Concepts9
Functional Graph
A functional graph is a directed graph where every node has exactly one outgoing edge, so repeatedly following edges from any start eventually loops into a cycle.
Block-Cut Tree
A Block-Cut Tree decomposes an undirected graph into biconnected components (blocks) and articulation points, forming a bipartite tree.
Virtual Tree (Auxiliary Tree)
A Virtual Tree (Auxiliary Tree) compresses a large tree into a much smaller tree that contains only the k important nodes and the LCAs needed to keep them connected.
LCA - Binary Lifting
Binary lifting precomputes 2^k ancestors for every node so we can jump upward in powers of two.
Bridge Tree
A bridge tree is built by contracting every 2-edge-connected component of an undirected graph into a single node, leaving only bridges as edges between nodes.
Tree Distances and Diameter
Tree diameter is the longest simple path in a tree and can be found with two BFS/DFS runs.
Lowest Common Ancestor (LCA)
The Lowest Common Ancestor (LCA) of two nodes in a rooted tree is the deepest node that is an ancestor of both.
Minimum Spanning Tree - Kruskal
Kruskal’s algorithm builds a minimum spanning tree (MST) by sorting all edges by weight and greedily picking the next lightest edge that does not form a cycle.
Centroid Decomposition
Centroid decomposition splits a tree around a special node (centroid) so that every remaining component has at most half the nodes.