A Space-Efficient Backtrack-Free Representation for Constraint Satisfaction Problems

Abstract

In this paper we present a radical approach to obtaining a backtrack-free representation for a constraint satisfaction problem: remove values that lead to dead-ends. This technique does not require additional space but has the drawback of removing solutions. We investigate a number of variations on the basic algorithm including the use of seed solutions, consistency techniques, and a variety of pruning heuristics. Our experimental results indicate that a significant proportion of the solutions to the original problem can be retained especially when an optimization algorithm that specifically searches for such "good" backtrack-free representations is employed. Further extensions increase solution retention by searching for high-coverage backtrack-free representations, by removing tuples rather than values, and by combining multiple backtrack-free representations. Our approach elucidates, for the first time, a three-way trade-off between space complexity, potential backtracks, and solution loss and enables algorithms that can actively reason about the trade-off between space, backtracks, and solution loss.

Backtracking-free Interactive Music Scores with Temporal Relations over Rythms

“…We believe that using space-efficient backtrack-free representation of constraint satisfaction problems [6] is appropriate for real-time interaction. This approach has never been used in real applications as far as we know altough the results with random generated CSPs are quite promising.…”

“…In this paper we propose an extension of the 1-BFR algorithm proposed in [6] to allow multiple seeds. Since the 1-BFR requires a known variable ordering, we propose a static variable ordering for the CSPs of a subclass of interactive scores.…”

“…It was introduced in [6] an algorithm to find a 1-BFR form of a CSP that guarantees that a known solution of the problem (a seed) will be part of the solutions of the 1-BFR form of the CSP. Its space complexity is polynomial in the number of variable because it only needs to represent the domains of each variable.…”

“…A simple extension to the 1-seeded algorithm into a multi-seeded algorithm. We present a deterministic version of the algorithm, which closer to the implementation than the 1-seeded algorithm presented in [6]. Function MultiSeeded-1-BFR enforces consistency on P n and delegates the rest of the computation to MultiSeeded-1-BFR-sons that recursively reduces a value from the domain from each solution of P i that do not extend PP i .…”

“…In the next section we present the results of executing the seeded and multiseeded algorithms for this problem. Results For non-seeded we got just one solution and preliminary studies in [6] showed that it is slower. For the seeded case: Seed i = 130, namely [0, 0, 1, 3, 5, 7, 8, 10], we got 150/603 solutions.…”