Algorithms for translational tiling

Algorithms for Translational Tiling

In this paper we study algorithms for tiling problems. We show that the conditions (T1) and (T2) of Coven and Meyerowitz [5], conjectured to be necessary and sufficient for a finite set A to tile the integers, can be checked in time polynomial in diam(A). We also give heuristic algorithms to find all non-periodic tilings of a cyclic group ZN . In particular we carry out a full classification of...

Simultaneous Translational and Multiplicative Tiling

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers a matrix. In this paper we consider the following simultaneous tiling problem: Given a...

Approximation algorithms for MAX-MIN tiling

The MAX-MIN tiling problem is as follows. We are given A[1, . . . , n][1, . . . , n], a two-dimensional array where each entry A[i][j ] stores a non-negative number. Define a tile of A to be a subarray A[ , . . . , r][t, . . . , b] of A, the weight of a tile to be the sum of all array entries in it, and a tiling of A to be a collection of tiles of A such that each entry A[i][j ] is contained in...

A Comparison of Compiler Tiling Algorithms

Linear algebra codes contain data locality which can be exploited by tiling multiple loop nests. Several approaches to tiling have been suggested for avoiding connict misses in low associativity caches. We propose a new technique based on intra-variable padding and compare its performance with existing techniques. Results show padding improves performance of matrix multiply by over 100% in some...

The Study of Translational Tiling with Fourier Analysis