Announcements. Solution to Assignment 1 is posted Assignment 2 is available Video automatically uploaded (see web page)

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1 Announcements Solution to Assignment 1 is posted Assignment 2 is available Video automatically uploaded (see web page) c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 1 / 27

2 Review: Searching A frontier is a set of paths Generic search algorithm: Repeatedly: select a path from the frontier stop of it is a path to a goal otherwise expand it in all ways, and add the resulting paths to the frontier Frontier is a stack depth-firt search Frontier is a queue breadth-firt search Frontier is a priority queue ordered by path cost least-cost-first search c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 2 / 27

3 Graph Search Algorithm Input: a graph a start node s Boolean procedure goal(n) that tests if n is a goal node frontier := { s } while frontier is not empty: select and remove path n 0,..., n k from frontier if goal(n k ) return n 0,..., n k Frontier := Frontier { n 0,..., n k, n : n k, n A} c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 3 / 27

4 Clicker Question Suppose the frontier contains the following paths. The paths are listed in order of being added to the frontier, (so that (i) was the first of these added (ii) was the second of these added, etc.) i) a, b, c with cost 6 ii) a, e, f with cost 5 iii) a, e, w with cost 5 iv) a, d, x with cost 9 v) a, d, z with cost 7 Which path will be expanded next for breath-first search A (i) because it was added first B (v) because it was added last C either (ii) or (iii) because they have least cost D (iv) because it has the greatest cost E we can t tell; any of them could be chosen c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 4 / 27

5 Clicker Question Suppose the frontier contains the following paths. The paths are listed in order of being added to the frontier, (so that (i) was the first of these added (ii) was the second of these added, etc.) i) a, b, c with cost 6 ii) a, e, f with cost 5 iii) a, e, w with cost 5 iv) a, d, x with cost 9 v) a, d, z with cost 7 Which path will be expanded next for least-cost-first search A (i) because it was added first B (v) because it was added last C either (ii) or (iii) because they have least cost D (iv) because it has the greatest cost E we can t tell; any of them could be chosen c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 5 / 27

6 Clicker Question Suppose the frontier contains the following paths. The paths are listed in order of being added to the frontier, (so that (i) was the first of these added (ii) was the second of these added, etc.) i) a, b, c with cost 6 ii) a, e, f with cost 5 iii) a, e, w with cost 5 iv) a, d, x with cost 9 v) a, d, z with cost 7 Which path will be expanded next for depth-first search A (i) because it was added first B (v) because it was added last C either (ii) or (iii) because they have least cost D (iv) because it has the greatest cost E we can t tell; any of them could be chosen c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 6 / 27

7 Clicker Question Z W X Y Suppose the start node is Z the goal is Y and children are expanded from left to right. Which of the following is true: A breadth-first, depth-first and least-cost-first (LCF) all halt B none of breadth-first, depth-first and LCF halt C depth-first and LCF halt, but breadth-first search doesn t halt D breadth-first and LCF searches halt, but depth-first search doesn t halt E none of the above c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 7 / 27

8 Clicker Question Z W X Y Suppose the start node is Z the goal is W and children are expanded from left to right. Which of the following is true: A breadth-first, depth-first and least-cost-first (LCF) all halt B none of breadth-first, depth-first and LCF halt C depth-first and LCF halt, but breadth-first search doesn t halt D breadth-first and LCF searches halt, but depth-first search doesn t halt E none of the above c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 8 / 27

9 Clicker Question Z W X Y Suppose the start node is W the goal is X and children are expanded from left to right. Which of the following is true: A breadth-first, depth-first and least-cost-first (LCF) all halt B none of breadth-first, depth-first and LCF halt C depth-first and LCF halt, but breadth-first search doesn t halt D breadth-first and LCF searches halt, but depth-first search doesn t halt E none of the above c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 9 / 27

10 Learning Objectives At the end of the class you should be able to: show how A search works on an example argue why A search always finds an optimal solution first explain how cycle pruning and multiple-path pruning can improve efficiency of search algorithms explain the complexity of cycle pruning and multiple-path pruning for different search algorithms justify why the monotone restriction is useful for A search c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 10 / 27

11 A Search A search uses both path cost and heuristic function. cost(p) is the cost of path p. h(n) is a nonnegative estimate of the cost of the shortest path from node n to a goal node. h( n 0,..., n k ) = h(n k ). Let f (p) = cost(p) + h(p). f (p) estimates the total path cost of going from a start node to a goal via p. start path p estimate n goal }{{}}{{} } cost(p) {{ h(p) } f (p) In A search, the frontier is a priority queue ordered by f (p). It always selects the path on the frontier with the lowest estimated cost from the start to a goal node constrained to go via that path. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 11 / 27

12 AIspace examples Vancouver neighbourhood graph Misleading heuristic demo c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 12 / 27

