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 costupdated

Dijkstra's Algorithm

• Finds shortest path to all other vertices– Can terminate early once goal is known

• Assumption:– No negative edges

• Big O – for the curious– O(V2) with table• V times do linear search for next vertex to visit

– O((E + V)logV) with priority queue (binary heap)• O(E + VlogV) possible with fibonacci heap

MST

• Minimum Spanning Tree– Spanning tree with minimal possible total weight

Prim's MST

• 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

Sound familiar?

Prim's MST ~ Dijkstra's

• 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 : 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– May be multiple equal cost

• Assumption:– Undirected

• Big O – for the curious– O(V2) with table• V times do linear search for next vertex to visit

– O((E + V)logV) with priority queue (binary heap)• O(E + VlogV) possible with fibonacci heap

A*

• A-Star : Dijkstra's algorithm, butinclude estimate of future cost for everyvertex

• Estimate must never underestimate cost

http://computerscience.chemeketa.edu/cs160Reader/NineAlgorithms/Search2.html