Graph Search Howie Choset

Transcription

1 Graph Search Howie Choset 16-11

2 Outline Overview of Search Techniques A* Search

3 Graphs Collection of Edges and Nodes (Vertices) A tree

4 Grids

5 Stacks and Queues Stack: First in, Last out (FILO) Queue: First in, First out (FIFO) 5

6 Depth First Search algorithm dft(x) visit(x) FOR each y such that (x,y) is an edge IF y was not visited yet THEN dft(y) Copied from wikipedia Worst case performance Worst case space complexity O( V + E ) for explicit graphs traversed without repetition, O( V ) if entire graph is traversed without repetition, O(longest path length searched) for implicit graphs without elimination of duplicate nodes 6

7 Breadth First Search visit(start node) queue <- start node WHILE queue is nor empty DO x <- queue FOR each y such that (x,y) is an edge and y has not been visited yet DO visit(y) queue <- y END wikipedia END 7

8 Depth First and Breadth First 8

9 Wavefront Planner: A BFS 9

10 Search Uninformed Search Use no information obtained from the environment Blind Search: BFS (Wavefront), DFS Informed Search Use evaluation function More efficient Heuristic Search: A*, D*, etc. 10

11 Uninformed Search Graph Search from A to N A B C D BFS E F G H I J K L M N 11

12 Informed Search: A* Notation n node/state c(n 1,n ) the length of an edge connecting between n 1 and n b(n 1 ) = n backpointer of a node n 1 to a node n. 1

13 Informed Search: A* Evaluation function, f(n) = g(n) + h(n) Operating cost function, g(n) Actual operating cost having been already traversed Heuristic function, h(n) Information used to find the promising node to traverse Admissible never overestimate the actual path cost Cost on a grid 1

14 A*: Algorithm Start O is empty? N Pick n best from O such that f(n best ) < f(n) Remove n best from O and add it to C Y End The search requires lists to store information about nodes N update b(x)=n best if (g(n best )+c(n best,x)<g(x)) N Expand all nodes x that are neighbors of n best and not in C x is not in O? n best = goal? Y Add x to O Y End 1) Open list (O) stores nodes for expansions ) Closed list (C) stores nodes which we have explored 1

15 Dijkstra s Search: f(n) = g(n) 1. O = {S}. O = {1,,, 5}; C = {S} (1,,,5 all back point to S). O = {1,, 5}; C = {S, } (there are no adjacent nodes not in C). O = {1, 5, }; C = {S,, } (1,, point to S; 5 points to ) 5. O = {5, }; C = {S,,, 1} 6. O = {, G}; C = {S,, 1} (goal points to 5 which points to which points to S) 7. O = {}; C = {S,,, 1} (Path found because Goal was popped) Pop lowest f first 15

16 Two Examples Running A* 0 Star t 1 1 D A C 1 1 B E F J G H I 1 GOAL L A B C K 6 E F G H I J L M N D K 6 16

17 Example (1/5) 0 Start 1 1 D A C B E F J L K Legend h(x) c(x) 1 1 G H I Priority = g(x) + h(x) 1 Note: g(x) = sum of all previous arc costs, c(x), from start to x GOAL Example: c(h) = 17

18 Example (/5) 0 Start 1 1 D A C 1 1 B E F J 1 1 G H I 1 GOAL L K First expand the start node B() If goal not found, A() expand the first node C() in the priority queue (in this case, B) H() Insert the newly expanded A() nodes into the priority que C() and continue until the goal I(5) found, or the priority queu G(7) empty (in which case no p Note: for exists) each expanded node, you also need a pointer to its respective parent. For example, nodes A, B and C point to Start 18

19 Example (/5) 0 Start 1 1 D A C B E F J 1 1 G H I L K B() A() C() H() A() C() I(5) G(7) No expansion E() GOAL(5) C() D(5) I(5) F(7) G(7) 1 We ve found a path to the goal: Start => A => E => Goal (from the pointers) GOAL Are we done? 19

20 0 Start 1 1 D A C L B E F J K 1 1 G H I 1 GOAL B() A() C() Example (/5) H() A() C() I(5) G(7) No expansion E() C() D(5) I(5) F(7) G(7) GOAL(5) There might be a shorter path, but assuming non-negative arc costs, nodes with a lower prio than the goal cannot yield a better path. In this example, nodes with a priority greater th equal to 5 can be pruned. Why don t we expand nodes with an equivalen (why not expand nodes D 0 and I?)

