Interactive algorithm learning

Algorithm Visualizer

See each decision, state change, and line of code. This English catalog contains 95 focused pages built from the same tested visual engines as the full Chinese catalog.

Start with Quick Sort Choose a learning path
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95 pages, one coherent learning catalog

Use the tools first, or jump directly into one of 92 translated data structure and algorithm pages.

Learning Toolkit

Start with a compact reference or follow a guided route through the catalog.
Algorithm Complexity Reference

Algorithm Complexity Reference

Compare the time and space complexity of every translated algorithm, with concise notes and direct links to each visualization.Algorithm Learning Paths

Algorithm Learning Paths

Follow focused learning routes through the translated catalog, from search and sorting fundamentals to graphs, dynamic programming, strings, and geometry.

Data Structures

Understand compact structures that make repeated updates and queries efficient.
Array

Array

Explore indexed access, middle insertion and deletion, and the amortized doubling strategy behind dynamic arrays.Linked List

Linked List

Follow next and previous references, compare traversal with constant-time rewiring, and preserve bidirectional links.Stack

Stack

Push, pop, and peek through one active end while preserving last-in, first-out ordering.Queue and Deque

Queue and Deque

Compare first-in, first-out processing with a double-ended queue that supports both ends.Binary Search Tree

Binary Search Tree

Insert, search, and traverse an ordered tree, then compare a balanced shape with a degenerate chain.Binary Heap

Binary Heap

Keep the maximum at the root while array and tree views trace every sift-up and sift-down swap.Hash Table

Hash Table

Resolve collisions with separate chaining or linear probing and observe how load factor changes lookup work.Graph

Graph

Traverse vertices and edges with a BFS queue or DFS stack and compare their frontier behavior.Trie

Trie

Trace exact and prefix searches through shared character paths and reveal autocomplete candidates.Disjoint Set Union

Disjoint Set Union

Unite components, find representative roots, and watch path compression flatten parent forests.LRU Cache

LRU Cache

Combine a hash table and doubly linked list to update recency and evict the least recently used key.Skip List

Skip List

Search a randomized hierarchy of linked express lanes by moving right and descending level by level.Segment Tree

Segment Tree

Decompose range sums into complete segments and propagate a point update from leaf to root.B+ Tree

B+ Tree

Descend through wide index nodes for exact lookup, then follow linked leaves for a range scan.Bloom Filter

Bloom Filter

Set and probe several hash-selected bits to distinguish definite absence from possible membership.Fenwick Tree

Fenwick Tree

Visualize lowbit jumps for prefix queries and point updates in a Fenwick Tree, with every affected range and tree value shown step by step.

Sorting

Compare adjacent swaps, divide and conquer, heaps, and non-comparison counting.
Bubble Sort

Bubble Sort

Watch adjacent comparisons move the largest remaining value to the sorted suffix on every Bubble Sort pass.Cocktail Shaker Sort

Cocktail Shaker Sort

Alternate forward and backward Bubble Sort passes so both large and small misplaced values move quickly.Bitonic Sort

Bitonic Sort

Build bitonic sequences and merge them through a fixed comparison network whose columns can run in parallel.Selection Sort

Selection Sort

Scan the unsorted suffix for its minimum and place exactly one value on every outer pass.Insertion Sort

Insertion Sort

Grow a sorted prefix by shifting larger values right and inserting each key into its final local position.Binary Insertion Sort

Binary Insertion Sort

Use binary search inside the sorted prefix to locate an insertion point before shifting values into place.Shell Sort

Shell Sort

Run insertion sort over shrinking gaps so distant inversions disappear before the final adjacent pass.Merge Sort

Merge Sort

Follow bottom-up Merge Sort as adjacent sorted runs are merged through an auxiliary array and copied back.Top-Down Merge Sort

Top-Down Merge Sort

Split recursively to single values, then merge sorted halves while the call stack exposes divide and conquer.Quick Sort

Quick Sort

Watch Lomuto partitioning place each pivot, inspect the explicit interval stack, change the input, and follow Quick Sort code in four languages.Three-Way Quick Sort

Three-Way Quick Sort

Partition values into less-than, equal-to, and greater-than regions so duplicate pivots finish together.Dual-Pivot Quick Sort

Dual-Pivot Quick Sort

Choose two pivots and partition one interval into three regions during a single scan.Heap Sort

Heap Sort

Build a max heap, extract its root into the sorted suffix, and watch array and tree views stay synchronized.Counting Sort

Counting Sort

Count a compact integer value range into buckets, then write values back in order without comparing elements.Radix Sort

Radix Sort

Distribute values by one digit at a time and collect buckets stably from least to most significant digit.Bucket Sort

Bucket Sort

Scatter values across numeric ranges, sort each bucket independently, and concatenate the ranges in order.

