples & Pattern Guessing**
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| Technique | Example Problems | Difficulty | Note |
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| ----------------------- | ----------------------------------------------------------------------------- | ---------- | ------------------------------ |
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| Test tiny inputs | [Subarray Sum Equals K](https://leetcode.com/problems/subarray-sum-equals-k/) | Medium | Reveals hidden patterns |
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| Explore all small cases | [Permutation Cycles](https://codeforces.com/problemset/problem/499/B) | Medium | Helps predict general behavior |
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---
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### **9. Count, Don't Simulate**
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| Technique | Example Problems | Difficulty | Note |
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| ------------------------------- | ---------------------------------------------------------------------- | ---------- | -------------------------------------------- |
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| Combinatorial counting | [Counting Rectangles](https://codeforces.com/problemset/problem/166/B) | Medium | Count possibilities instead of brute-forcing |
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| DP / math instead of simulation | [Unique Paths](https://leetcode.com/problems/unique-paths/) | Medium | Mathematical shortcuts |
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---
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### **10. Transform Operations**
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| Technique | Example Problems | Difficulty | Note |
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| ------------------------------------ | ----------------------------------------------------------------------------- | ---------- | ---------------------------------------------------- |
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| Replace swaps with position tracking | [Minimum Swaps to Sort](https://leetcode.com/problems/minimum-swaps-to-sort/) | Medium | Track positions instead of performing actual swaps |
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| Simplify operations | [Interval Addition](https://leetcode.com/problems/range-addition/) | Medium | Convert complicated operations to simpler arithmetic |
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---
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### **11. Problem Duality**
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| Technique | Example Problems | Difficulty | Note |
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| --------------------- | -------------------------------------------------------------------------------------------- | ---------- | ---------------------------- |
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| Complement sets | [Number of Subsets](https://leetcode.com/problems/count-of-subsets/) | Medium | Solve “not A” instead of “A” |
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| Min → Max / Max → Min | [Minimize Max Distance](https://leetcode.com/problems/minimize-max-distance-to-gas-station/) | Medium | Flip perspective |
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---
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### **12. Memory-less Thinking / State Reduction**
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| Technique | Example Problems | Difficulty | Note |
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| ------------------------- | -------------------------------------------------------------------------- | ---------- | ---------------------------- |
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| Reduce DP state | [House Robber III](https://leetcode.com/problems/house-robber-iii/) | Medium | Only store essential info |
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| Track key parameters only | [Game State Optimization](https://codeforces.com/problemset/problem/117/C) | Medium | Avoid storing entire history |

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