Dijkstras Algorithm and Heuristic Graph Search David Johnson

Dijkstra’s Algorithm and Heuristic Graph Search David Johnson

DFS and BFS • Examples of uninformed graph search – Only metric of quality is number of vertices in path • Want to look at difficulty of moving along an edge – Weighted graph – Want the minimum cost path

Dijkstra’s Algorithm • Always finds the lowest cost path – For positive edge weight graphs • Some subtle differences with breadth-first search • Maintains a cost to visit every vertex • Looks at the successors of the current lowest cost vertex in the wavefront

Dijkstra’s Outline • • Initialize a cost array to infinity Set the start vertex cost to 0. 0, put in search list Find the minimum cost vertex in search. List For each successor – Compute tentative cost = current cost + edge – If tentative cost < existing cost then overwrite • Mark min cost vertex as visited/locked – This has a different meaning than breadth-first • Terminate when goal is minimum cost vertex

Example • Look at graph in matlab, run through by hand.

Heuristic Search • More powerful techniques employ a heuristic – Estimate cost to goal – Expand most promising vertices first – Try to avoid the expanding ring of BFS and Dijkstra

Search Heuristics • Heuristic – Informal approach to solving a problem – “rule-of-thumb” – Estimates • On a terrain map, what would be an example heuristic for choosing which node to expand?

Greedy Best-First Search • Expands the vertex in the search list with the smallest valued estimated cost to goal – Can only use when a meaningful heuristic is available – Ignores the cost so far

Dijkstra vs Best-First

Problem Solved?

A* • Combines – Dijkstra’s current path cost G(n) – Best-first’s cost to goal heuristic H(n) • Finds shortest paths – Under some conditions • Doesn’t search as much as BFS • Pretty standard

A* • Expands the node with smallest F(n) = G(n) + H(n) • Need to choose H(n) – If H(n) < true cost, finds shortest path • What if G(n) >> H(n)? • What if H(n) >> G(n)?

Some Example Heuristics • 4 -point connectivity (Manhattan distance)

Some Example Heuristics • 8 -point connectivity

Some Example Heuristics • Euclidean Distance – Cannot move to match heuristic – H(n) does not match G(n)

One complication • Multiple paths may have same cost • A* will explore them all What kind of terrain would this map represent?

Add a tie-breaker • Makes the costs not exactly the same

Implementation • Just like Dijkstra – Need a cost-so-far plus heuristic cost summed array – Select current vertex based on that

More resources • Amit’s A* page – http: //theory. stanford. edu/~amitp/Game. Progra mming/Heuristics. html – Applet – http: //www. vision. ee. ethz. ch/~cvcourse/astar/ AStar. html