Dynamically improved bounds bidirectional search

Abstract

This paper presents a bidirectional search algorithm that dynamically improves the bounds during its execution. It has the property that it always terminates on or before the forward search meets the backward search. Computational experiments on the pancake problem, the sliding tile puzzle, and the topspin problem demonstrate that it is capable of solving problems using significantly fewer node expansions than A^⁎ or state-of-the-art bidirectional algorithms such as MM_ϵ and GBFHS.

Introduction

When bidirectional heuristic search (BHS) was first introduced by Pohl [15] in the early 1970s, there was optimism that it could significantly outperform unidirectional search. While there have been a number of success stories for BHS, overall the computational results have been disappointing in comparison with unidirectional search, in particular, in comparison with unidirectional A^⁎. This paper presents a BHS algorithm, Dynamically Improved Bounds Bidirectional Search (DIBBS), that improves upon previous BHS algorithms.

Pohl's algorithm, Bidirectional Heuristic Path Algorithm (BHPA) [15], essentially performs two interleaved A^⁎ searches, one in the forward direction, which we denote as FA^⁎, and one in the backward direction, which we denote as BA^⁎. One shortcoming of BHPA is that some nodes may be expanded twice, once in each direction. Kaindl and Kainz [12] proved that because of this shortcoming, BHPA may need to expand nearly all of the nodes that are expanded by FA^⁎ and by BA^⁎. In such cases, BHPA is inferior to A^⁎. They also proved that if all the f-values (defined in the next section) are distinct, then BHPA must expand at least as many nodes as either FA^⁎ or BA^⁎ expands. We prove, theoretically and demonstrate computationally, that DIBBS is capable of breaking through this theoretical barrier. Holte et al. [10], [11], [19] recently introduced a family of “meet in the middle” algorithms which also are capable of breaking through this theoretical barrier, but their approach is quite different than the approach taken here.

In 1989, Kwa [14] presented BS^⁎, which improved upon BHPA by eliminating the necessity of expanding any nodes in both directions. While this was an important theoretical advance, computational results continued to be disappointing. We prove that DIBBS, like BS^⁎, never expands a node in both directions. In fact, DIBBS goes one step better — it terminates before the first node is eligible to be expanded in both directions. Loosely speaking, DIBBS terminates on or before the two searches meet (defined to mean that a node that has been expanded in one direction is eligible to be expanded in the opposite direction). The “meet in the middle” algorithms by Holte et al. [11] mentioned above, when using a consistent heuristic, also have this property, but again their approach is quite different than the approach taken here.

In the same paper where Kaindl and Kainz [12] demonstrated some of the shortcomings of BHPA, they also laid the foundation for improving BHS algorithms. They developed a BHS algorithm that dynamically improves the bounds provided by the heuristics during the search. In 2004, Auer and Kaindl [1] built two BHS algorithms based on this foundation. The first, Max-BS^⁎, uses the improved bounds as the priority function in one of the search directions, but not the other. It does not use the improved bounds for pruning. The second, BiMax-BS${}_F^{⁎}$, uses the improved bounds for pruning in both directions, but does not use them in the priority function. Our work further builds on the foundation laid by Kaindl and Kainz. First, we show how to refine and improve the dynamic bounds. Second, we show how to seamlessly build the dynamic bound improvement method into a new priority function that permits DIBBS to fully utilize the improved bounds for pruning in both directions and in the priority function.

More recently, Barker and Korf [2] investigated the limitations of front-to-end BHS algorithms. They arrived at two main conclusions. First, “Adding a weak heuristic to a bidirectional bruteforce search cannot prevent it from expanding additional nodes.” Second, “With a strong heuristic, a bidirectional heuristic search expands more nodes than a unidirectional heuristic search.” In addition, they conjectured that the possible contribution of Kaindl and Kainz's BHS algorithm was to avoid generating nodes with f equal to the cost of an optimal path. We demonstrate computationally that DIBBS can overcome their second conclusion and their conjecture.

