On representing the degree sequences of sublogarithmic-degree Wheeler graphs

TL;DR

This paper introduces a space-efficient data structure for storing degree sequences of Wheeler graphs with sublogarithmic maximum degree, enabling constant-time pattern matching queries.

Contribution

It presents a novel storage method for degree sequences of Wheeler graphs with sublogarithmic degree, achieving near-optimal space and constant query time.

Findings

Stores degree sequences in n H_k(S) + o(n) bits for k in o(log n / (log log n)^2)

Enables constant-time pattern matching queries on Wheeler graphs with sublogarithmic degree

Provides a theoretical foundation for efficient Wheeler graph representations

Abstract

We show how to store a searchable partial-sums data structure with constant query time for a static sequence $S$ of $n$ positive integers in $o \left( \frac{\log n}{(\log \log n)^2} \right)$, in $n H_k (S) + o (n)$ bits for $k \in o \left( \frac{\log n}{(\log \log n)^2} \right)$. It follows that if a Wheeler graph on $n$ vertices has maximum degree in $o \left( \frac{\log n}{(\log \log n)^2} \right)$, then we can store its in- and out-degree sequences $\Din$ and $\Dout$ in $n H_k (\Din) + o (n)$ and $n H_k (\Dout) + o (n)$ bits, for $k \in o \left( \frac{\log n}{(\log \log n)^2} \right)$, such that querying them for pattern matching in the graph takes constant time.