A label setting algorithm for solving the Elementary Resource Constrained Shortest Path Problem, using node resources to forbid repetition of nodes on the path, is implemented. A state-space augmenting approach for accelerating run times is considered. Several augmentation strategies are suggested and compared numerically.
We have produced extended series for prudent self-avoiding walks on the square lattice. These are subsets of self-avoiding walks. We conjecture the exact growth constant and critical exponent for the walks, and show that the (anisotropic) generating function is almost certainly not differentiably-finite
We have produced extended series for two-dimensional prudent polygons, based on a transfer matrix algorithm of complexity O(n5), for a series of n-step polygons. For prudent polygons in two dimensions we find the growth constant to be smaller than that for the corresponding walks, and by considering three distinct subclasses of prudent walks and polygons, we find that the growth constant for polygons varies with class, while for walks it does not. We give exact values for the critical exponents γ and α for walks and polygons, respectively. We have extended the definition of prudent walks to three dimensions and produced series expansions, using a back-tracking algorithm, for both walks and polygons. In the three-dimensional case we estimate the growth constant for both walks and polygons and also estimate the usual critical exponents γ, ν and α.
Abstract.We give an expository account of our computational proof that every position of Rubik's Cube can be solved in 20 moves or less, where a move is defined as any twist of any face. The roughly 4.3 × 10 19 positions are partitioned into about two billion cosets of a specially chosen subgroup, and the count of cosets required to be treated is reduced by considering symmetry. The reduced space is searched with a program capable of solving one billion positions per second, using about one billion seconds of CPU time donated by Google. As a byproduct of determining that the diameter is 20, we also find the exact count of cube positions at distance 15. 1. Introduction. In this paper we give an expository account of our proof that every position of Rubik's Cube 1 can be solved in 20 moves or less, where a move is defined as any twist of any face. This manner of counting moves, known as the half-turn metric (HTM), is by far the most popular move-count metric for the cube and is understood to be the underlying metric when discussions around this problem, also known as finding "God's number" for the puzzle, arise in various online forums devoted to the cube. The problem has been of keen interest ever since the cube appeared on shelves three decades ago. In group theory language, the problem we solve is to determine the diameter, i.e., maximum edge-distance between vertices, of the HTM-associated Cayley graph of the Rubik's Cube group. As summarized in the next section, many researchers have found increasingly tight upper and lower bounds for the HTM diameter of the cube. The present work explains the computational aspects of our proof that it equals 20.While the technological improvements of increased memory capacity and greater CPU power were necessary for this result, it was primarily a series of mathematical and algorithmic improvements that allowed us to settle the question in the current time frame. Our decomposition of the problem, along with certain insights, let us solve it with a program capable of doing about 65 billion group operations per second on a single common desktop CPU, which works out to be about 23 Rubik's Cube group operations per CPU cycle.The decomposition is based on a subgroup H of the Cube group. Following the section on the problem's history, our paper recommences with a description of this subgroup and its properties, including the notion of an H-wise relabeling of the cube for illustration purposes. We then introduce the notion of canonical sequences,
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