# Bidirectional Search That Is Guaranteed to Meet in the Middle

AAAI, pp. 3411-3417, 2016.

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Abstract:

We present MM, the first bidirectional heuristic search algorithm whose forward and backward searches are guaranteed to \"meet in the middle\", i.e. never expand a node beyond the solution midpoint. We also present a novel framework for comparing MM, A*, and brute-force search, and identify conditions favoring each algorithm. Finally, we ...More

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Introduction

- A central assumption made by Barker and Korf is that the forward and backward searches comprising the bidirectional search never expand a node whose g-value exceeds C∗/2.
- The authors say that a bidirectional search “meets in the middle” if it has this property.
- This assumption raises a difficulty in applying their theory, because no known Bi-HS algorithm is guaranteed to meet in the middle under all circumstances.
- Papers on “front-to-front”1 Bi-HS typically claim their searches meet

Highlights

**Introduction and overview**

Bidirectional search algorithms interleave two separate searches, a normal search forward from the start state, and a search backward from the goal.

Barker and Korf (2015)’s comparison of unidirectional heuristic search (Uni-HS, e.g. A*), bidirectional heuristic search (Bi-HS), and bidirectional brute-force search (Bi-BS) has two main conclusions: BK1: Uni-HS will expand fewer nodes than bidirectional heuristic search if more than half of the nodes expanded by Uni-HS have g ≤ C∗/2, where C∗ is the optimal solution cost.

BK2: If fewer than half of the nodes expanded by Uni-HS using heuristic h have g ≤ C∗/2, adding h to bidirectional brute-force search will not decrease the number of nodes it expands.

A central assumption made by Barker and Korf is that the forward and backward searches comprising the bidirectional search never expand a node whose g-value exceeds C∗/2- We present a new framework for comparing MM0, unidirectional brute-force search (Uni-BS), MM, and A* that allows a precise characterization of the regions of the state space that will be expanded by one method but not another
- In this paper we introduced MM, the first bidirectional heuristic search algorithm guaranteed to meet in the middle
- (2) A thorough experimental comparison should be done on more domains and with more bidirectional search algorithms

Methods

- The purpose of the experiments is to verify the correctness the general rules (GR1–GR3).
- Since some rules refer to a heuristic’s relative accuracy, the authors used at least two heuristics of different accuracy in each domain.
- The two domains used in the study are the 10-Pancake Puzzle and Rubik’s Cube.
- In both domains all problems are bi-friendly.
- Because GR1–GR3 make predictions about the number of nodes expanded, that is the only quantity the authors measure in the experiments

Conclusion

- In this paper the authors introduced MM, the first Bi-HS algorithm guaranteed to meet in the middle.
- The authors introduced a framework that divides the state-space into disjoint regions and allows a careful analysis of the behavior of the different algorithms in each of the regions.
- The authors studied the various types of algorithms and provided some general rules that were confirmed by the experiments.
- This paper initiated this direction.
- Future work will continue as follows: (1) A deeper analysis on current and new MM variants may further deepen the knowledge in this issue. (2) A thorough experimental comparison should be done on more domains and with more bidirectional search algorithms. (3) Heuristics that are designed for MM, i.e., that only return values larger than C∗/2 are needed

Summary

## Introduction:

A central assumption made by Barker and Korf is that the forward and backward searches comprising the bidirectional search never expand a node whose g-value exceeds C∗/2.- The authors say that a bidirectional search “meets in the middle” if it has this property.
- This assumption raises a difficulty in applying their theory, because no known Bi-HS algorithm is guaranteed to meet in the middle under all circumstances.
- Papers on “front-to-front”1 Bi-HS typically claim their searches meet
## Methods:

The purpose of the experiments is to verify the correctness the general rules (GR1–GR3).- Since some rules refer to a heuristic’s relative accuracy, the authors used at least two heuristics of different accuracy in each domain.
- The two domains used in the study are the 10-Pancake Puzzle and Rubik’s Cube.
- In both domains all problems are bi-friendly.
- Because GR1–GR3 make predictions about the number of nodes expanded, that is the only quantity the authors measure in the experiments
## Conclusion:

In this paper the authors introduced MM, the first Bi-HS algorithm guaranteed to meet in the middle.- The authors introduced a framework that divides the state-space into disjoint regions and allows a careful analysis of the behavior of the different algorithms in each of the regions.
- The authors studied the various types of algorithms and provided some general rules that were confirmed by the experiments.
- This paper initiated this direction.
- Future work will continue as follows: (1) A deeper analysis on current and new MM variants may further deepen the knowledge in this issue. (2) A thorough experimental comparison should be done on more domains and with more bidirectional search algorithms. (3) Heuristics that are designed for MM, i.e., that only return values larger than C∗/2 are needed

