Graph and Heuristic Search. Lecture 3. Ning Xiong. Mälardalen University. Agenda

Transcription

1 Graph and Heuristic earch Lecture 3 Ning iong Mälardalen University Agenda Uninformed graph search - breadth-first search on graphs - depth-first search on graphs - uniform-cost search on graphs General informed (heuristic) search algorithms - greedy best-first - A* search Local search algorithm: hill climbing 1

2 Loop with graph search Depth-first a b d c Initial state: a Goal state: e d In a general graph, there exists the risk of loop or Cycle making the search process continue for ever a e b d Method to avoid loop Depth-first a b d c Initial state: a Goal state: e d If we memorize the nodes that previously have been expanded, we can avoid picking state a for expansion for the second time. Consequently the search can terminate and find the goal e. b a d e 2

3 Avoiding Loop Memorize all expanded nodes in a Closed list All nodes in Closed list should be blocked for processing again The graph search according to Russel s book: 1. Remove the first node from the Fringe list, assign it to s 2. If s is the goal, terminate with found solution 3. If s already in the Closed list, go to step 1, otherwise do the following: i) add s to the Closed list ii) expand s, inset the children of s into Fring, order the nodes in Fringe Idea of Russel s approach: not block a successor node already expanded into Fringe, skip such a node later when it appear first in Fringe A mall Modification A small modification is to prevent nodes that are already in Closed list from entering fringe Discard successor nodes that are already in Closed list Modified Graph earch: Function Graph-EARCH (problem, fringe) fringe = [initial state]; Closed = [ ]; loop do If fringe is empty then return failure Assign s as the first node in fringe and remove s from fringe If goal-test on s succeeds then return solution(s) Expand s by creating its successor nodes Put s on Closed list Discard successors of s if already in Closed list Insert the remaining successors of s into fringe and locate nodes in fringe according to search strategy 3

4 Multiple Paths in Graph Multiple paths mean redundant information. It is desired to reduce redundant information to increase computational Efficiency. 6 A 1 7 A B 8 A C D A E A F A A G 4 H A A I J A K A L A M A N A O A Breadth-First Graph earch A child node of B has the state already in fringe, meaning a second path, we should discard this path. o this child is not added into fringe, else C will be expanded second time at level 2. C B D A C Fringe=(B, C) Fringe=(C, D) Function Breadth-First-EARCH (problem, fringe) fringe = [initial state]; Closed = [ ]; loop do If fringe is empty then return failure Assign s as the first node in fringe and remove s from fringe If goal-test on s succeeds then return solution(s) Expand s by creating its successor nodes Put s on Closed list Discard successors of s if already in Closed or Fringe list Insert the remaining successors of s in the right end of Fringe list 4

5 Depth-First Graph earch A child node of B has the state already in fringe, meaning a second path, we can discard this path. o this child is not added into fringe. B A C Fringe=(B, C) C D Fringe=(D, C) Function Depth-First-EARCH (problem, fringe) fringe = [initial state]; Closed = [ ]; loop do If fringe is empty then return failure Assign s as the first node in fringe and remove s from fringe If goal-test on s succeeds then return solution(s) Expand s by creating its successor nodes Put s on Closed list Discard successors of s if already in Closed or Fringe list Insert the remaining successors of s into in the front of Fringe list Uniform-Cost Graph earch A child node of B has the state already in fringe, meaning a second path. We may discard one path based on path cost. The node and path with larger cost can be removed without altering final solution C Function Uniform-Cost-EARCH (problem, fringe) fringe = [initial state]; Closed = [ ]; loop do If fringe is empty then return failure Assign s as the first node in fringe and remove s from fringe If goal-test on s succeeds then return solution(s) Expand s by creating its successor nodes Put s on Closed list Discard successors of s if already in Closed list Insert the remaining successors of s into fringe in increasing order of path cost, check nodes with same states and remove those having larger path costs. B D A C Fringe=(B, C) 5

6 Worked Example C A B G2 0 G1 0 6 D 2 E 4 5 Initial: Goal: G1, G2 Other states: A, B, C, D, E Arc from C to : 2 Arc from to A: 3 Arc from A to C: 1 Arc from A to D: 6 Arc from C to D: 4 Arc from D to G1: 6 Arc from G1 to C: 2 Arc from to B: 7 Arc from D to B: 3 Arc from B to E: 1 Arc from B to G2: 9 Arc from E to G2: 5 Arc from G2 to B: 8 Breadth-First Graph earch A B C D E G2 1. Expand : generate A and B 2. Expand A: generate C and D 3. Expand B: generate E and G2 6

7 Breadth-First Graph earch A B C D E G2 G1 4. C can not be expanded 5. Expand D: generate G1 6. E can not be expanded 7. G2 is found as goal, trace back from G2 to get the path Depth-First Graph earch A B C D 1. Expand : generate A and B 2. Expand A: generate C and D 7

8 Depth-First Graph earch A B C D G1 3. C can not be expanded 4. Expand D: generate G1 5. G1 is checked as the goal, trace back to get the path Uniform-Cost Graph earch A(3) B(7) C(4) D(9) 1. Expand : generate A(3) and B(7) 2. Expand A: generate C(4) and D(9) Fringe=[C(4), B(7), D(9)] 8

9 Uniform-Cost Graph earch A(3) B(7) C(4) D(9) D(8) 3. Expand C: generate D(8) and remove D(9) in Fringe Fringe=[B(7), D(8)] Uniform-Cost Graph earch C(4) A(3) D(9) B(7) E(8) G2(16) D(8) 4. Expand B: generate E(8) and G2(16) Fringe=[D(8), E(8), G2(16)] 9

10 Uniform-Cost Graph earch C(4) A(3) D(9) B(7) E(8) G2(16) D(8) G1(14) 5. Expand D: generate G1(14) Fringe=[E(8), G1(14), G2(16)] Uniform-Cost Graph earch A(3) C(4) D(9) B(7) E(8) G2(16) D(8) G2(13) G1(14) 6. Expand E: generate G2(13), remove G2(16) in Fringe Fringe=[G2(13), G1(14)] G2(13) is checked as goal, optimal path: B E G2 10

11 Heuristic (Informed) earch The main idea is to utilize heuristic knowledge from the domain in the search procedure The heuristic knowledge is embedded into an evaluation function f(n), which assesses the "desirability of nodes The evaluation function is used to choose the most desirable leaf node for expansion Order the nodes in fringe in terms of the evaluation function (most desirable node appears the first) General approach for informed search is also called best-first search Two categories of the best-first search strategy: greedy best-first search A * search Greedy best-first search Evaluation function f(n) = h(n) (heuristic function) h(n): estimate of cost of the cheapest path from n to goal e.g., h(n) = straight-line distance from n to goal Greedy best-first search expands the node that is estimated to be closest to goal 11

12 Greedy best-first search example Expand Arad: generate ibiu(253), Timisoara(329), Zerind(374) Greedy best-first search example Expand ibiu: generate Fagaras(176), Oradea(380), Rimnicu Vilcea(193) 12

13 Greedy best-first search example Expand Fagaras: generate Bucharest(0) Check Bucharest as the goal A * search Purpose: avoid expanding paths that are already expensive, combine known cost and predicted cost Evaluation function f(n) = g(n) + h(n) -- g(n) = cost so far to reach n -- h(n) = estimated cost of the cheapest path from n to goal Order nodes in Fringe according the values of the f function. When h(n)=0, A * search turns to be uniform cost search 13

14 A * search example A * search example 14

15 A * search example A * search example 15

16 A * search example Another Example: A* earch A(8) B(12) C(7) D(11) 1. Expand : generate A(8) and B(12) 2. Expand A: generate C(7) and D(11) h(a)=5, h(b)=5, h(c)=3, h(d)=2, h(e)=4 Fringe=[C(7), D(11), B(12)] h(g1)=h(g2)=0 16

17 Another Example: A* earch A(8) B(12) C(7) D(11) D(10) 3. Expand C: generate D(10), remove D(11) from Fringe Fringe=[D(10), B(12)] Another Example: A* earch A(8) B(12) C(7) D(11) D(10) B(16) G1(14) 4. Expand D: generate G1(14) and B(16), B(16) is then discarded Fringe=[B(12), G1(14)] 17

