Coloring triangle-free rectangular frame intersection graphs with O(log log n) colors

Recently, Pawlik et al. have shown that triangle-free intersection graphs of line segments in the plane can have arbitrarily large chromatic number. Specifically, they construct triangle-free segment intersection graphs with chromatic number Θ(log log n). Essentially the same construction produces Θ(loglogn)-chromatic triangle-free intersection graphs of a variety of other geometric shapes-those belonging to any class of compact arc-connected subsets of ℝ2 closed under horizontal scaling, vertical scaling, and translation, except for axis-aligned rectangles. We show that this construction is asymptotically optimal for the class of rectangular frames (boundaries of axis-aligned rectangles). Namely, we prove that triangle-free intersection graphs of rectangular frames in the plane have chromatic number O(loglogn), improving on the previous bound of O(logn). To this end, we exploit a relationship between off-line coloring of rectangular frame intersection graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem, and colors these subgraphs with O(loglogn) colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition). © 2013 Springer-Verlag

Scalable Query Processing on Spatial Networks

Spatial networks (e.g., road networks) are general graphs with spatial information (e.g., latitude/longitude) information associated with the vertices and/or the edges of the graph. Techniques are presented for query processing on spatial networks that are based on the observed coherence between the spatial positions of the vertices and the shortest paths between them. This facilitates aggregation of the vertices into coherent regions that share vertices on the shortest paths between them. Using this observation, a framework, termed SILC, is introduced that precomputes and compactly encodes the N^2 shortest path and network distances between every pair of vertices on a spatial network containing N vertices. The compactness of the shortest paths from source vertex V is achieved by partitioning the destination vertices into subsets based on the identity of the first edge to them from V. The spatial coherence of these subsets is captured by using a quadtree representation whose dimension-reducing property enables the storage requirements of each subset to be reduced to be proportional to the perimeter of the spatially coherent regions, instead of to the number of vertices in the spatial network. In particular, experiments on a number of large road networks as well as a theoretical analysis have shown that the total storage for the shortest paths has been reduced from O(N^3) to O(N^1.5). In addition to SILC, another framework, termed PCP, is proposed that also takes advantage of the spatial coherence of the source vertices and makes use of the Well Separated Pair decomposition to further reduce the storage, under suitably defined conditions, to O(N). Using these frameworks, scalable algorithms are presented to implement a wide variety of operations such as nearest neighbor finding and distance joins on large datasets of locations residing on a spatial network. These frameworks essentially decouple the process of computing shortest paths from that of spatial query processing as well as also decouple the domain of the participating objects from the domain of the vertices of the spatial network. This means that as long as the spatial network is unchanged, the algorithm and underlying representation of the shortest paths in the spatial network can be used with different sets of objects

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios

Probabilistic methods for distributed information dissemination

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 457-484).The ever-increasing growth of modern networks comes with a paradigm shift in network operation. Networks can no longer be abstracted as deterministic, centrally controlled systems with static topologies but need to be understood as highly distributed, dynamic systems with inherent unreliabilities. This makes many communication, coordination and computation tasks challenging and in many scenarios communication becomes a crucial bottleneck. In this thesis, we develop new algorithms and techniques to address these challenges. In particular we concentrate on broadcast and information dissemination tasks and introduce novel ideas on how randomization can lead to powerful, simple and practical communication primitives suitable for these modern networks. In this endeavor we combine and further develop tools from different disciplines trying to simultaneously addresses the distributed, information theoretic and algorithmic aspects of network communication. The two main probabilistic techniques developed to disseminate information in a network are gossip and random linear network coding. Gossip is an alternative to classical flooding approaches: Instead of nodes repeatedly forwarding information to all their neighbors, gossiping nodes forward information only to a small number of (random) neighbors. We show that, when done right, gossip disperses information almost as quickly as flooding, albeit with a drastically reduced communication overhead. Random linear network coding (RLNC) applies when a large amount of information or many messages are to be disseminated. Instead of routing messages through intermediate nodes, that is, following a classical store-and-forward approach, RLNC mixes messages together by forwarding random linear combinations of messages. The simplicity and topology-obliviousness of this approach makes RLNC particularly interesting for the distributed settings considered in this thesis. Unfortunately the performance of RLNC was not well understood even for the simplest such settings. We introduce a simple yet powerful analysis technique that allows us to prove optimal performance guarantees for all settings considered in the literature and many more that were not analyzable so far. Specifically, we give many new results for RLNC gossip algorithms, RLNC algorithms for dynamic networks, and RLNC with correlated data. We also provide a novel highly efficient distributed implementation of RLNC that achieves these performance guarantees while buffering only a minimal amount of information at intermediate nodes. We then apply our techniques to improve communication primitives in multi-hop radio networks. While radio networks inherently support broadcast communications, e.g., from one node to all surrounding nodes, interference of simultaneous transmissions makes multihop broadcast communication an interesting challenge. We show that, again, randomization holds the key for obtaining simple, efficient and distributed information dissemination protocols. In particular, using random back-off strategies to coordinate access to the shared medium leads to optimal gossip-like communications and applying RLNC achieves the first throughput-optimal multi-message communication primitives. Lastly we apply our probabilistic approach for analyzing simple, distributed propagation protocols in a broader context by studying algorithms for the Lovász Local Lemma. These algorithms find solutions to certain local constraint satisfaction problems by randomly fixing and propagating violations locally. Our two main results show that, firstly, there are also efficient deterministic propagation strategies achieving the same and, secondly, using the random fixing strategy has the advantage of producing not just an arbitrary solution but an approximately uniformly random one. Both results lead to simple, constructions for a many locally consistent structures of interest that were not known to be efficiently constructable before.by Bernhard Haeupler.Ph.D

Computer science: the hardware software and heart of IT

1st edition, 201