Download presentation

Presentation is loading. Please wait.

Published byGavin Baker Modified over 4 years ago

1
Towards theoretical frameworks for comparing constraint satisfaction models and algorithms Peter van Beek, University of Waterloo CP 2001 · Paphos, Cyprus November 2001

2
2 Acknowledgements Joint work with: Fahiem Bacchus Xinguang Chen Grzegorz Kondrak Toby Walsh

3
3 Constraint programming methodology Model problem ·specify in terms of constraints on acceptable solutions ·define/choose constraint model: variables, domains, constraints Solve model ·define/choose algorithm ·define/choose heuristics Verify and analyze solution

4
4 Previous work Many models, algorithms, and heuristics proposed Choice of model, algorithm, and heuristic can greatly influence efficiency ( e.g., Nadel, 1990; Ginsberg et al., 1990; Frost and Dechter, 1994; Tsang et al., 1995 Smith et al., 2000; Beacham et al., 2001)

5
5 How to compare? Empirical studies ·benchmark sets ·random problems Theoretical studies ·worst case ·average case (e.g. Brown & Purdom, 1981; Nadel, 1983; Wilf, 1984) ·pair-wise comparisons -partial orders -bounded worse Performance measures ·nodes visited ·constraint checks ·CPU time

6
6 Part I Comparing backtracking algorithms

7
7 Some backtracking algorithms Chronological Backtracking (BT) Backjumping (BJ) (Gaschnig, 1978) Conflict-Directed Backjumping (CBJ) (Prosser, 1993) Forward Checking (FC) (Haralick & Elliott, 1980; McGregor, 1979) Maintaining arc consistency (MAC) (Haralick & Elliott, 1980; McGregor, 1979; Sabin & Freuder, 1994)

8
8 Example: 4-queens 4 3 2 1 x1x1 x2x2 x3x3 x4x4 x i x j and | x i - x j | | i - j |

9
9 Search tree for 4-queens x1x1 x2x2 x3x3 x4x4 1234 (1,1,1,1)(4,4,4,4)(2,4,1,3)(3,1,4,2)

10
10 4 3 2 1 {x 1 1} Backjumping Q Q {x 1 1, x 2 3} x1x1 x2x2 x1x1 x2x2 {x 1 1, x 2 3, x 4 2} Q x3x3 x1x1 x3x3 x4x4 x2x2

11
11 New algorithms: BJ 1, …, BJ n BJ k : ·allowed to backjump at most k times from a leaf ·after that it must chronologically backtrack Special cases: ·BJ 1 equivalent to BJ ·BJ n equivalent to CBJ

12
12 k-consistency Given ·any k-1 distinct variables, ·any consistent assignment to those variables, ·any additional variable x there exists a consistent assignment to x 4 3 2 1 x1x1 x2x2 x3x3 x4x4 2-consistent ? Q Q 1-consistent ? 3-consistent ? strongly 2-consistent yes no

13
13 Maintaining consistency { x 1 1, x 2 4 } 4 3 2 1 x1x1 x2x2 x3x3 x4x4 Q Q x 3 x 4 |x 3 - x 4 | 1 Binary constraints Unary constraints x 3 1, 3, 4 x 4 1, 2, 4 Induced CSP

14
14 Maintaining consistency 4 3 2 1 x1x1 x2x2 x3x3 x4x4 Q Q { x 1 1, x 2 4 } x 3 x 4 |x 3 - x 4 | 1 Binary constraints Unary constraints x 3 1, 3, 4 x 4 1, 2, 4 FC: strong 1-consistency

15
15 Maintaining consistency Q Q { x 1 1, x 2 4 } x 3 x 4 |x 3 - x 4 | 1 Binary constraints Unary constraints x 3 1, 3, 4 x 4 1, 2, 4 4 3 2 1 x1x1 x2x2 x3x3 x4x4 MAC: strong 2-consistency Q Q

16
16 New algorithms: MC 1, …, MC n MC k : ·maintains strong k-consistency on induced CSP at each node in backtrack tree Special cases: ·MC 1 equivalent to FC ·MC 2 equivalent to MAC for binary CSPs

17
17 Comparing backtracking algorithms Proposal: for all CSP models for all variable orderings algorithm a 1 algorithm a 2 Desirable features: ·still holds if use best possible CSP model for a 2 ·still holds if use best possible variable ordering for a 2

18
18 Theoretical methodology Using local consistency concepts: ·formulate necessary and sufficient conditions for a node to be visited by a backtracking algorithm Using conditions: ·construct partial orders of the algorithms

19
19 4 3 2 1 x1x1 x2x2 x3x3 x4x4 {x 1 1} {x 1 1, x 2 4} ? Example sufficient condition: FC If a node is consistent and its parent is strongly 1-consistent, then FC visits the node Q Q

20
20 Example necessary condition: MAC-CBJ If MAC-CBJ visits a node, then it is consistent and its parent is strongly 2-consistent 4 3 2 1 x1x1 x2x2 x3x3 x4x4 {x 1 1} {x 1 1, x 2 4} ? Q

21
21 Example conclusion: MAC-CBJ vs FC FC visits p MAC-CBJ visits p p is consistent and parent(p) strongly 1-consistent p is consistent and parent(p) strongly 2-consistent

22
22 Partial order for nodes visited MC k+1 -CBJMC k -CBJMC 1 -CBJMC n -CBJMC k+1 MC k MC 1 MC n BJ k+1 BJ k BJ 1 BJ n BT.................................... less than or equal

23
23 Partial order for nodes visited MC k+1 -CBJMC k -CBJMC 1 -CBJMC n -CBJMC k+1 MC k MC 1 MC n BJ k+1 BJ k BJ 1 BJ n BT.................................... incomparable

24
24 Partial order for nodes visited BT MC k+1 -CBJ MC k -CBJ MC 1 -CBJ MC n -CBJ MC k+1 MC k MC 1 MC n BJ k+1 BJ k BJ 1 BJ n.................................... incomparable

25
25 Partial order for nodes visited MC k+1 -CBJ MC k -CBJ MC 1 -CBJ MC n -CBJ MC k+1 MC k MC 1 MC n BJ k+1 BJ k BJ 1 BJ n BT.................................... incomparable

26
26 Partial order for nodes visited MC k+1 -CBJMC k -CBJMC 1 -CBJMC n -CBJMC k+1 MC k MC 1 MC n CBJBT........................

