Best-First Search Minimizing Space or Time. IDA* Save space, take more time

Transcription

1 Best-First Search Minimizing Space or Time IDA* Save space, take more time IDA*-1

2 A* space complexity» What does the space complexity of A* depend upon? IDA*-2

3 A* space complexity 2» What does the space complexity of A* depend upon? > Saves all found nodes IDA*-3

4 A* space complexity 3» What does the space complexity of A* depend upon? > Saves all found nodes» What does the number of found nodes depend upon? IDA*-4

5 A* space complexity 4» What does the space complexity of A* depend upon? > Saves all found nodes» What does the number of saved nodes depend upon? > Depends upon the branching factor (B) and height of tree (H) IDA*-5

6 A* space complexity 5» What is the space complexity of A*? IDA*-6

7 A* space complexity 6» What is the space complexity of A*? > Approximately B H IDA*-7

8 Space saving» How can we save space? IDA*-8

9 Space saving 2» How can we save space? > Not keep all the found nodes IDA*-9

10 Space saving 3» How can we save space? > Not keep all the found nodes» Which ones do we keep? IDA*-10

11 Space saving 4» How can we save space? > Not keep all the found nodes» Which ones do we keep? > The ones in the current path IDA*-11

12 Space saving 5» How can we save space? > Not keep all the found nodes» Which ones do we keep? > The ones in the current path» How do we get the nodes we threw away? IDA*-12

13 Space saving 6» How can we save space? > Not keep all the found nodes» Which ones do we keep? > The ones in the current path» How do we get the nodes we threw away? > By regenerating them when a different path is to be extended IDA*-13

14 Iterative deepening» How does iterative deepening work? IDA*-14

15 Iterative deepening 2» How does iterative deepening work? > By doing depth-first search with increasing depth IDA*-15

16 Iterative deepening 3» How does iterative deepening work? > By repeating depth-first search with increasing depth» What can we use instead of depth? IDA*-16

17 Iterative deepening 4» How does iterative deepening work? > By repeating depth-first search with increasing depth» What can we use instead of depth? What analogous feature is the A* algorithm based on? IDA*-17

18 Iterative deepening 5» How does iterative deepening work? > By repeating depth-first search with increasing depth» What can we use instead of depth? What analogous feature is the A* algorithm based on? > The f(n) function IDA*-18

19 Iterative deepening 6» How does iterative deepening work? > By repeating depth-first search with increasing depth» What can we use instead of depth? What analogous feature is the A* algorithm based on? > The f(n) function» How do we use the f(n) function? IDA*-19

20 Iterative deepening 7» How does iterative deepening work? > By repeating depth-first search with increasing depth» What can we use instead of depth? What analogous feature is the A* algorithm based on? > The f(n) function» How do we use the f(n) function? > Do depth-first search with increasing f-limit IDA*-20

21 A view of iterative f(n) deepening S f = Bi f = Bj f = Bk Bi < Bj < Bk IDA*-21

22 IDA* algorithm bound = f(start_node) found = false while not found do depth-first search from start_node for nodes N expand nodes with f(n) bound if goal_found then found ß true else bound = min {N : generated by search f(n) > bound f(n) } fi such that end members of the set IDA*-22

23 Bratko Figure 12.2 f = 7 f = 9 A E f = 8 f = 10 f = 12 B C D S T f = 1 Assume f(s) = 5 Depth-first search limited by f 5 F G f = 11 f = 11 f = 11 Find f(a) = 7 > Bound f(e) = 9 > Bound New bound = min {7, 9} = 7 Depth-first search limited by f 7 Find nodes B and E outside the bounds New bound = min {8, 9} = 8 From now on find new bounds as follows 9 from f(e) 10 from f(c) 11 from f(f) Find the goal. IDA*-23

24 Exercise question Trace the execution of A* for the tree. How many nodes are generated by A* and IDA*? Count all re-generated nodes. A f = 1 f = 2 B C f = 1 f = 10 f = 10 D E F f = 1 G f = 2 f = 10 f = 10 f = 3 H I J K f = 4 L f = 4 M f = 5 Created Gunnar by Bratko Gotshalks IDA*-24

