Graphs

Fold digits into squared sums; reaching 1 is happy, meeting a repeat is a cycle, so use tortoise and hare.

Count land cells, union adjacent land, subtract one per merge: the remainder is the island count.

All courses are finishable exactly when the prerequisite directed graph is acyclic; detect a cycle with DFS path-marking or with Kahn's in-degree processing.

Run Kahn's algorithm, appending each course as its in-degree hits zero; the emitted sequence is a valid order, or empty if a cycle blocks some course.

Start with n components and union each edge's endpoints, decrementing the count on each merge; the remaining count is the number of connected components.

One Kahn's pass gives both answers: feasibility is emitted-count equals total, and the order is the emitted sequence, empty if a cycle blocks completion.

Scan the grid; each unvisited land cell starts a new island and a flood fill sinks its whole connected region so it is counted once.

BFS explores a graph ring by ring using a queue, so it finds the fewest-hops path first.

Flood Fill spreads a new value outward from a start cell to all connected cells sharing the original value, using BFS or DFS, each cell touched once.

DFS from each matching start, mark cells visited along the path, unmark on backtrack, succeed when the word is fully matched.

Treat words as nodes with edges between one-letter-apart words and BFS from begin; the level at which end is first reached is the shortest ladder length.

Seed DFS or BFS from every border 'O' to mark safe cells with a sentinel, then sweep the board flipping unmarked 'O' to 'X' and restoring sentinels to 'O'.

Seed a BFS queue with all initially rotten cells at time zero, process level by level counting minutes, and return the elapsed time or negative one if any fresh cell remains unreachable.

Sweep the grid; on each unvisited land cell, count one island and flood-fill its entire connected component as visited, giving O(rows times cols) time.

From the starting cell, DFS or BFS recoloring every connected cell of the original color to the new color, guarding against the no-op case where new equals old.

A happy number reaches 1 when you repeatedly replace it with the sum of the squares of its digits; Floyd's fast and slow pointers detect whether the sequence cycles or terminates.

Process edges in input order using Union-Find; the first edge whose two endpoints already share a root is the redundant edge that closes a cycle.

Colour each node white, grey, or black during DFS; reaching a grey node means a back edge exists and there is a cycle.

Run K+1 rounds of Bellman-Ford relaxation, reading from a frozen previous-round snapshot so each round adds exactly one hop.

Two pointers at different speeds traverse the list: if they meet there is a cycle; if the fast one reaches null there is not.

Count a jump when the index reaches the current window boundary, then extend the boundary to the farthest reachable index, giving O(n) time and O(1) space.

Build a Trie for insertion; search with DFS where a dot fans out to every child and a real character follows one exact child, succeeding when any path ends at a word marker.

A graph is a data structure made of vertices connected by edges, where edges can be directed or undirected and optionally carry weights.