Weighted Graphs & Shortest Paths

Weighted Graphs Edges have an associated weight or cost

BFS? BFS only gives shortest path in terms of edge count, not edge weight

Dijkstra's Shortest Path Algorithm Conceptual: V = all vertices T = included vertices Pick starting vertex, include in T Pick element not in T with minimal cost to reach from T Move to T Update costs of remaining vertices

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Dijkstra's Find paths from u to all others:

Details Implementation Known once visited Cost starts at  Update all neighbors at each visit Path marked when cost updated

Dijkstra's Algorithm Finds shortest path to all other vertices Can terminate early once goal is known Assumption: No negative edges

A* A-Star : Dijkstra's algorithm, but include estimate of future cost for every vertex Estimate must never overestimate cost https://qiao.github.io/PathFinding.js/visual/

MST Minimum Spanning Tree Spanning tree with minimal possible total weight

Prim's MST Conceptual: Sound familiar? V = all vertices T = included vertices Pick starting vertex, include in T Pick element not in T with minimal cost to reach from T Move to T Update costs of remaining vertices Sound familiar?

Prim's MST ~ Dijkstra's Conceptual: One big difference… V = all vertices T = included vertices Pick starting vertex, include in T Pick element not in T with minimal cost to reach from T Move to T Update costs of remaining vertices : cost = just current edge One big difference…

Prim's Implementation Same as Dijkstra's but… Cost of vertex = min( all known edges to it )

Prim's Algorithm Finds a MST Assumption: May be multiple equal cost Undirected