Reachability and Distances under Multiple Changes

TL;DR

This paper extends dynamic graph algorithms to handle multiple edge changes simultaneously, demonstrating that reachability and distance queries can be maintained efficiently under certain conditions in the dynamic descriptive complexity framework.

Contribution

It generalizes previous work by allowing multiple edge updates and shows how reachability and distance can be maintained in DynFO with additional quantifiers.

Findings

Reachability maintained in DynFO$(+, imes)$ for O(log n / log log n) node changes.

With majority quantifiers, reachability and distance maintained for O(log^c n) node changes.

Preliminary results suggest distances are in DynFO.

Abstract

Recently it was shown that the transitive closure of a directed graph can be updated using first-order formulas after insertions and deletions of single edges in the dynamic descriptive complexity framework by Dong, Su, and Topor, and Patnaik and Immerman. In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes. We show that the former problem can be maintained in DynFO$(+, \times)$ under changes affecting O($\frac{\log n}{\log \log n}$) nodes, for graphs with $n$ nodes. If the update formulas may use a majority quantifier then both Reachability and Distance can be maintained under changes that affect O($\log^c n$) nodes, for fixed $c \in \mathbb{N}$. Some preliminary results towards showing that distances are in DynFO are discussed.