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14
Sublogarithmic Distributed MIS Algorithm for Sparse Graphs using NashWilliams Decomposition
 In Journal of Distributed Computing Special Issue of selected papers from PODC
, 2008
"... We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on gr ..."
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We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graphtheoretic structure, called NashWilliams forestsdecomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearlytight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.
Small InducedUniversal Graphs and Compact Implicit Graph Representations
 In Proc. 43’rd annual IEEE Symp. on Foundations of Computer Science
, 2002
"... We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound ..."
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Cited by 18 (1 self)
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We show that there exists a graph G with n 2 nodes, where any forest with n nodes is a nodeinduced subgraph of G. Furthermore, the result implies existence of a graph with n nodes that contains all nnode graphs of fixed arboricity k as nodeinduced subgraphs. We provide a lower bound of the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 17 (3 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Lineartime compression of boundedgenus graphs into informationtheoretically optimal number of bits
 In: 13th Symposium on Discrete Algorithms (SODA
, 2002
"... 1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer ..."
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Cited by 17 (1 self)
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1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer Graphics [12, 1720] and compact routing table of Computer Networks [7}. For brevity, let a ~rgraph stand for a graph with property n. The informationtheoretically optimal number of bits required to represent an nnode ngraph is [log 2 N~(n)], where N,~(n) is the number of distinct nnode *rgraphs. Although determining or approximating the close forms of N ~ (n) for nontrivial classes of n is challenging, we provide a lineartime methodology for graph compression schemes that are informationtheoretically optimal with respect to continuous uperadditive functions (abbreviated as optimal for the rest of the extended abstract). 1 Specifically, if 7r satisfies certain properties, then we can compress any nnode medge 1rgraph G into a binary string Z such that G and Z can be computed from each other in O(m + n) time, and that the bit count of Z is at most fl(n) + o(fl(n)) for any continuous uperadditive function fl(n) with log 2 N~(n) < fl(n) + o(fl(n)). Our methodology is applicable to general classes of graphs; this extended abstract focuses on graphs with sublinear genus. 2 For example, if the input nnode,rgraph G is equipped with an embedding on its genus surface, which is a reasonable assumption for graphs arising from 3D model compression, then our methodology is applicable to any 7r satisfying the following statements:
Dynamic Representations of Sparse Graphs
 In Proc. 6th International Workshop on Algorithms and Data Structures (WADS
, 1999
"... We present a linear space data structure for maintaining graphs with bounded arboricity  a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth  under edge insertions, edge deletions, and adjacency queries. The data structure supports adjacency queries in wors ..."
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We present a linear space data structure for maintaining graphs with bounded arboricity  a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth  under edge insertions, edge deletions, and adjacency queries. The data structure supports adjacency queries in worst case O(c) time, and edge insertions and edge deletions in amortized O(1) and O(c+log n) time, respectively, where n is the number of nodes in the graph, and c is the bound on the arboricity.
Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures
"... Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outde ..."
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Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outdegree at most ⌈(1 + ε)d ∗ ⌉, where d ∗ is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is ⌈d ∗ ⌉. Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2approximation algorithms by Aichholzer et al. [1] (for orientation / pseudoarboricity), by Arikati et al. [3] (for arboricity) and by Charikar [5] (for maximum density). 1
Orienting Fully Dynamic Graphs with WorstCase Time Bounds ⋆
"... Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all outdegrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a ..."
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Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all outdegrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low outdegree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph’s arboricity, which is very natural in this context. Their solution achieves a constant outdegree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excludedminor graphs. It remained an open question – first proposed by Brodal and Fagerberg, later by Erickson and others – to obtain similar bounds with worstcase update time. We address this 15 year old question by providing a simple algorithm with worstcase bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combinatorial invariant, and achieves a logarithmic outdegree with logarithmic worstcase update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worstcase update time compared to a similar amortized update time of Neiman and Solomon (2013). 1