Recent Progress in Heuristic Search: A Case Study of the Four-Peg Towers of Hanoi Problem

We integrate a number of new and recent advances in heuristic search, and apply them to the fourpeg Towers of Hanoi problem. These include frontier search, disk-based search, parallel processing, multiple, compressed, disjoint, and additive pattern database heuristics, and breadth-first heuristic search. New ideas include pattern database heuristics based on multiple goal states, a method to reduce coordination among multiple parallel threads, and a method for reducing the number of heuristic calculations. We perform the first complete breadth-first searches of the 21 and 22-disc fourpeg Towers of Hanoi problems, and extend the verification of “presumed optimal solutions” to this problem from 24 to 30 discs. Verification of the 31-disc problem is in progress. 1 Towers of Hanoi Problems The standard Towers of Hanoi problem consists of three pegs, and n different sized discs, initially stacked in decreasing order of size on one peg. The task is to transfer all the discs from the initial peg to a goal peg, by only moving one disc at any time, and never placing a larger disc on top of a smaller disc. To move the largest disc from the initial peg to the goal peg, none of the smaller discs can be on either peg, but must all be on the remaining auxiliary peg. This generates a subproblem of the same form as the original problem, allowing a simple recursive solution. It is easily proven that the shortest solution to this problem requires 2n − 1 moves. The problem becomes more interesting if we add another peg (see Figure 1). While the four-peg Towers of Hanoi problem is 117 years old [Hinz, 1997], the optimal solution length is not known in general. The difficulty is that moving the largest disc from the initial to the goal peg requires that the remaining discs be distributed over the two auxiliary pegs, but we don’t know a priori how to distribute them in an optimal solution. In 1941, a recursive strategy was proposed that constructs a valid solution [Frame, 1941; Stewart, 1941], and optimality proofs of this “presumed optimal solution” were offered, but they contained an unproven An auxiliary peg is any peg other than the initial or goal peg. assumption [Dunkel, 1941], and the conjecture remains unproven. Absent a proof, the only way to verify optimality of this solution for a given number of discs is by a systematic search for shorter solutions. Previously this had been done for up to 24 discs [Korf, 2004], and we extend it here to 30 discs. The size of the problem space is 4n, where n is the number of discs, since each disc can be on any of four pegs, and the discs on any peg must be in sorted order. Thus, each new disc multiplies the size of the problem space by four. Figure 1: Four-Peg Towers of Hanoi Problem 2 Prior Work on Four-Peg Towers of Hanoi The simplest search is a brute-force search, with no heuristics. Complete breadth-first searches, generating all states from a given initial state, have previously been done for up to 20 discs [Korf, 2004]. There are two different symmetries that can be used to speed up this search. The first is that transferring the discs from an initial peg to any of the other three pegs are equivalent problems. Thus, given any state, we sort the three noninitial pegs in order of their largest disc, reducing the number of states by almost a factor of six [Bode & Hinz, 1999]. The reduction is slightly less than six because if two non-initial pegs are empty, permuting them has no effect. The other is a symmetry of the solution path [Bode & Hinz, 1999]. Assume that we find a path to a middle state, where all discs but the largest are distributed over the two auxiliary pegs. We can then move the largest disc from the initial to the goal peg. If we then execute the moves made to reach the middle state in reverse order, we will return all but the largest disc to the initial peg. If, however, while executing these moves, we interchange the initial and goal pegs, we will