Tree Searching Strategies

1. Tree Searching Strategies 2010/12/03

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

3. Sir William Rowan Hamilton (4 August 1805–2 September 1865) was an Irish physicist, astronomer, and mathematician

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

5. Fig. 6-8 The tree representation of whether there exists a Hamiltonian circuit of the graph in Fig. 6-6

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

7. The Traveling Salesperson Problem (TSP) • The traveling salesperson problem asks for a Hamiltonian circuit of the minimum length. (NP-complete problem)

8. 8-Puzzle Problem Initial State: Goal State:

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

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

11. Breadth-First Search Scheme • In breadth-first search, all the nodes on one level of the tree are examined before the nodes on the next level are examined. • It can be accomplished with the help of the queue.

12. 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. • Step3: Remove the first element from the queue. Add all descendants of the first element, if any, to the end of the queue one by one. • Step4: If the queue is empty, then signal failure. Otherwise, go to Step 2.

13. 1 2 3 4 6 5 7 Goal Node

14. Depth-First Search Scheme • The depth-first search always selects the deepest node for further expansion (visiting-and-expanding). • It can be accomplished with the help of the stack.

15. 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 stack is a goal node. If it is, stop. • Step3: Remove the top element from the stack. Add all descendants of the first element, if any, to the top of the stack one by one. • Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

16. Example: The sum of subset problem solved by depth-first search • The 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

17. 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 is f(n) = w(n), where w(n) is the number of misplaced tiles in node n.

18. 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 stack is a goal node. If it is, stop. • Step3: Remove the top element from the stack. Add the first element’s descendants, if any, to the top of the stack according to the order computed by the evaluation function. • Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

19. An 8-puzzle problem solved by the hill climbing method

20. 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.

21. 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. • Step3:Remove the root element from the heap and expand the element, i.e., add all descendants of the element into the heap. • Step4:If the heap is empty, then signal failure. Otherwise, go to Step 2.

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

23. Q&A