This document summarizes a presentation on a new bidirectional A* search algorithm with shorter post-processing for solving 8-puzzle problems. It introduces bidirectional A* search and balanced heuristics. The new algorithm uses an inequality to reject nodes during search and trim the post-processing phase. Experimental results on the 8-puzzle and 15-puzzle show that the symmetric heuristic outperforms the balanced heuristic, reducing the number of states generated and solving problems faster.

1. CMPUT 657
Heuristic Search
Final Project Presentation
Topic
A new Bi-directional A* search with Shorter Post
Processing on 8/15 puzzle
Presented By
Mohamad Saiful Islam

2. Outline
 Bi – Directional A* Search
 Balanced Algorithm
 New Bi-Directional A*
 Problem Domain
 Experimental Results
 Conclusion / Discussion

3. Bi-Directional A* Search
 Bi-Directional Search [1]
O Two simultaneous search processes starting from two
end points, root and goal
O Pohl [2] applied this to Dijkstra and A*
 Two Phases
O Main Phase: Two searches continue untill a meeting
point is found
O Post Phase: Make sure that path found is truely the
optimal one
 Post phase is needed as the first meeting point does
not gurantee optimal path

4. Bi-Directional A* Search
 Advantage
O A* space complexity two bi-dir search each
O Total much smaller than A*
 Pohl's [2] Conjecture – Bi-Directional A* can miss
the meeitng point, causing number of nodes
searched twice as big as A* [Missile Metaphor]
 Kaindl et. al [3] proved that the conjecture was
wrong, deviced an efficient implementation
Obd
 Obd/2


5. Balanced Heuristics
 Two search processes need two heuristics
O
 If A* reduces to Dijkstra
 Ikeda et. al[4] proved that A* becomes Dijkstra when
distance d is replaced by d' where
O
O So Bi-Directional A* becomes bi-directional Dijkstra
when
O
O Balanced Heuristics
O Bi-Directional A* using balanced heuristic considered to
be most effecient shortest path algorithms [4, 5, 6]
hu, hu
d'u,v=du,vhv−hu
huhu=constant
hu=0

6. New Bi-Directional Algorithm
 Node m scanned on both sides,
 For a node , the path from source to
destination through w is P

 Proved using the propety of Balanced heuristics
 So in post-phase, nodes that are only in the closed list
of opposite side are needed to be examined
 Bi-directional A* with balanced heuristics thus refrains
from examining huge number of nodes in post-
processing phase
L=gmgm
w∉S∪S
P ≥L

7. New Bi-directional Algorithm
 Let u is a node that will be added to the closed list and
the path through this node is P
O If then length of P is not smaller than L
O Then u can be rejected
O Proved using the propety of Balanced heuristics
 Algorithm using this in-equality can be relaxed from
using balanced heuristincs but they maintain the
short post phases of balanced heuristics
 New Algorithm proposed by Pijls at el[7] and
implemented in this project
guF−hu≥L

8. New Bi-directional Algorithm
1:forall v∈V do
2: gv=∞
3:endfor
4: S=∅
5: L=∞
6: gs=0 //sbecomesLABELED
7:boolean cand−found=true
8:while cand−found=true do

9. New Bi-directional Algorithm
9: C={v∣vislabelled∧gvhv−htL} //Cset ofall candidates
10: cand− found= false
11: while C≠∅∧cand− found=false do
12: uo=argmin{gvhv∣v∈C}
13: if uo∉S∧guoF−huo≥L then
14: C=C−{uo} // uo becomesREJECTED
15: else
16: cand−found=true //asuitablecandidateisfound
Trimming
Explicit Condition
Node not in opposite
closed list

10. 17: endif
18: endwhile
19: if cand−found=true then
20: S=S{uo} // uo becomesSCANNED
21: F=guohuo
22: if uo∉S then
23: forall edgesuo,v∈E with vLABELED∨UNREACHED do
24: if gvguoduo ,v then
Nipping = Doubly scanned nodes
can not Expand more new nodes

11. 25: gv=guoduo ,v //vbecomesrelabeled
26: predv=uo
27: L=min{L,gvgv}
28: endif
29: end for
30: end if
31: endif
32:endwhile