18 Another Example: A* earch A(8) C(7) D(11) E(12) B(12) G2(16) D(10) B(16) G1(14) 5. Expand B: generate E(12) and G2(16) Fringe=[E(12), G1(14), G2(16)] Another Example: A* earch A(8) C(7) D(11) B(12) E(12) G2(16) D(10) G2(13) B(16) G1(14) 6. Expand E: generate G2(13), remove G2(16) in Fringe Fringe=[G2(13), G1(14)] Recognize G2 as the goal, Path: B E G2 18

19 Optimality of A*? Whether an A* algorithm is optimal depends on the used heuristic function h. ome specific property of h is required to ensure optimality We will discuss such property for tree and graph searches respectively When Is A* Optimal on Trees Theorem 1: If h(n) is admissible, A * using TREE-EARCH is optimal A heuristic h(n) is admissible if for every node n, h(n) h * (n), where h * (n) is the true least cost to reach the goal state from n. An admissible heuristic never overestimates the least cost to reach the goal. 19

20 Proof of Optimality of A* on trees? (only for CDT312) uppose a goal state G appears as the first element in the fringe, the search will terminate and return the path from the initial to G. As there is only one path to G, we only need to prove that G is the optimal goal state Proof of Optimality on Trees with admissible heuristics (only CDT312) Prove: a suboptimal goal G2 will never appear first in Fringe uppose some suboptimal goal G 2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G. We have: 1. f(g 2 )=g(g 2 )>g(g) 2. g(g)=g(n)+h*(n)>g(n)+h(n)=f(n) Hence we have f(g 2 )>f(n) 20

21 When is A* Optimal on Graphs Theorem 2: If h(n) is consistent, A* using GRAPH-EARCH is optimal A heuristic is consistent if for every node n, every successor n' of n generated by any action a, h(n) c(n,a,n') + h(n') Consistence stronger than Admissibility Consistence is a sufficient condition for admissibility: if h is consistent, then h is admissible. Exercise: Please prove the admissibility of heuristic function when it is consistent (only for CDT312) Ning will announce suggested solution in blackboard one week later 21

22 Proof of Optimality for A* on graphs? (only for CDT312) uppose a goal state G appears as the first element in the fringe, the search will terminate and return the path from the initial to G. We have to prove: 1. G is optimal goal state ince consistence ensures admissibility, this can be proved in the same way as for A* tree search 2. It is not possible to find another path to G yielding lower cost? Proof of optimality on graphs with consistent heuristics (only for CDT312) To Prove: any other path to G will have more cost uppose n is a node in the Fringe and appears on another path to G, and n is the successor of n on the path, we have f(n )= g(n') + h(n') = g(n) + c(n,a,n') + h(n') g(n) + h(n) = f(n) Therefore f(n) is non-decreasing along the path 22

23 Hill Climbing earch In many optimization problems in engineering applications, we want to find the good possible solution, but we don t care the path to goal In such cases, we can try to use a simple local search strategy: hill climbing. It only selects among the children of a current node and moves to the child with the best evaluation, terminates when no child improves over the current state. Hill-climbing search Loop: 1. Generate and evaluate successors of Current node 2. Neighbor <---- Best successor of Current 3. If Neighbor is not better than Current, then terminate and return Current as the solution. 4. Current <--- Neighbor, go to step 1 23

24 Feature election Task Complex process Feature selection Fuzzy modelling Fuzzy model tate (node):feature subset Operator: select one more feature Hill-Climbing for Feature election Finding the best feature subset with hill climbing, no back tracking , 1 2, 3 2, 4 2, 3, 1 2, 3, 4 Ning iong, Peter Funk, Construction of Fuzzy Knowledge Bases Incorporating Feature election, Journal of oft Computing, vol 10, nr 9, p , pring Verlag,

25 Gradient earch as Hill-climbing If you want to optimize a math function f(x 1, x 2,,x n ), you may use gradient information to locate the best solution in the continuous space.. Gradient gradient Gradient Gradient method used for optimizing weights of a neural network Gradient method used for learning the conditional probabilities of a Bayesian net Hill-climbing search Problem: depending on initial state, can get stuck in local maxima 25

26 Reading Guidance Read the sections 3.5, 4.1 of Russel s book Read the introduction remark and Hillclimbing search in section 4.3 of Russel s book You may need to read chapter 3 of Luger s book to better understand various search algorithms on graph. 26