One might guess that Barker and Korf's pessimistic pronouncements might have stifled further research on BHS algorithms, but it actually had the opposite effect. In their paper, they postulated the existence of a balanced BHS algorithm that never expands nodes deeper than the solution midpoint, even though such an algorithm did not exist at the time. Holte et al. [10] filled the void by developing the MM algorithm that is guaranteed to meet in the middle. Sharon et al. [19] modified MM to create MM_ϵ, which exploits knowledge about the smallest cost of an edge. A detailed analysis of both algorithms is presented in Holte et al. [11]. Computational results presented in these papers established that it is possible for BHS algorithms to expand fewer nodes than either bidirectional bruteforce search or unidirectional heuristic search. See [21] for a recent survey of bidirectional heuristic search.

MM and MM_ϵ spawned a flurry of research, resulting in several new BHS algorithms and a deeper understanding of the minimum number of nodes that must be expanded by a BHS algorithm. Eckerle et al. [7] began investigating the minimum number of nodes that must be expanded. They extended the assumptions from undirectional heuristic search to define Deterministic, Expansion-based, Black Box (DXBB) bidirectional algorithms. They characterized pairs of nodes that must be expanded by any DXBB BHS algorithm. Chen et al. [5] continued this vein of research by proving that the minimum number of node expansions for a DXBB BHS algorithm can be determined by finding a Minimum Vertex Cover (MVC) in an auxiliary bipartite graph called $G_{MX}$. They also developed a BHS algorithm, named Near-Optimal Bidirectional Search (NBS), that makes use of this knowledge. They proved that NBS will never expand more than twice the number of nodes as the size of a MVC of $G_{MX}$ and that no DXBB BHS algorithm can provide a stronger guarantee than this. Shaham et al. [17] constructed an efficient algorithm for finding an MVC of $G_{MX}$. They also created Fractional MM (fMM), which allows flexibility in choosing the meeting point of the forward and backward searches. Shaham et al. [18] extend the theory regarding the minimum number of node expansions to the case where the cost of the smallest edge is known and constructed the Meet at the Threshold (MT) algorithm that controls the meeting point of the forward and backward search via a threshold parameter. Shperberg et al. [20] created the Dynamic Vertex Cover Bidirectional Search (DVCBS) algorithm, which, similar to NBS, uses information about $G_{MX}$ to guide the search, but unlike NBS, does not have a theoretical guarantee about its worst case performance. Barley et al. [3] created an algorithm named Generalized Breadth-First Heuristic Search (GBFHS), which has great flexibility in controlling where the forward and backward searches meet.

Although the DXBB assumptions do not explicitly state that $h_f$ is only used in the forward direction and $h_b$ is only used in the backward direction, the analysis in [7] implicitly assumes this. Furthermore, DIBBS assumes that the heuristics are consistent, whereas DXBB algorithms do not. DIBBS does not satisfy these assumptions, so it is not subject to the theoretical minimum number of node expansions provided by the MVC of $G_{MX}$. In fact, in Sections 5 and 6 it is shown that DIBBS often is capable of solving problems while expanding fewer nodes than in a MVC of $G_{MX}$. It should be emphasized that while DIBBS shares information between the forward and backward searches, it is a front-to-end algorithm, which means that the heuristics are only evaluated from a node to either the start node or the goal node. This is in contrast with front-to-front algorithms, where the heuristics are evaluated from a node to every other node in the frontier of the opposite search direction. Front-to-front evaluations dynamically improve the bounds, but at a very high computational cost. DIBBS dynamically improves the bounds with very little additional computational cost. This paper only addresses front-to-end BHS algorithms.

After this paper was accepted for publication, the authors were made aware of a paper by Sadhukhan [16], in which a similar algorithm was presented. This paper provides more mathematically rigorous proofs of the correctness of the algorithm, a proof that no node is expanded twice, a more careful analysis of which nodes will not be expanded, and more computational testing, including a computational investigation of the minimum number of nodes that must be expanded.