25 IDA* performance» What would we examine in thinking about IDA* performance? IDA*-25

26 IDA* performance 2» What would we examine in thinking about IDA* performance? > Space > TIme IDA*-26

27 IDA* space performance» Space is not a consideration, why? IDA*-27

28 IDA* space performance 2» Space is not a consideration, why? > Only one path is kept at a time IDA*-28

29 IDA* time performance Need to look at time.» What is the problem if only one path is kept at any time? IDA*-29

30 IDA* time performance 2 Need to look at time.» What is the problem if only one path is kept at any time? > Have to regenerate paths that are to be extended IDA*-30

31 IDA* regeneration performance» Under what conditions is the overhead of re-generating nodes > High? IDA*-31

32 IDA* regeneration performance 2» Under what conditions is the overhead of re-generating nodes > High? When there are many different f values IDA*-32

33 IDA* regeneration performance 3» Under what conditions is the overhead of re-generating nodes > High? When there are many different f values Extreme case have one new node per path regenerated IDA*-33

34 IDA* regeneration performance 4» Under what conditions is the overhead of re-generating nodes > High? When there are many different f values Extreme case have one new node per path regenerated Unacceptable overhead IDA*-34

35 IDA* regeneration performance 5» Under what conditions is the overhead of re-generating nodes > Low? IDA*-35

36 IDA* regeneration performance 5» Under what conditions is the overhead of re-generating nodes > Low? When there are equal f values IDA*-36

37 IDA* regeneration performance 6» Under what conditions is the overhead of re-generating nodes > Low? When there are equal f values Each path generates many new nodes IDA*-37

38 IDA* regeneration performance 7» Under what conditions is the overhead of re-generating nodes > Low? When there are equal f values Each path generates many new nodes Regenerated nodes are a small fraction of total generated nodes IDA*-38

39 Monotonic function» What is a monotonic function? IDA*-39

40 Monotonic function» What is a monotonic function? > A function that is either entirely non-increasing or non-decreasing IDA*-40

41 f function monotonicity» For the A* algorithm does It matter if the h function is non-monotonic? IDA*-41

42 f function monotonicity 2» For the A* algorithm does It matter if the f function is non-monotonic? > No IDA*-42

43 f function monotonicity 3» For the A* algorithm does It matter if the f function is non-monotonic? > No» Why? IDA*-43

44 f function monotonicity 4» For the A* algorithm does It matter if the f function is non-monotonic? > No» Why? > The A* algorithm has all the paths and will always expand the best one first IDA*-44

45 f function monotonicity 5» For the IDA* algorithm does It matter if the f function is non-monotonic? IDA*-45

46 f function monotonicity 5» For the IDA* algorithm does It matter if the f function is non-monotonic? > Yes IDA*-46

47 f function monotonicity 6» For the IDA* algorithm does It matter if the f function is non-monotonic? > Yes» Why? IDA*-47

48 f function monotonicity 7» For the IDA* algorithm does It matter if the f function is non-monotonic? > Yes» Why? > The IDA* algorithm always expands paths from the start with a monotonically increasing f function it expands nodes in best-first order. IDA*-48

49 Non-monotonic f function problem In the following if f-bound = 3, then the B node is expanded before node C. The tree rooted at B could be large before the bound become 10, when C would be expanded, where the solution could be in the tree rooted at G. A f = 5 f = 3 B C f = 10 F f = 4 G f = 2 IDA*-49

50 IDA* Problem» What is a major problem with IDA*? IDA*-50

51 IDA* Problem 2» What is a major problem with IDA*? > In unfavourable situations the cost of regenerating nodes becomes unacceptable IDA*-51

52 IDA* Problem 3» What is a major problem with IDA*? > In unfavourable situations the cost of regenerating nodes becomes unacceptable» How do we solve the problem? IDA*-52

53 IDA* Problem 4» What is a major problem with IDA*? > In unfavourable situations the cost of regenerating nodes becomes unacceptable» How do we solve the problem? > Create a different algorithm IDA*-53

54 IDA* Problem 5» What is a major problem with IDA*? > In unfavourable situations the cost of regenerating nodes becomes unacceptable» How do we solve the problem? > Create a different algorithm RBFS Recursive Best-First Search IDA*-54