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 x1, x2, …,xn, then there are 2n possible assignments.

4. Satisfiability problem
An instance:
-x1……..……(1)
x1…………..(2)
x2 v x5….….(3)
x3…….…….(4)
-x2…….…….(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. 8-Puzzle Problem 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 Initial State:
Goal State: 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5

9. Tree Representation of the solution of 8-puzzle problem

10. How to expand the tree ? Breadth-First Search Depth-First Search
Hill Climbing
Best-First Search

11. Breadth-First Search Scheme
Step1: Form a one-element queue consisting of the root node.
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: Remove the first element from the queue. Add the first element’s descendants, if any, to the end of the queue.
Step4: If the queue is empty, then signal failure. Otherwise, go to Step 2.

12. 2 3 1 8 4 7 6 5 1 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 2 3 2 8 3 1 4 7 6 5 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 1 2 3 7 8 4 6 5 4 6 5 7
Goal Node

13. Depth-First Search Scheme
Step1: Form a one-element stack consisting of the root node.
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: Remove the top element from the stack. Add the first element’s descendants, if any, to the top of the stack.
Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

14. 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.

15. 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.

16. Hill Climbing Search Scheme
Step1: Form a one-element stack consisting of the root node.
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: 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: If the stack is empty, then signal failure. Otherwise, go to Step 2.

17. An 8-Puzzle Problem Solved by the Hill Climbing Method

18. 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.

19. Best-First Search Scheme
Step1:Consturct a heap by using the evaluation function. First, form a 1-element heap consisting of the root node.
Step2:Test to see if the root element in the heap is a goal node. If it is, stop; otherwise, go to Step 3.
Step3:Remove the root element from the heap and expand the element. Add the descendants of the element into the heap.
Step4:If the heap is empty, then signal failure. Otherwise, go to Step 2.

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

21. Q&A