[PPT] - PREDICTING THE EFFECTIVENESS OF BIDIRECTIONAL HEURISTIC SEARCH PowerPoint Presentation

SLIDE 1

Moving AI Lab

PREDICTING THE EFFECTIVENESS OF BIDIRECTIONAL HEURISTIC SEARCH

Nathan R. Sturtevant, Canada CIFAR AI Chair, Amii, University of Alberta Shahaf Shperberg , Ben-Gurion University Ariel Felner , Ben-Gurion University Jingwei Chen, University of Alberta

SLIDE 2

Q: When does bidirectional (heuristic) search perform well? A: Performance of bidirectional search is positively correlated with the number of states that have heuristics that are both low and inaccurate.

BIDIRECTIONAL SEARCH

Caveat: Not enough time in talk to be completely precise.

SLIDE 3

A strong heuristic expands a majority

f the states in the first half of the

search. BK2: Unidirectional search

utperforms bidirectional search with

a strong heuristic.

BARKER AND KORF (2015)

SLIDE 4

MM is guaranteed to meet in the middle. If |FF| > |RN|, A* will expand more states than MM if the heuristic is weak, fewer if the heuristic is accurate.

HOLTE ET AL (2017)

Start Goal

SLIDE 5

MM is guaranteed to meet in the middle. If |FF| > |RN|, A* will expand more states than MM if the heuristic is weak, fewer if the heuristic is accurate.

HOLTE ET AL (2017)

FF Start Goal

SLIDE 6

MM is guaranteed to meet in the middle. If |FF| > |RN|, A* will expand more states than MM if the heuristic is weak, fewer if the heuristic is accurate.

HOLTE ET AL (2017)

FF RN Start Goal

SLIDE 7

Front-to-end bidirectional search

admissible heuristic
not necessarily consistent

Corresponds to finding a minimum vertex cover on a bipartite graph

ECKERLE ET AL (2017)

SLIDE 8

Necessary expansions for a pair of states: ff(a) < C* fb(b) < C* gf(a) + gb(b) < C*

ECKERLE ET AL (2017)

SLIDE 9

Necessary expansions for a pair of states: ff(a) < C* fb(b) < C* gf(a) + gb(b) < C*

ECKERLE ET AL (2017)

a

SLIDE 10

Necessary expansions for a pair of states: ff(a) < C* fb(b) < C* gf(a) + gb(b) < C*

ECKERLE ET AL (2017)

b a

SLIDE 11

Necessary expansions for a pair of states: ff(a) < C* fb(b) < C* gf(a) + gb(b) < C*

ECKERLE ET AL (2017)

?

b a

SLIDE 12

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13

SLIDE 13

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C*

SLIDE 14

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C*

SLIDE 15

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C*

SLIDE 16

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C*

SLIDE 17

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C*

SLIDE 18

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C*

SLIDE 19

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C* gf(a) + gb(b) < C*

SLIDE 20

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C* gf(a) + gb(b) < C*

SLIDE 21

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C* gf(a) + gb(b) < C*

SLIDE 22

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C* gf(a) + gb(b) < C*

SLIDE 23

EXPLANATION

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13 ff < C* fb < C*

SLIDE 24

EXPLANATION

36,364 36,363 36,369 36,664 41,842 72,956 123,442 148,657 148,757 141,219 138,538 138,324 138,328 138,330 1

1

19

2

342

3

5,391

4

31,924

5

53,120

6

31,819

7

11,686

8

3,159

9

715

10

128

11

23

12

3 1

1

19

2

342

3

3,396

4

10,697

5

11,586

6

6,604

7

2,634

8

810

9

213

10

47

11

13

12

2 gF gB

C* = 13

SLIDE 25

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 26

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 27

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 28

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 29

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 30

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 31

RUBIK’S CUBE 7-TILE PDB

C* = 12

1

18

2

237

3

1,201

4

1,981

5

2,670

6

3,291

7

3,567

8

3,638

9

3,647

10

3,740

11

3,138 1

1

18

2

237

3

1,330

4

2,438

5

3,123

6

3,615

7

3,904

8

4,156

9

4,273

10

4,371

11

3,749 gF gB

SLIDE 32

1

19

2

259

3

839

4

534

5

173

6

31

7

3 1

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2 gF gB

PANCAKE GAP HEURISTIC

C* = 13

SLIDE 33

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 34

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 35

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 36

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 37

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 38

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 39

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 40

PANCAKE GAP HEURISTIC

2,666 2,667 2,686 2,945 3,784 4,316 4,460 4,368 3,965 3,048 2,129 1,879 1,860 1,859 1

1

19

2

259

3

839

4

534

5

173

6

31

7

3

8 9 10 11 12

1

19

2

250

3

919

4

917

5

406

6

123

7

29

8

2

9 10 11 12

gF gB

C* = 13

SLIDE 41

CRITICAL STATES

Sample states near goal and

find heuristic inaccuracies

Goal

SLIDE 42

CRITICAL STATES

Sample states near goal and

find heuristic inaccuracies

Goal

SLIDE 43

CRITICAL STATES

Sample states near goal and

find heuristic inaccuracies

Sample the heuristic to find

how many low heuristic values there are

Goal

SLIDE 44

CRITICAL STATES

Sample states near goal and

find heuristic inaccuracies

Sample the heuristic to find

how many low heuristic values there are

Goal

SLIDE 45

ASYMMETRY

Sample states near goal and

find heuristic inaccuracies

Sample the heuristic to find

how many low heuristic values there are

Look at the asymmetry of the

state space

SLIDE 46

EXAMPLE: TOH WITH PDB HEURISTIC

is close to the goal and has a low heuristic value is far from the goal Low heuristics are inaccurate Bidirectional heuristic search outperforms unidirectional search

ϕ(s) s

5 4 3 7 6 2 1 5 4 7 6

Problem State s PDB Abstraction ϕ(s)

SLIDE 47

EXAMPLE: UNINFORMED SEARCH

Low and inaccurate heuristics for almost all states

SLIDE 48

CONCLUSION

Critical states have both low and inaccurate heuristics Need critical states for bidirectional search to perform well More critical states → bidirectional search will do better

https://webdocs.cs.ualberta.ca/~nathanst/papers/sturtevant2020unibidi.pdf