1 Tree Searching Strategies

2 The procedure of solving many problems may be represented by trees. Therefore the solving of these problems becomes a tree searching problem.

3 Satisfiability problem Tree Representation of Eight Assignments. If there are n variables x 1, x 2, …,x n, then there are 2 n possible assignments.

4 Satisfiability problem An instance: -x 1 …….. …… (1) x 1 …………..(2) x 2 v x 5 …. ….(3) x 3 ……. …….(4) -x 2 ……. …….(5) A Partial Tree to Determine the Satisfiability Problem. We may not need to examine all possible assignments.

5 Hamiltonian circuit problem E.g. the Hamiltonian circuit problem A Graph Containing a Hamiltonian Circuit

6 Fig. 6-8 The Tree Representation of Whether There Exists a Hamiltonian Circuit of the Graph in Fig. 6-6

7 A tree showing the non-existence of any Hamiltonian circuit.

8 How to expand the tree ? Breadth-First Search Depth-First Search Hill Climbing Best-First Search Branch-and-Bound Strategy (for optimization problems)

9 Breadth-First Search Scheme Step1 Step1: Form a one-element queue consisting of the root node. Step2 Step2: Test to see if the first element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. Step3 Step3: Remove the first element from the queue. Add the first element ’ s descendants, if any, to the end of the queue. Step4 Step4: If the queue is empty, then signal failure. Otherwise, go to Step 2.

Goal Node

11 Depth-First Search Scheme Step1 Step1: Form a one-element stack consisting of the root node. Step2 Step2: Test to see if the top element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. Step3 Step3: Remove the top element from the stack. Add the first element ’ s descendants, if any, to the top of the stack. Step4 Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

12 E.G.: the depth-first search E.g. sum of subset problem Given a set S={7, 5, 1, 2, 10}, answer if  S ’  S  sum of S ’ = 9. The Sum of Subset Problem Solved by Depth-First Search.

13 Hill climbing A variant of depth-first search The method selects the locally optimal node to expand. E.g. for the 8-puzzle problem, evaluation function f(n) = w(n), where w(n) is the number of misplaced tiles in node n.

14 Hill Climbing Search Scheme Step1 Step1: Form a one-element stack consisting of the root node. Step2 Step2: Test to see if the top element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. Step3 Step3: Remove the top element from the stack. Add the first element ’ s descendants, if any, to the top of the stack according to order computed by the evaluation function. Step4 Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

15 An 8-Puzzle Problem Solved by the Hill Climbing Method

16 Best-first search strategy Combing depth-first search and breadth-first search Selecting the node with the best estimated cost among all nodes. This method has a global view.

17 Best-First Search Scheme Step1 Step1:Form a one-element list consisting of the root node. Step2 Step2:Remove the first element from the list. Expand the first element. If one of the descendants of the first element is a goal node, then stop; otherwise, add the descendants into the list. Step3 Step3:Sort the entire list by the values of some estimation function. Step4 Step4:If the list is empty, then signal failure. Otherwise, go to Step 2.

18 An 8-Puzzle Problem Solved by the Best-First Search Scheme Goal Node

19 The branch-and-bound strategy This strategy can be used to solve optimization problems without an exhaustive search. (DFS, BFS, hill climbing and best-first search can not be used to solve optimization problems.) E.g. A Multi-Stage Graph Searching Problem.

20 E.G.:A Multi-Stage Graph Searching Problem

21 Solved by branch-and-bound

22 Solved by branch-and-bound

23 Branch-and-bound strategy 2 mechanisms: A mechanism to generate branches A mechanism to generate a bound so that many braches can be terminated. Although it is usually very efficient, a very large tree may be generated in the worst case. It is efficient in the sense of average case.

24 Exercise Use BFS, DFS, Hill-Climbing and Best-First Search schemes to solve the following 8-puzzle problem with the evaluation function being the number of misplaced tiles Initial State: Goal State: