Blind Search in Graphs. Dr. Asaad Sabah Hadi

Transcription

1 Blind Search in Graphs Dr. Asaad Sabah Hadi 1

2 Depth-First Search without the unique parent assumption 2

3 Trees and Graphs Trees: Consists of nodes connected by links (or edges). every node has a single parent. The root has no parents. No cycles exist. Graphs: Consists of nodes connected by links (or edges). Nodes have any number of parents. Cycles may exists 3

4 Types of Graphs Graphs are divided based on the type of edges they have: Undirected graphs: are graphs where the edges between nodes have no direction. Directed graphs: are graphs where the edges have a direction. In this case, we can move from a node to another only in the edge direction. 4

5 Directed Graph Example Figure: A labeled directed graph. 5

6 6 Graphs: An Example

7 Graphs: An Example Figure : The city of Königsberg. 7

8 Figure : Graph of the Königsberg bridge system. 8

9 A General Tree Search Algorithm Input: State Space Ouput: failure or path from a start state to a goal state. Assumptions: L is a list of nodes that have not yet been examined. The state space is a tree where each node has a single parent. 1. Set L to be a list of the initial nodes in the problem. 2. While L is not empty 1. Pick a node n from L. 2. If n is a goal node 1. stop and return it and the path from the initial node to n. Else 1. remove n from L and add all of n s children to L labelling each with its path from the initial node. End while Return failure 9

10 Breadth-First Graph Search Algorithm Input: State Space Ouput: failure or path from a start state to a goal state. Assumptions: Open is a list of nodes that have not yet been examined. Closed is the list of states that have been examined. 1. Set Open to be a list of the initial nodes in the problem. At any given point in time. 2. While Open is not empty 1. Pick a node n from the front of Open. 2. If n is a goal node 1. stop and return it and the path from the initial node to n. Else 1. remove n from Open 2. add n to Closed 3. get all n s children 4. discard n s children that are in the Closed or Open lists. 5. add the remaining children to the end of Open labelling each with its path from the initial node. End while return failure 10

11 Breadth-First Search in a Graph Iter atn n Open Closed 0 [A] [] 1 A [B,C,D] [A] 2 B [C,D,E,F] [B,A] 3 C [D,E,F,G,H] [C,B,A] 4 D [E,F,G,H,I,J] [D,C,B,A] 5 E [F,G,H,I,J,K,L] [E,D,C,B,A] 6 F [G,H,I,J,K,L,M] [F,E,D,C,B,A] 11

12 Breadth-First Worst Case Space Usage Closed List 0 1 Open List... 2 : :... d 12 1+b+b b d = (b d+1-1)/(b-1) which is O(b d+1 )

13 Depth-First Graph Search Algorithm Input: State Space Ouput: failure or path from a start state to a goal state. Assumptions: Open is a list of nodes that have not yet been examined. Closed is the list of states that have been examined. 1. Set Open to be a list of the initial nodes in the problem. At any given point in time. 2. While Open is not empty 1. Pick a node n from the front of Open. 2. If n is a goal node 1. stop and return it and the path from the initial node to n. Else 1. remove n from Open 2. add n to Closed 3. get all n s children 4. discard n s children that are in the Closed or Open lists. 5. add the remaining children to the front of Open labelling each with its path from the initial node. End while return failure 13

14 Depth-First Search in a Graph Iter atn 14 n Open Closed 0 [A] [] 1 A [B,C,D] [A] 2 B [E,F,C,D] [B,A] 3 E [K,L,F,C,D] [E,B,A] 4 K [S,L,F,C,D] [K,E,B,A] 5 S [L,F,C,D] [S,K,E,B,A] 6 L [T,F,C,D] [L,S,K,E,B,A] 7 T [F,C,D] [T,L,S,K,E,B,A] 8 F [M,C,D] [F,T,L,S,K,E,B,A] 9 M [C,D] [M,F,T,L,S,K,E,B,A]

15 Depth-First Worst Case Space Usage Closed List 0 1 Open List... 2 : :... d 1+b+b b d = (b d+1-1)/(b-1) which is O(b d+1 ) 15

16 Depth-First or Breadth-First? Tree Search Times Algorithm Worst Run time Worst Space Usage Depth-First Search Breadth-First Search 16 Guaranteed to find a path? O(b d+1 ) O(db) No (Unless the depth of the tree is limited) Guaranteed to find shortest Path? O(b d+1 ) O(b d ) Yes Yes Both searches have running times that grow exponentially in the worst case. Depth-First search uses a lot less space provided there are no infinitely deep paths in the tree. This is particularly important for AI search problems where there might be a huge number of states to explore. No

17 Depth-First or Breadth-First? Graph Search Times Algorithm Worst Run time Worst Space Usage Depth-First Search Breadth-First Search Guaranteed to find a path? O(b d+1 ) O(b d+1 ) No (Unless the depth of the tree is limited) Guaranteed to find shortest Path? O(b d+1 ) O(b d+1 ) Yes Yes When using a Closed list of visited nodes, Both searches have running times and space usage that grow exponentially in the worst case. No 17

18 Data-Driven and Goal-Driven Search So far we have been assuming that the start state contains the current state of our problem and is at the root of our search and the goal state is at the fringe. Direction of Search Start Goal

19 19 Data-Driven Search Example:

20 Data-Driven and Goal-Driven Search Alternatively: we could start with the goal and work our way back to the initial state Goal... Direction of Search... Start 20

21 Goal-Driven Search 1. Find what states could have produced the goal 2. Consider those states as subgoals 3. Repeat for each subgoal until an initial state is found 21

22 Data-Driven or Goal-Driven Search? Problem Structure Example Goal- Driven Ease of Determining Goal Branching Factor Availability of Data 22 Goal is given Difficult to form a goal or hypothesis Large number of rules producing many possible goals Large number of potential goals and few ways to use the information Data not given but must be acquired by system Data is given Theorem Prover DENDRAL expert System Theorem Prover DENDRAL expert System Medical Diagnosis Systems Geological data analysis by PROSPECTOR X X X Data Driven X X X

23 Data-Driven or Goal-Driven Search? Fig State space in which goal-directed search effectively prunes extraneous search paths. 23

24 Data-Driven or Goal-Driven Search? Data Direction of Reasoning Goal Data-driven search examines at least 13 nodes before reaching the goal Goal-driven search examines 10 nodes before reaching the goal 24

25 Data-Driven or Goal-Driven Search? Fig 13 State space in which data-directed search prunes irrelevant data and their consequents and determines one of a number of possible goals. 25

26 Data-Driven or Goal-Driven Search? Data-driven search examines 11 nodes before finding a goal Goal-driven search starting from potential goals examines most of the nodes before reaching the proper data 26

27 In Conclusion There are different types of blind search algorithms. Which one is more appropriate for the problem on hand depends on: The algorithm s characteristics. The computing time and memory resources available. The characteristics of the solution needed: importance of finding the shortest path. whether we want all solutions or a single solution The problem being solved: The size of the search space including its branching factor. The depth at which the solution is expected to be found. 27