Computer Science and Software Engineering University of Wisconsin - Platteville. 3. Search (Part 1) CS 3030 Lecture Notes Yan Shi UW-Platteville

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1 Computer Science and Software Engineering University of Wisconsin - Platteville 3. Search (Part 1) CS 3030 Lecture Notes Yan Shi UW-Platteville Read: Textbook Chapter ,3.12, 4.

2 Problem Solving as Search Problem: State Space Initial state to start A description of all possible actions at any state Transition model: what each action does The goal states A path cost function that assign a cost to each path. Problem solution: an action sequence that leads from the initial state to a goal state.

3 The river crossing puzzle Initial state: PFCG Actions: Person can take one item to cross the river Transition model: fox eats chicken, chicken eats grain if left alone Goal state: PFCG Path cost function: equal efforts State space is usually represented as a graph. Search tree is more often used for problem solving. All possible paths through a graph. Can you construct the state space and search tree for the river crossing puzzle?

4 Search in Search Tree Start from the initial state (or the goal state) Expand a state by applying the search actions to that state, generating ALL of its successor states. These successors are in the next level down of the search tree The order in which we choose states for expansion is determined by the search strategy Different strategies result in different behavior KEY: We want to find the solution while realizing in memory as few as possible of the nodes in the search space.

5 More Examples The 8-squares puzzle Initial state: Goal state: Actions: up, down, left, right Maze: Initial state: entrance Goal state: exit Actions: north, south, east, west

6 A quick review on graph Graph: G = {V, E} V: a set of vertices (nodes) E: a set of edges Directed graph Path, loop Connected graph and disconnected graph Complete graph Root Tree: Root, leaf, parent, child, siblings, ancestors, decedents, branching factor

7 How to represent a graph? Adjacency list Intuitive A E Less memory for sparse graph Adjacency matrix B C D Faster access: O(1) vs. O(n)

8 Graph Problem Examples Konigsberg bridges problem: a walk that crosses each bridge once? Traveling salesperson: a route that visits each city once? Map coloring: can map be painted so no two adjacent countries have the same color?

9 Data-Driven or Goal-Driven Search Data-driven search: Start from an initial state and move forward until a goal is reached Top-down approach A.K.A. forward chaining When initial data is available and goal is not clear Goal-driven search: Start at the goal and work back toward a start state Bottom-up approach A.K.A. backward chaining When goal is clear: exit a maze, medical diagnosis

10 Example: The Towers of Hanoi Initial State: (123)()() Actions: Op1: Move disk from peg 1 to peg 2 Op2: Move disk from peg 1 to peg 3 Op3: Move disk from peg 2 to peg 1 Op4: Move disk from peg 2 to peg 3 Op5: Move disk from peg 3 to peg 1 Op6: Move disk from peg 3 to peg 2 Goal test: ()()(123) Path cost: 1 per step

11 Example: The Towers of Hanoi First five levels of search tree:

12 Goal Tree A.K.A. and-or tree Goal: solution Subgoal: each step along the way And-node: a goals can be achieved only by solving all its subgoals. Or-node: a goals can be achieved by solving any of its subgoals. Leaf nodes are either success nodes or failure nodes.

13 Example: The Towers of Hanoi with 4 disks

14 Example: The Towers of Hanoi with 4 disks

15 Properties of Search Methods Complexity Time and space Completeness Is it guaranteed to find a goal state if one exists? Optimality (often used to mean admissibility) Is it guaranteed to find a solution in the quickest time? Admissibility Is it guaranteed to find the best solution? Irrevocability No backtracking Called tentative if there is backtracking

16 P, NP and NP-hard In real-world, the search problem can be classified to two classes: P and NP. class P is the set of all problems for which solutions with polynomial time behavior have been found. class NP is the set of all problems for which solutions with exponential behavior have been found. If an optimization of the problem cannot be solved in polynomial time, it is called NP-hard. If a decision problem is both NP and NP-hard, it is NP-complete.

17 Uninformed Search Brute-force search (exhaustive search, blind search, generate and test) Examine every node until it finds a goal Simplest form of search Assume no additional knowledge other than how to traverse the search tree and detect a leaf and goal node How many possible states do we have? TSP of n cities Sliding-block puzzle (8-squares) Rubik s cube (3 by 3)

18 Breadth-First Search Check all siblings before children ;; breadth_first_search: StartState->SUCCESS FAILURE Open <- [Start] // states to be considered Closed <- [] // states that have been considered while open!= [] Next first(open) Open rest(open) // remove first item from open if isgoal(next), return SUCCESS let Kids = children(next) - (Open union Closed) Closed Closed union [Next] Open append (Open, Kids) end-while return FAILURE Use a queue

19 Depth-First Search Check all decedents before siblings ;; depth_first_search: StartState->SUCCESS FAILURE Open <- [Start] // states to be considered Closed <- [] // states that have been considered while open!= [] Next first(open) Open rest(open) // remove first item from open if isgoal(next), return SUCCESS let Kids = children(next) - (Open union Closed) Closed Closed union [Next] Open append (Kids, Open) end-while return FAILURE Use a stack

20 Example: TSP 200 Eau Claire Appleton Dodgeville Brookfield Cuba City Can we solve TSP using BFS and DFS? How to order siblings? Initial State:? Actions: Travel from one city to another Transition model: roads Goal test: visit each city exactly once & return to A Path cost: traveling distance

21 Breadth-first vs. Depth-first Complexity BFS DFS Completeness Optimality Admissibility Irrevocability b is the branching factor of the tree d is the depth of the shallowest goal state reached m is the depth of the deepest goal state reached

22 Breadth-first vs. Depth-first BFS Complexity Time: O(b d ) Space: O(b d ) DFS Completeness Yes No Optimality No No Admissibility Yes (if no weight) No Irrevocability Yes Yes Time: O(b m ) Space: O(bm) b is the branching factor of the tree d is the depth of the shallowest goal state reached m is the depth of the deepest goal state reached

23 Breadth-first vs. Depth-first When to use which? Some paths are extremely long All path are of similar length All path are of similar length and all lead to a goal state High branching factor: a state may lead to many different states Internet search engine?

24 Variations of BFS and DFS Uniform-cost search: Instead of expanding the shallowest node, expand the node with the smallest path cost. e.g. Dijkstra s Algorithm If all steps are equal, it is the same as BFS The first goal found is an optimal solution Depth-limited search: DFS with depth limit x Avoid getting stuck in infinitely deep path or loops will find a solution if it is within the depth limit

25 Review: Dijkstra s algorithm Dijkstra s algorithm: single source shortest path algorithm G = (V,E} S = {vertices whose shortest paths from the source is determined} di = best estimate of shortest path to vertex i pi = predecessors Initialize di and pi, Set S to empty, While there are still vertices in V-S, Sort the vertices in V-S according to the current best estimate of their distance from the source, Add u, the closest vertex in V-S, to S, Update all the vertices still in V-S connected to u to a better estimation if possible

26 Dijkstra s algorithm example

27 Dijkstra s algorithm example 0+7= =10 4 5

28 Dijkstra s algorithm example 0+7= = =10 7+6=13

29 Dijkstra s algorithm example 0+7= = =10 7+6=13

30 Dijkstra s algorithm example 0+7= = = =10 7+6=13

31 Dijkstra s algorithm example 0+7= = = =10 7+6=13

32 Dijkstra s algorithm example 0+7= = = =10 7+6=13

33 Depth-First Iterative Deepening DFID, a.k.a. Iterative Deepening Search or IDS Exhaustive search technique that combines depth-first with breadth-first search: repeatedly carrying out depth-limit search on the tree, starting with a depth-first search limited to a depth of 1, then a depth-first search of depth 2, 3, and so on, until a goal node is found. Combines the benefit of BFS that it will always find the shortest-step path and of DFS in the efficiency of memory use. Avoid the problem of DFS that it may be trapped in infinitely deep path.

34 Is DFID too time consuming? It is almost as efficient as DFS and BFS because for most trees, the majority of nodes are in the deepest level All three approaches spend most of their time examining these node For a tree of depth d and with a branching factor of b, the total number of nodes is 1 + b + b^2 + b^ b^d = (1-b d+1 )/(1-b) O(b d ) The total # of nodes DFID checks is (d + 1) + b(d) + b^2 (d-1) + b^3(d-2) b^d O(b d )

35 Properties of DFID Complete? Optimal? Time complexity Space complexity Backtracking? Yes Yes if no weight O(b d ) O(bd) Yes