13 Admissibility of A If there is a solution, A always finds an optimal solution the first path to a goal selected if the branching factor is finite arc costs are bounded above zero (there is some ɛ > 0 such that all of the arc costs are greater than ɛ), and h(n) is nonnegative and an underestimate of the cost of the least-cost path from n to a goal node: 0 h(n) cost of shortest path from n to a goal c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 13 / 27

14 Why is A admissible? If a path p to a goal is selected from the frontier, can there be a lower cost path to a goal? h(p) = 0 Suppose path p is on the frontier. Because p was chosen before p, and h(p) = 0: Because h is an underestimate: cost(p) cost(p ) + h(p ). cost(p ) + h(p ) cost(p ) for any path p to a goal that extends p. So cost(p) cost(p ) for any other path p to a goal. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 14 / 27

15 Why is A admissible? A can always find a solution if there is one: The frontier always contains the initial part of a path to a goal, before that goal is selected. A halts, as the costs of the paths on the frontier keeps increasing, and will eventually exceed any finite number. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 15 / 27

16 How do good heuristics help? Suppose c is the cost of an optimal solution. What happens to a path p from start, where cost(p) + h(p) < c It will be expanded cost(p) + h(p) > c It will not be expanded cost(p) + h(p) = c It might or might not be expanded. How can a better heuristic function help? c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 16 / 27

17 Complexity of A Search Does A search guarantee to find the least-cost path or the path with fewest arcs? What happens on infinite graphs or on graphs with cycles if there is a solution? What happens on infinite graphs or on graphs with cycles if there is no solution? What is the time complexity as a function of length of the path selected? What is the space complexity as a function of length of the path selected? How does the goal affect the search? c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 17 / 27

18 Summary of Search Strategies Strategy Frontier Selection Complete Halts Space Depth-first Last node added No No Linear Breadth-first First node added Yes No Exp Lowest-cost-first Minimal cost(p) Yes No Exp Best-first Minimal h(p) No No Exp A Minimal f (p) Yes No Exp Complete if there a path to a goal, it can find one, even on infinite graphs. Halts on finite graph (perhaps with cycles). Space as a function of the length of current path c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 18 / 27

19 Cycle Pruning s A searcher can prune a path that ends in a node already on the path, without removing an optimal solution. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 19 / 27

20 Graph searching with cycle pruning Input: a graph, a set of start nodes, Boolean procedure goal(n) that tests if n is a goal node. frontier := { s : s is a start node} while frontier is not empty: select and remove path n 0,..., n k from frontier if n k {n 0,..., n k 1 } : if goal(n k ): return n 0,..., n k Frontier := Frontier { n 0,..., n k, n : n k, n A} c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 20 / 27

21 Cycle Pruning s In depth-first search, checking for cycles can be done in constant time in path length. For other methods, checking for cycles can be done in linear time in path length. With cycle pruning, which algorithms halt on finite graphs? c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 21 / 27

22 Multiple-Path Pruning s Multiple path pruning: prune a path to node n that the searcher has already found a path to. What needs to be stored? Lowest-cost-first search with multiple-path pruning is Dijkstra s algorithm. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 22 / 27

23 Graph searching with multiple-path pruning Input: a graph, a set of start nodes, Boolean procedure goal(n) that tests if n is a goal node. frontier := { s : s is a start node} expanded := {} while frontier is not empty: select and remove path n 0,..., n k from frontier if n k expanded : add n k to expanded if goal(n k ): return n 0,..., n k Frontier := Frontier { n 0,..., n k, n : n k, n A} c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 23 / 27

24 Multiple-Path Pruning How does multiple-path pruning compare to cycle pruning? Which search algorithms with multiple-path pruning always halt on finite graphs? What is the time overhead of multiple-path pruning? What is the space overhead of multiple-path pruning? Is it better for depth-first or breadth-first searches? Can multiple-path pruning prevent an optimal solution being found? c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 24 / 27

25 Multiple-Path Pruning & Optimal Solutions Problem: what if a subsequent path to n has a lower cost than the first path to n? remove all paths from the frontier that use the longer path. change the initial segment of the paths on the frontier to use the lower-cost path. ensure this doesn t happen. Make sure that the lower-cost path to a node is expanded first. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 25 / 27

26 Multiple-Path Pruning & A Suppose path p to n was selected, but there is a lower-cost path to n. Suppose this lower-cost path is via path p on the frontier. Suppose path p ends at node n. p was selected before p, so: cost(p) + h(n) cost(p ) + h(n ). Suppose cost(n, n) is the actual cost of a path from n to n. The path to n via p has a lower cost that p so: cost(p ) + cost(n, n) < cost(p). cost(n, n) < cost(p) cost(p ) h(n ) h(n). We can ensure this doesn t occur if h(n ) h(n) cost(n, n). c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 26 / 27

27 Monotone Restriction Heuristic function h satisfies the monotone restriction if h(m) h(n) cost(m, n) for every arc m, n. If h satisfies the monotone restriction, A with multiple path pruning always finds a least-cost path to a goal. This is a strengthening of the admissibility criterion. c D. Poole and A. Mackworth 2017 CPSC 322 Lecture 4 27 / 27