Results 1 
5 of
5
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 ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
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:
On approximate distance labels and routing schemes with affine stretch
 IN INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING (DISC
, 2011
"... For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
For every integral parameter k> 1, given an unweighted graph G, we construct in polynomial time, for each vertex u, adistance label L(u) of size Õ(n2/(2k−1)). For any u, v ∈ G, givenL(u),L(v) we can return in time O(k) an affine approximation ˆ d(u, v) on the distance d(u, v) between u and v in G such that d(u, v) � ˆ d(u, v) � (2k − 2)d(u, v) +1. Hence we say that our distance label scheme has affine stretch of (2k − 2)d +1.Fork=2our construction is comparable to the O(n 5/3) size, 2d +1 affine stretch of the distance oracle of Pǎtraşcu and Roditty (FOCS ’10), it incurs a o(log n) storage overhead while providing the benefits of a distance label. For any k>1, givena restriction of o(n 1+1/(k−1) ) on the total size of the data structure, our construction provides distance labels with affine stretch of (2k − 2)d +1 which is better than the stretch (2k − 1)d scheme of Thorup and Zwick (J. ACM ’05). Our second contribution is a compact routing scheme with polylogarithmic addresses that provides affine stretch guarantees. With Õ(n 3/(3k−2))bit routing tables we obtain affine stretch of (4k − 6)d +1, for any k>1. Given a restriction of o(n 1/(k−1) ) on the table size, our routing scheme provides affine stretch which is better than the stretch (4k − 5)d routing scheme of Thorup and Zwick (SPAA ’01).
NodeDisjoint Multipath Spanners and their Relationship with FaultTolerant Spanners
, 2011
"... Motivated by multipath routing, we introduce a multiconnected variant of spanners. For that purpose we introduce the pmultipath cost between two nodes u and v as the minimum weight of a collection of p internally vertexdisjoint paths between u and v. Given a weighted graph G, a subgraph H is a p ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Motivated by multipath routing, we introduce a multiconnected variant of spanners. For that purpose we introduce the pmultipath cost between two nodes u and v as the minimum weight of a collection of p internally vertexdisjoint paths between u and v. Given a weighted graph G, a subgraph H is a pmultipath sspanner if for all u, v, the pmultipath cost between u and v in H is at most s times the pmultipath cost in G. The s factor is called the stretch. Building upon recent results on faulttolerant spanners, we show how to build pmultipath spanners of constant stretch and of Õ(n1+1/k) edges 1, for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds. Additionally, we give an improved construction for the case p = k = 2. Our spanner H has O(n 3/2) edges and the pmultipath cost in H between any two node is at most twice the corresponding one in G plus O(W), W being the maximum edge weight.
Compact routing schemes with improved stretch
 In Panagiota Fatourou and Gadi Taubenfeld, editors, PODC
, 2013
"... ABSTRACT We consider the problem of compact routing in weighted general undirected graphs, in which the goal is to construct local routing tables that allow information to be sent on short paths in the network. In this paper the first improvement to the work of Thorup and Zwick [SPAA'01] is pr ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT We consider the problem of compact routing in weighted general undirected graphs, in which the goal is to construct local routing tables that allow information to be sent on short paths in the network. In this paper the first improvement to the work of Thorup and Zwick [SPAA'01] is presented. Specifically, we construct an improved routing scheme obtaining for every k routing tables of sizