m-Nearly k-Universal Words -- Investigating Simon Congruence

Abstract

Determining the index of the Simon congruence is a long outstanding open problem. Two words u and v are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., oath is a scattered factor of logarithm. Following the idea of scattered factor k-universality, we investigate m-nearly k-universality, i.e., words where m scattered factors of length k are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for m = 1. For m = 1, we show some results if in addition w is (k − 1)universal as well as some further insights for different m.

Combinatorial Algorithms for Subsequence Matching: A Survey

“…For instance, it can be used to directly compare two words w. r. t. their universality index, or serve as a starting point for the analysis of the set of missing subsequences of words. See, for instance, [7,17,22,21,18,39].…”