- Table1: Table 1
- Table2: Rubik’s Cube results. M=million, B=billion

Funding

- Financial support for this research was in part provided by Canada’s Natural Science and Engineering Research Council (NSERC) and by Israel Science Foundation (ISF) grant #417/13
- This material is based upon work supported by the National Science Foundation under Grant No 1551406

Reference

- Arefin, K. S., and Saha, A. K. 2010. A new approach of iterative deepening bi-directional heuristic front-to-front algorithm (IDBHFFA). International Journal of Electrical and Computer Sciences (IJECS-IJENS) 10(2).
- Auer, A., and Kaindl, H. 2004. A case study of revisiting bestfirst vs. depth-first search. In Proc. 16th European Conference on Artificial Intelligence (ECAI), 141–145.
- Barker, J. K., and Korf, R. E. 2015. Limitations of front-to-end bidirectional heuristic search. In Proc. 29th AAAI Conference on Artificial Intelligence, 1086–1092.
- Davis, H. W.; Pollack, R. B.; and Sudkamp, T. 198Towards a better understanding of bidirectional search. In Proc. National Conference on Artificial Intelligence (AAAI), 68–72.
- de Champeaux, D., and Sint, L. 1977. An improved bidirectional heuristic search algorithm. J. ACM 24(2):177–191.
- de Champeaux, D. 1983. Bidirectional heuristic search again. J. ACM 30(1):22–32.
- Eckerle, J. 1994. An optimal bidirectional search algorithm. In Proc. KI-94: Advances in Artificial Intelligence, 18th Annual German Conference on Artificial Intelligence, 394.
- Helmert, M., and Roger, G. 200How good is almost perfect? In Proc. 23rd AAAI Conference on Artificial Intelligence, 944–949.
- Helmert, M. 2010. Landmark heuristics for the pancake problem. In Proc. 3rd Annual Symposium on Combinatorial Search, (SoCS).
- Holte, R. C., and Burch, N. 2014. Automatic move pruning for single-agent search. AI Communications 27(4):363–383.
- Holte, R. C.; Felner, A.; Sharon, G.; and Sturtevant, N. R. 2015. Bidirectional search that is guaranteed to meet in the middle: Extended Version. Technical Report TR15-01, Computing Science Department, University of Alberta.
- Ikeda, T.; Hsu, M.-Y.; Imai, H.; Nishimura, S.; Shimoura, H.; Hashimoto, T.; Tenmoku, K.; and Mitoh, K. 1994. A fast algorithm for finding better routes by AI search techniques. In Proc. Vehicle Navigation and Information Systems Conference, 291–296.
- Kaindl, H., and Kainz, G. 1997. Bidirectional heuristic search reconsidered. J. Artificial Intelligence Resesearch (JAIR) 7:283– 317.
- Kaindl, H., and Khorsand, A. 1994. Memory-bounded bidirectional search. In Proc. 12th National Conference on Artificial Intelligence (AAAI), 1359–1364.
- Korf, R. E. 1997. Finding optimal solutions to Rubik’s Cube using pattern databases. In Proc. 14th National Conference on Artificial Intelligence (AAAI), 700–705.
- Korf, R. E. 2004. Best-first frontier search with delayed duplicate detection. In Proc. 19th National Conference on Artificial Intelligence (AAAI), 650–657.
- Kwa, J. B. H. 1989. BS*: An admissible bidirectional staged heuristic search algorithm. Artificial Intelligence 38(1):95–109.
- Nicholson, T. A. J. 1966. Finding the shortest route between two points in a network. The Computer Journal 9(3):275–280.
- Nilsson, N. J. 1982. Principles of Artificial Intelligence. Springer.
- Pohl, I. 1969. Bi-directional and heuristic search in path problems. Technical Report 104, Stanford Linear Accelerator Center.
- Politowski, G., and Pohl, I. 1984. D-node retargeting in bidirectional heuristic search. In Proc. National Conference on Artificial Intelligence (AAAI), 274–277.
- Rokicki, T.; Kociemba, H.; Davidson, M.; and Dethridge, J. 2013. The diameter of the Rubik’s Cube group is twenty. SIAM J. Discrete Math. 27(2):1082–1105.
- Sadhukhan, S. K. 2012. A new approach to bidirectional heuristic search using error functions. In Proc. 1st International Conference on Intelligent Infrastructure at the 47th Annual National Convention COMPUTER SOCIETY of INDIA (CSI-2012).

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