On the Use of Suffix Arrays for Memory-Efficient Lempel-Ziv Data Compression

Much research has been devoted to optimizing algorithms of the Lempel-Ziv (LZ) 77 family, both in terms of speed and memory requirements. Binary search trees and suffix trees (ST) are data structures that have been often used for this purpose, as they allow fast searches at the expense of memory usage. In recent years, there has been interest on suffix arrays (SA), due to their simplicity and low memory requirements. One key issue is that an SA can solve the sub-string problem almost as efficiently as an ST, using less memory. This paper proposes two new SA-based algorithms for LZ encoding, which require no modifications on the decoder side. Experimental results on standard benchmarks show that our algorithms, though not faster, use 3 to 5 times less memory than the ST counterparts. Another important feature of our SA-based algorithms is that the amount of memory is independent of the text to search, thus the memory that has to be allocated can be defined a priori. These features of low and predictable memory requirements are of the utmost importance in several scenarios, such as embedded systems, where memory is at a premium and speed is not critical. Finally, we point out that the new algorithms are general, in the sense that they are adequate for applications other than LZ compression, such as text retrieval and forward/backward sub-string search.Comment: 10 pages, submited to IEEE - Data Compression Conference 200

The positive semidefinite Grothendieck problem with rank constraint

Given a positive integer n and a positive semidefinite matrix A = (A_{ij}) of size m x m, the positive semidefinite Grothendieck problem with rank-n-constraint (SDP_n) is maximize \sum_{i=1}^m \sum_{j=1}^m A_{ij} x_i \cdot x_j, where x_1, ..., x_m \in S^{n-1}. In this paper we design a polynomial time approximation algorithm for SDP_n achieving an approximation ratio of \gamma(n) = \frac{2}{n}(\frac{\Gamma((n+1)/2)}{\Gamma(n/2)})^2 = 1 - \Theta(1/n). We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial time algorithm which approximates SDP_n with a ratio greater than \gamma(n). We improve the approximation ratio of the best known polynomial time algorithm for SDP_1 from 2/\pi to 2/(\pi\gamma(m)) = 2/\pi + \Theta(1/m), and we show a tighter approximation ratio for SDP_n when A is the Laplacian matrix of a graph with nonnegative edge weights.Comment: (v3) to appear in Proceedings of the 37th International Colloquium on Automata, Languages and Programming, 12 page

Grothendieck inequalities for semidefinite programs with rank constraint

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.Comment: 22 page