
A Lower Bound for Dynamic Fractional Cascading
We investigate the limits of one of the fundamental ideas in data struct...
read it

Lower Bounds for Planar Electrical Reduction
We improve our earlier lower bounds on the number of electrical transfor...
read it

Fullydynamic Planarity Testing in Polylogarithmic Time
Given a dynamic graph subject to insertions and deletions of edges, a na...
read it

A universal predictorcorrector type incremental algorithm for the construction of weighted straight skeletons based on the notion of deforming polygon
A new predictorcorrector type incremental algorithm is proposed for the...
read it

The Power of Vertex Sparsifiers in Dynamic Graph Algorithms
We introduce a new algorithmic framework for designing dynamic graph alg...
read it

Universal Slope Sets for Upward Planar Drawings
We prove that every set S of Δ slopes containing the horizontal slope i...
read it

Planar Cycle Covering Graphs
We describe a new variational lowerbound on the minimum energy configur...
read it
WorstCase Polylog Incremental SPQRtrees: Embeddings, Planarity, and Triconnectivity
We show that every labelled planar graph G can be assigned a canonical embedding ϕ(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 ϕ(G) to ϕ(G') is O(log n). In contrast, there exist embedded graphs where Ω(n) changes are necessary to accommodate one inserted edge. We provide a matching lower bound of Ω(log n) local changes, and although our upper bound is worstcase, our lower bound hold in the amortized case as well. Our proof is based on BC trees and SPQR trees, and we develop presplit variants of these for general graphs, based on a novel biased heavypath 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 presplit 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 Poutré and Westbrook from 1998.
READ FULL TEXT
Comments
There are no comments yet.