Lecture 13 Dynamic Programming Action Costs Uniform Cost Search General principle Divide and conquer A path that passes from Y in going to X to Z can be broken into two parts X to Y and Y to Z Shortest path principle The shortest path from X to Z that goes through Y is made up of the shortest path from X to Y and the shortest path from Y to Z Some actions can be more costly than others so we need to modify search algorithms to account for action costs BFS with dynamic programming fails if path costs are not equal Associate action costs with actions Enumerate paths in order of their total path cost Find the path with the smallest path cost which is the sum of action costs along a path Implement agenda with priority queue Priority queue same basic operations as stacks and queues with two differences Items are pushed with numeric score the cost Popping returns the item with the smallest cost Dynamic Programming for Uniform Cost Searches Since costs associated with each child can doffer we must look at all children before we can decide which has the lowest cost The next child that we visit may not represent the shortest distance to that state However the next parent that is expanded will always be the shortest distance to the parent state So essentially we are substituting the notion of expansions for visits Heuristics So far our searches only consider the cost of the path from start to current We can add heuristics to consider current to goal Heuristics estimate cost of going from current state to the goal A Uniform Cost Search with Heuristics A heuristic function h s returns the estimated cost from the input state s to the goal A consistent heuristic satis es two conditions h goal 0 h a satis es the following triangle inequality if b is a child of a then h a is less than or equal to the sum of h b and the distance from a to b If a heuristic is consistent the the heuristic will never overestimate the actual distance and A will nd the shortest path An admissible heuristic never overestimates the actual distance If a heuristic is not admissible the the heuristic cannot be consistent and A is not guaranteed to nd the shortest path Check Yourself Heuristics Correct answer Let r0 co and r1 c1 represent the current and goal locations How many of the following heuristics are admissible abs r0 r1 abs c0 c1 min abs r0 r1 abs c0 c1 max abs r0 r1 abs c0 c1 2 min abs r0 r1 abs c0 c1 2 max abs r0 r1 abs c0 c1 Summary Developed a new class of search algorithms uniform cost Allows solutions of problems with different action costs Developed a new class of optimizations heuristics Focuses search toward the goal
View Full Document
Unlocking...