12. Problem Domain
 8/15 puzzle problem
O Class of sliding puzzle problem
 8 puzzle
O States, 181,440 unique states
 15 puzzle
O 10,461,394,944,000 unique states
 Most of the puzzle instances are hard to solve using A*
3.1×103

13. Experimental Result
 Heuristics
 Uni-directional
O Number of mis-placed tiles
O Manhattan Distance
O Manhattan Distance + Linear Conflict
 Third one performs the best [8] so used in the
Bi-Directional search as the base heuristic

14. Experimental Result
 Bi-directional
 Symmetric
O
 Balanced
O
hv=v ,t , hv=v ,t =s ,v
u ,v=under estimate of du ,v
hv=
1
2
{v ,t −s ,v}
hv=
1
2
{s ,v−v ,t}

15. Experimental Result
5 10 15 20 25 30 35
0
50000
100000
150000
200000
250000
300000
Number of States Generated
Misplaced Manhattan Balanced Symmetric
Solution Length
#States

16. Experimental Result
5 10 15 20 25 30 35
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
Number of unique states generated
Misplaced
Manhattan
Balanced
Symmetric
Solution Length
#States

17. Experimental Result
Number States Number of Unique States Solution
Length
Misplac-
ed
Manha-
ttan
Balanc
ed
Symme-
tric
Misplaced Manhattan Balanced Symmetric
31189 5167 3557 855 27295 4814 3350 731 24
121498 12012 9323 2670 94075 11115 8676 2255 28
76238 3644 6555 518 63014 3452 6133 492 26
65565 7501 5615 2516 54493 6889 5258 2264 26
65750 9936 5514 1576 54794 9155 5160 1447 26
49548 6370 6024 1547 42021 5897 5632 1418 25
51165 11538 5957 1796 43087 10417 5574 1516 25
67277 8878 5663 1745 55841 8171 5305 1449 26
65715 5667 5400 1039 54647 5320 5066 896 26
49704 3680 5487 935 42520 3448 5139 872 25

18. Experimental Result
Manhattan Symmetric
# States # Unique States # States # Unique States Solution
Length
332474 313769 64963 52145 32
571221 526604 66103 52928 33
432482 407974 27726 22569 34
334703 310299 30438 26108 35
671954 625793 136149 110414 36
211219 199339 16620 12934 37
96297 80029 38
18102 15356 39
59120 48104 40

19. Conclusion
 Bi-Directional Search better than Uni-
Directional Search
 Symmetric Heuristics works better than
Balanced Heuristic
 Optimal Results
 Future work
O New Heuristics (landmark heuristic in Shortest
Path)
O New Domain

20. References
 [1] T.A.J. Nicholson, Finding the shortest route between two points in
a network, Computer Journal 9 (1966) 275–289.
 [2] I. Pohl, Bi-directional search, Machine Intelligence 6 (1971) 124–
140.
 [3] H. Kaindl, G. Kainz, Bidirectional heuristic search reconsidered, Journal
of Artificial Intelligence 38 (1) (1989) 95–109.
 [4] T.K. Ikeda, M. Hsu, H. Inai, S. Nishimura, H. Shimoura, T. Hashimoto,
K.Tenmoku, K. Mitoh, A fast algorithm for finding better routes by AI
search techniques, in: Proceedings Vehicle Navigation and Information
Systems Conference, IEEE, 1994.
 [5] A.V. Goldberg, C. Harrelson, Computing the shortest path: A* search
meets graph theory, in: 16th Annual ACM–SIAM Symposium on Discrete
Algorithms (SODA’05), 2005.
 [6] G.A. Klunder, H.N. Post, The shortest path problem on large scale real
road networks, Networks 48 (4) (2006) 182–194
 [7] W. Pijls, H. Post, A new bidirectional search algorithm with shortened
postprocessing, European Journal of Operational Research 198 (2009) 363–
369

21. References Cont..
 [8] O. Hansson, A. Mayer, and M. Yung, "Criticizing Solutions to Relaxed
Models Yields Powerful Admissible Heuristics," Information Sciences, Vol.
63, Issue 3, pp. 207-227, 1992.