Graph Algorithms

Trace shortest paths, spanning structures, and dependency order one edge at a time.
Dijkstra's Shortest Path

Dijkstra's Shortest Path

Trace Dijkstra's algorithm as it settles the nearest vertex, relaxes weighted edges, and builds a shortest-path tree on a directed graph.Kruskal's Minimum Spanning Tree

Kruskal's Minimum Spanning Tree

Process weighted edges from lightest to heaviest, use disjoint sets to reject cycles, and watch Kruskal build a minimum spanning tree.Prim's Minimum Spanning Tree

Prim's Minimum Spanning Tree

Grow one connected tree from a starting vertex by repeatedly taking the lightest edge that crosses from the tree to an outside vertex.Bellman-Ford Shortest Paths

Bellman-Ford Shortest Paths

Relax every directed edge for V-1 rounds, follow distances through negative edges, and reveal the final shortest-path tree.Topological Sort

Topological Sort

Run Kahn's algorithm on a directed acyclic graph, remove zero-indegree vertices, and watch each dependency update the final order.Floyd-Warshall

Floyd-Warshall

Update an all-pairs distance matrix by allowing one additional intermediate vertex per round.Strongly Connected Components

Strongly Connected Components

Trace Tarjan discovery and low-link values as one DFS stack emits maximal mutually reachable components.2-SAT

2-SAT

Convert two-literal clauses into implications, group them by SCC, and derive a satisfying assignment.Maximum Flow

Maximum Flow

Find augmenting paths in a residual network, push each bottleneck, and use reverse edges to reroute flow.Bipartite Matching

Bipartite Matching

Use augmenting paths to reassign occupied partners and increase a bipartite matching one edge at a time.Lowest Common Ancestor

Lowest Common Ancestor

Build binary-lifting ancestors, align node depths, and jump together to find their deepest shared ancestor.Eulerian Path

Eulerian Path

Run Hierholzer stack traversal, consume every edge once, and append vertices while dead ends unwind.

Dynamic Programming

Build reusable state transitions and compare competing choices.
Edit Distance

Edit Distance

Fill the Levenshtein dynamic-programming matrix to find the minimum insertions, deletions, and substitutions between two strings.0/1 Knapsack

0/1 Knapsack

Fill the 0/1 Knapsack dynamic-programming table cell by cell and compare the skip and take choices that produce the optimal value.Unbounded Knapsack

Unbounded Knapsack

Fill capacities while allowing each item to be chosen repeatedly, exposing the loop-order difference from 0/1 Knapsack.Longest Common Subsequence

Longest Common Subsequence

Build the LCS table from matching and mismatching character pairs, then trace backward to reconstruct an actual common subsequence.Longest Increasing Subsequence

Longest Increasing Subsequence

Use one-dimensional dynamic programming to extend increasing subsequences, preserve predecessors, and reconstruct one longest result.Coin Change

Coin Change

Build the minimum number of reusable coins for every amount and reconstruct one optimal combination.Stone Merging

Stone Merging

Fill an interval-DP triangle by trying every final split and adding the sum of the merged interval.Traveling Salesperson DP

Traveling Salesperson DP

Use a bitmask for visited cities and end-position states to assemble the shortest Hamiltonian tour.Tree Dynamic Programming

Tree Dynamic Programming

Process nodes postorder and combine selected versus unselected child states for maximum independent set.Digit DP

Digit DP

Count valid numbers by scanning upper-bound digits with tight and free suffix states.Rerooting DP

Rerooting DP

Compute subtree answers bottom-up, then transfer the whole-tree answer across every edge in a second pass.

Backtracking and Search

Explore a decision space, reject invalid branches, and undo choices safely.
N-Queens

N-Queens

Place queens column by column, reject attacked squares, and watch recursive backtracking recover from choices that block the board.Subsets

Subsets

Traverse an include-or-exclude decision tree depth first and record every leaf to generate the complete power set without duplicates.Permutations

Permutations

Choose one unused value per depth, record complete arrangements, and undo choices while exploring the decision tree.Combination Sum

Combination Sum

Choose reusable candidates in nondecreasing order, prune sums above the target, and avoid duplicate combinations.Maze Solving with DFS

Maze Solving with DFS

Explore a grid depth first, mark visited cells, retreat from dead ends, and watch backtracking preserve the final path to the goal.Number of Islands

Number of Islands

Flood-fill each unseen land component and count how many independent searches begin.Word Search

Word Search

Backtrack through adjacent grid cells, match one character per depth, and restore visited cells after failure.Sudoku Solver

Sudoku Solver

Try legal digits in an empty cell, propagate row-column-box constraints, and backtrack from contradictions.A* Search

A* Search

Order frontier cells by path cost plus heuristic and reconstruct an optimal route when the goal is settled.

