Homomorphic Hashing for Sparse Coefficient Extraction

TL;DR

This paper introduces homomorphic hashing techniques to enhance space efficiency in dynamic programming algorithms, enabling faster solutions for problems like SAT and subset sum by exploiting sparsity in DP tables.

Contribution

It systematically applies homomorphic hash functions to combine DP and coefficient extraction methods, resulting in improved space-efficient algorithms for several combinatorial problems.

Findings

Algorithms run in time proportional to nonzero DP entries

Achieves strict improvement over sparse DP methods

Provides improved algorithms for sparse CNF SAT

Abstract

We study classes of Dynamic Programming (DP) algorithms which, due to their algebraic definitions, are closely related to coefficient extraction methods. DP algorithms can easily be modified to exploit sparseness in the DP table through memorization. Coefficient extraction techniques on the other hand are both space-efficient and parallelisable, but no tools have been available to exploit sparseness. We investigate the systematic use of homomorphic hash functions to combine the best of these methods and obtain improved space-efficient algorithms for problems including LINEAR SAT, SET PARTITION, and SUBSET SUM. Our algorithms run in time proportional to the number of nonzero entries of the last segment of the DP table, which presents a strict improvement over sparse DP. The last property also gives an improved algorithm for CNF SAT with sparse projections.