[Paper Review] The Shift-Match Number and String Matching Probabilities for Binary Sequences

Arman Bilge, Ayşe Erzan+1 more|ArXiv.org|Sep 21, 2004

Algorithms and Data Compression3 citations

TL;DR

This paper introduces the 'shift-match number'—a novel invariant for binary strings that classifies strings based on their subsequence matching probabilities in longer binary sequences. It proves that matching probabilities depend not only on string length but also on this shift-match number, enabling recursive computation of occurrence probabilities and revealing a spectral structure in string degree distributions within network models of genomic sequences.

ABSTRACT

We define the ``shift-match number'' for a binary string and we compute the probability of occurrence of a given string as a subsequence in longer strings in terms of its shift-match number. We thus prove that the string matching probabilities depend not only on the length of shorter strings, but also on the equivalence class of the shorter string determined by its shift-match number.

Motivation & Objective

• To identify a structural invariant in binary strings that determines their subsequence matching probabilities in longer sequences.
• To resolve the discrepancy in matching probabilities for strings of equal length, which vary due to internal structure.
• To develop a recursive framework for computing the probability of a string's occurrence as a subsequence in longer binary strings.
• To apply the shift-match number to compute degree distributions in network models of genomic sequences, where nodes represent strings and edges represent subsequence relationships.

Proposed method

• Define the shift-match number s(a) as an n-digit binary number derived from pairwise comparisons of adjacent bits in a string a, using a delta function to detect transitions.
• Establish an equivalence relation ≅ on binary strings based on identical shift-match numbers, showing that strings in the same class have identical subsequence occurrence probabilities.
• Derive a recursive formula for N(a,L), the number of length-L binary strings containing a as a subsequence, in terms of s(a) and L, using conditional counts N(a,L,m) for partial matches.
• Use the shift-match number to compute the expected degree distribution d(s,L) in a network of strings, where degree is the number of longer strings containing a given string as a subsequence.
• Apply the framework to compute average degrees and spectral line strengths in genomic network models, with degeneracy determined by the number of strings sharing the same shift-match number.
• Validate the model numerically and analytically, showing agreement with prior simulations and revealing a discrete spectral structure in degree distributions.

Experimental results

Research questions

• RQ1 How do subsequence matching probabilities for binary strings of fixed length vary across different string structures, despite equal length?
• RQ2 What structural property of a binary string determines its subsequence occurrence probability in longer random binary strings?
• RQ3 Can the variation in matching probabilities be explained by a single invariant, and if so, what is its mathematical form?
• RQ4 How does the shift-match number relate to the degree distribution in network models of genomic sequences where edges represent subsequence relationships?
• RQ5 To what extent do the spectral lines in the degree distribution correspond to distinct shift-match numbers, and how are their strengths determined?

Key findings

• Strings of equal length but different internal structure exhibit distinct subsequence matching probabilities, which are fully explained by their shift-match number.
• The shift-match number s(a) is an n-digit binary number derived from the transition structure of a, and it classifies strings into equivalence classes with identical matching probabilities.
• For L ≥ 7, the number of equivalence classes stabilizes, and the probability P(a,L) depends only on s(a), not on the specific string a.
• The expected degree of a string in a network model is determined solely by its shift-match number s, and decreases monotonically with increasing s.
• The degree distribution splits into discrete spectral lines corresponding to distinct shift-match numbers, with line strength ν(d(s)) proportional to the number of strings sharing the same s.
• The recursive formula for N(a,L) in terms of s(a) and L enables exact computation of matching probabilities, resolving prior numerical observations of structural dependence.