Worst-Case Polylog Incremental SPQR-trees: Embeddings, Planarity, and Triconnectivity

TL;DR

This paper introduces a worst-case polylogarithmic bound on the number of local changes needed to update planar graph embeddings after edge modifications, and develops efficient data structures for incremental planarity and connectivity testing.

Contribution

It presents a novel approach using pre-split BC and SPQR trees with biased heavy-path decomposition to achieve efficient updates in planar graph embeddings.

Findings

Achieves $O("log n)

local changes for embedding updates after edge modifications

Provides deterministic $O("log^3 n)

Abstract

We show that every labelled planar graph $G$ can be assigned a canonical embedding $\phi(G)$, such that for any planar $G'$ that differs from $G$ by the insertion or deletion of one edge, the number of local changes to the combinatorial embedding needed to get from $\phi(G)$ to $\phi(G')$ is $O(\log n)$. In contrast, there exist embedded graphs where $\Omega(n)$ changes are necessary to accommodate one inserted edge. We provide a matching lower bound of $\Omega(\log n)$ local changes, and although our upper bound is worst-case, our lower bound hold in the amortized case as well. Our proof is based on BC trees and SPQR trees, and we develop \emph{pre-split} variants of these for general graphs, based on a novel biased heavy-path decomposition, where the structural changes corresponding to edge insertions and deletions in the underlying graph consist of at most $O(\log n)$ basic operations of a particularly simple form. As a secondary result, we show how to maintain the pre-split trees under edge insertions in the underlying graph deterministically in worst case $O(\log^3 n)$ time. Using this, we obtain deterministic data structures for incremental planarity testing, incremental planar embedding, and incremental triconnectivity, that each have worst case $O(\log^3 n)$ update and query time, answering an open question by La Poutr\'e and Westbrook from 1998.