27
27 Closer look: nodes visited, binary CSPs MAC-CBJFC-CBJMACFCCBJBJ 2 BJ BT less than or equal

28
28 Closer look: constraint checks, binary CSPs MAC-CBJFC-CBJMACFCCBJ BJ 2 BJ BT less than or equal at most O(f) more O(nd) O(n2d2)O(n2d2)

29
29 Experiments: lookahead FC MAC O(nd) Path BT O(nd)

30
30 Example: FC vs MAC Adapted from J. Ullman (1988); suggested by D. De Haan Odd-even relation variables: x 1, …, x n, n odd domains: {1, 2, 3, 4} x1x1 x2x2 xnxn... FC: 2 n+1 - 4 MAC: 4

31
31 Increasing lookahead Time (sec.) to solve 6 x 6 puzzles MC 4 MC 3 MC 2 MC 1

32
32 Experiments: lookback FC-CBJFC

33
33 Experiments: lookahead vs lookback Increasing the level of consistency decreases effectiveness of CBJ ·Prosser (1993) ·Bacchus & van Run (1995) ·Bessiere & Regin (1996) MAC much better than FC-CBJ MAC better than MAC-CBJ MAC-CBJMAC FC-CBJFC O(nd) CBJBT O(nd) CBJ becomes useless ·Jussien, Debruyne, Boizumault (2000) MAC-DBT much better than MAC (binary) ·Chen & van Beek (2001) MGAC-CBJ much better than MAC (non-binary)

34
34 Theory: lookahead vs lookback MC k+1 -CBJ MC k -CBJ MC k+1 MC k BJ k+1 BJ k BJ k+1 better only if there exists a (k+1)-level backjump that is not a chronological backtrack Number of (k+1)-level backjumps number of k-level backjumps Therefore, as k increases, effectiveness of backjumping decreases

35
35 Part II Comparing CSP models

36
36 Conversion to binary Any non-binary CSP can be converted into one with only binary constraints ·dual graph method (Dechter & Pearl, 1989; Rossi et al. 1989) ·hidden variable method (Peirce, 1933; Rossi et al. 1989; Dechter 1990)

37
37 Crossword puzzles 123 6 4 7 5 8 10 9 20 11 22 12 21 13 17 14 181615 2319 a aardvark aback abacus abaft abalone abandon... Mona Lisa monarch monarchy monarda... zymurgy zyrian zythum

38
38 Non-binary model (original) 123 6 4 7 5 8 10 9 20 11 22 12 21 13 17 14 181615 2319 variables: one for each unknown letter (cell) domains: a, …, z constraints: contiguous letters must form words in dictionary

39
39 Dual model (binary) 123 6 4 7 5 8 10 9 20 11 22 12 21 13 17 14 181615 2319 variables: one for each unknown word across and down domains: words from dictionary constraints intersecting words must agree on common letter

40
40 Hidden model (binary) 123 6 4 7 5 8 10 9 20 11 22 12 21 13 17 14 181615 2319 variables: one for each unknown letter (cell) and one for each unknown word across and down domains: letters, words constraints: letter and word variable must agree

41
41 Comparing models One proposal: for all backtracking algorithms for all variable orderings model m 1 model m 2 Objections: ·effectiveness of a model is algorithm dependent ·models may contain different variables Desirable features: ·meets objections ·still holds if use best possible variable ordering for m 2 Another proposal: given backtrack algorithm for all v 2 for m 2 there exists a v 1 for m 1 s. t. model m 1 model m 2

42
42 Theoretical methodology Bound difference in number of nodes visited ·compare pruning power of local consistencies of alternative models ( e.g., Debruyne & Bessiere, 1997; Stergiou & Walsh, 1999; Prosser et al., 2000; Walsh, 2000) ·establish correspondence between nodes Bound difference in cost at each node: ·local consistency ·variable ordering

43
43 Some relationships MGAC-dual MGAC-hidden MGAC-orig FC-dual FC-hidden FC-orig BT-dual BT-hidden BT-orig polynomialnot polynomial

44
44 Part III Comparing variable ordering heuristics Future work

45
45 Conclusion Technical: ·comparing backtracking algorithms partial order ·comparing CSP models bounded worse relation on algorithm-model combinations Big picture: ·theory can guide, inform experimentation

Similar presentations

OK

1 Click here to End Presentation Software: Installation and Updates Internet Download CD release NACIS Updates.

1 Click here to End Presentation Software: Installation and Updates Internet Download CD release NACIS Updates.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on pelton turbine Ppt on central ac plant Ppt on quality education academy Ppt on chemical bonding and structures Ppt on jawaharlal nehru in hindi language Ppt on abstract art for kids Ppt on machine translation post Raster scan display and random scan display ppt online Ppt on leadership styles theories Ppt on crime file system