Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Optimal Online Edge Coloring of Planar Graphs with Advice

Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree $\Delta$, it follows from Vizing's Theorem that $O(m\log \Delta)$ bits of advice suffice to achieve optimality, where $m$ is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only $O(m)$ bits of advice are needed to compute an optimal solution online, independently of how large $\Delta$ is. On the other hand, we show that $\Omega (m)$ bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least $\Omega(m\log \Delta)$ bits of advice to achieve optimality.Comment: CIAC 201

Recognizing Planar Laman Graphs

Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}). To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Fast branching algorithm for Cluster Vertex Deletion

In the family of clustering problems, we are given a set of objects (vertices of the graph), together with some observed pairwise similarities (edges). The goal is to identify clusters of similar objects by slightly modifying the graph to obtain a cluster graph (disjoint union of cliques). Hueffner et al. [Theory Comput. Syst. 2010] initiated the parameterized study of Cluster Vertex Deletion, where the allowed modification is vertex deletion, and presented an elegant O(2^k * k^9 + n * m)-time fixed-parameter algorithm, parameterized by the solution size. In our work, we pick up this line of research and present an O(1.9102^k * (n + m))-time branching algorithm

Preprocessing Imprecise Points for Delaunay Triangulation: Simplified and Extended

Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection R of input regions known in advance. Building on recent work by Löffler and Snoeyink, we show how to leverage our knowledge of R for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, e.g., overlapping disks of different sizes and fat regions. Keywords: Delaunay triangulation - Data imprecision - Quadtree