Master's thesis: Permutations With Restricted Movement

TL;DR

This thesis explores restricted permutations with geometric structure, generalizing existing models to graphs, analyzing their entropy, and developing algorithms for pattern counting, with applications in coding and network technologies.

Contribution

It introduces a generalized model of restricted permutations on graphs, establishes a connection with perfect matchings, and provides entropy calculations and efficient counting algorithms.

Findings

Exact entropy computed for specific 2D cases.

Polynomial-time algorithm developed for counting admissible patterns.

Entropy depends only on the size of the restriction set with full affine dimension.

Abstract

We study restricted permutations of sets which have a geometrical structure. The study of restricted permutations is motivated by their application in coding for flash memories, and their relevance in different applications of networking technologies and various channels. We generalize the model of $\mathbb{Z}^d$-permutations with restricted movement suggested by Schmidt and Strasser in 2016, to restricted permutations of graphs, and study the new model in a symbolic dynamical approach. We show a correspondence between restricted permutations and perfect matchings. We use the theory of perfect matchings for investigating several two-dimensional cases, in which we compute the exact entropy and propose a polynomial-time algorithm for counting admissible patterns. We prove that the entropy of $\mathbb{Z}^d$-permutations restricted by a set with full affine dimension depends only on the size of the set. We use this result in order to compute the entropy for a class of two-dimensional cases. We discuss the global and local admissibility of patterns, in the context of restricted $\mathbb{Z}^d$-permutations. Finally, we review the related models of injective and surjective restricted functions.