Concepts318
Category
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.
MST Properties and Applications
An MST minimizes total edge weight over all spanning trees and has powerful properties such as the cut and cycle properties that guide correct, greedy construction.
Minimum Spanning Tree - Prim
Prim's algorithm builds a Minimum Spanning Tree (MST) by growing a tree from an arbitrary start vertex, always adding the lightest edge that connects the tree to a new vertex.
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.
Floyd-Warshall Algorithm
Floyd–Warshall computes the shortest distances between all pairs of vertices in O(n^3) time using dynamic programming.
Johnson's Algorithm
Johnson's Algorithm computes all-pairs shortest paths on sparse graphs by first removing negative edges via reweighting, then running Dijkstra from every vertex.
Dijkstra's Algorithm
Dijkstra's algorithm finds shortest path distances from one source to all vertices when all edge weights are non-negative.
Bellman-Ford Algorithm
Bellman–Ford finds single-source shortest paths even when some edge weights are negative.
Dijkstra - Variations and Applications
Dijkstra’s algorithm can be adapted to track the second shortest path by keeping the best and second-best distances per vertex.
Topological Sort - DP on DAG
Topological sort orders vertices of a directed acyclic graph (DAG) so every edge goes from earlier to later, which is perfect for dynamic programming (DP).
Breadth-First Search (BFS)
Breadth-First Search (BFS) explores a graph level by level, visiting all vertices at distance d from the source before any at distance d+1.
DFS - Tree and Graph Properties
Depth-First Search (DFS) assigns each vertex a discovery time and a finish time that capture a neat nesting structure of recursion.