21 1 1 D A C B E F J 1 0 Start 1 If the priority queue still wasn t empty, we would continue expanding while throwing away nodes with priority lower than. (remember, lower numbers = higher priority) G H I 1 GOAL L K B() A() C() Example (5/5) H() A() C() I(5) G(7) No expansion E() C() D(5) I(5) F(7) G(7) GOAL(5) We can continue to throw away nodes with priority levels lower than the lowest goal found. As we can see from this example, there was a shorter path through node K. To find the path, simply follow the back pointers. Therefore the path would be: Start => C => K => Goal K() L(5) J(5) 1 GOAL()

22 Monotonic never overestimates the cost of getting from a node to its neighbor. for all paths x,y where y is a successor of x, i.e., h(x) g(y)- g(x) h(y) h(a) = g(a) = 1 h(e) = 1 g(e) = h(a) g(e) -g(a) h(e) -1 1

23 A*: Example (1/6) A B C 6 E F G H I J L M N operating cost Legend D K 6 Heuristics A = 1 H = 8 B = 10 I = 5 C = 8 J = D = 6 K = E = 8 L = 6 F = 7 M =

24 A*: Example (/6) 6 A B C E F G D 6 Closed List A(0) Open List - Priority Queue E(11) B(1) Expand H(1) H I J L M Heuristics A = 1, B = 10, C = 8, D = 6, E = 8, F = 7, G = 6 H = 8, I = 5, J =, K =, L = 6, M =, N = 0 N K A(x) = A(g(n)) E(y) = E(f(n)) where f(n) = g(n) + h(n) = + 8 = 11

25 A*: Example (/6) Closed List Open List - Priority Queue 6 A B C E F G D 6 A(0) E() B(16) > H(15) > I(11) B(1) H(1) F(1) H I J K L M N Update Add new node Since A B is smaller Heuristics than A E B, the f- A = 1, B = 10, C = 8, D = 6, E = 8, F = 7, cost G = value 6 of B in an H = 8, I = 5, J =, K =, L = 6, M =, open N = list 0 needs not be updated 5

26 A*: Example (/6) 6 A H B C E F G I J D K 6 Closed List A(0) E() I(6) Open List - Priority Queue J(1) M(1) B(1) H(18) > H(1) F(17) > F(1) L(15) L M N Update Add new node Heuristics A = 1, B = 10, C = 8, D = 6, E = 8, F = 7, G = 6 H = 8, I = 5, J =, K =, L = 6, M =, N = 0 6

27 A*: Example (5/6) Closed List Open List - Priority Queue 6 A H B C E F G I J L M N D K 6 A(0) E() I(6) J(10) Update M(16) > M(1) N(1) B(1) F(1) > H(1) F(1) L(15) K(16) G(19) Add new node Heuristics A = 1, B = 10, C = 8, D = 6, E = 8, F = 7, G = 6 H = 8, I = 5, J =, K =, L = 6, M =, N = 0 7

28 A*: Example (6/6) Closed List Open List - Priority Queue 6 A B C E F G D 6 A(0) E() I(6) J(10) M(10) N(1) L() > > N(1) B(1) H(1) F(1) L(15) K(16) G(19) Goal H I J L M N K Update Add new node Since the path to N from Heuristics M is greater than that A = 1, B = 10, C = 8, D = 6, E = 8, F = 7, from G = 6 J, the optimal path H = 8, I = 5, J =, K =, L = 6, M =, to N N = is 0 the one traversed from J 8

29 A*: Example Result 6 A H B C E F G I J D K 6 Generate the path from the goal node back to the start node through the backpointer attribute L M N 9

30 Non-opportunistic 1. Put S on priority Q and expand it. Expand A because its priority value is 7. The goal is reached with priority value 8. This is less than B s priority value which is 1 0