String Algorithms

Reuse prefix and palindrome structure to avoid repeated comparisons.
KMP String Matching

KMP String Matching

Follow KMP character comparisons and LPS jumps to see why the text pointer never moves backward and matching stays linear.Rabin-Karp String Matching

Rabin-Karp String Matching

Slide a rolling-hash window across the text, skip unequal hashes in constant time, and verify characters only after a hash match.Boyer-Moore String Matching

Boyer-Moore String Matching

Compare pattern characters right to left and shift by bad-character and good-suffix information after a mismatch.Manacher's Longest Palindromic Substring

Manacher's Longest Palindromic Substring

Transform odd and even palindromes into one form, reuse mirror radii inside the rightmost box, and find the longest palindrome in linear time.Suffix Array

Suffix Array

Rank suffixes by doubling prefix lengths until their lexicographic order becomes unique.LCP Array

LCP Array

Use inverse suffix ranks and Kasai reuse to compute adjacent longest common prefixes in linear time.Aho-Corasick Automaton

Aho-Corasick Automaton

Build a trie of patterns, add failure links by BFS, and report overlapping matches during one text scan.Z Function

Z Function

Reuse a rightmost prefix-matching box to compute every suffix-prefix match length in linear time.

Math and Number Theory

Turn divisibility and prime structure into efficient iterative procedures.
Sieve of Eratosthenes

Sieve of Eratosthenes

Mark composite multiples from each prime square, stop after the square-root boundary, and reveal every prime in a numeric range.Linear Sieve

Linear Sieve

Generate primes while marking each composite exactly once by its smallest prime factor.Euclidean Algorithm

Euclidean Algorithm

Apply repeated remainders to compute a greatest common divisor while a matching square tiling gives the recurrence a geometric meaning.Binary Exponentiation

Binary Exponentiation

Read exponent bits, square the base each round, and multiply only when the current bit is set.Extended Euclidean Algorithm

Extended Euclidean Algorithm

Propagate Euclidean remainders backward to find coefficients satisfying ax + by = gcd(a, b).Chinese Remainder Theorem

Chinese Remainder Theorem

Merge modular congruences with inverses to reconstruct the unique solution modulo the product.Euler's Totient Function

Euler's Totient Function

Factor an integer and apply the multiplicative product formula to count smaller coprime values.Miller-Rabin Primality Test

Miller-Rabin Primality Test

Decompose n-1 into an odd part and powers of two, then test modular witnesses for compositeness.Fast Fourier Transform

Fast Fourier Transform

Bit-reverse the input and run logarithmic layers of butterfly operations for a fast discrete Fourier transform.Pollard's Rho Factorization

Pollard's Rho Factorization

Race two iterates through a modular pseudo-random sequence until a gcd reveals a nontrivial factor.

Computational Geometry

Use geometric invariants to reason about boundaries and distances.
Convex Hull

Convex Hull

Build lower and upper hulls with Andrew's monotone chain, using cross products to remove right turns and reveal the final convex polygon.Rotating Calipers

Rotating Calipers

Advance antipodal support lines around a convex polygon to find its diameter without checking every pair.Closest Pair of Points

Closest Pair of Points

Divide a planar point set at its median x-coordinate, solve both halves, and inspect a narrow strip for a closer cross-boundary pair.Line Segment Intersection

Line Segment Intersection

Use orientation cross products and bounding checks to classify proper, endpoint, collinear, and disjoint cases.Bentley-Ottmann Sweep Line

Bentley-Ottmann Sweep Line

Process endpoint and crossing events while a balanced status order tests only neighboring segments.

Searching

Maintain a precise candidate interval and discard impossible answers.
Binary Search

Binary Search

See Binary Search discard half of a sorted array at every probe, including both a successful lookup and a clean not-found result.Lower and Upper Bound

Lower and Upper Bound

Use half-open Binary Search templates to find the first value at least a target and the first value greater than it.Search in a Rotated Sorted Array

Search in a Rotated Sorted Array

Identify the sorted half at each binary-search probe and keep the half that can contain the target.Binary Search on the Answer

Binary Search on the Answer

Turn an optimization problem into a monotone feasibility predicate and search the numeric answer range.Ternary Search

Ternary Search

Compare two interior probes and discard one third of a unimodal search interval each round.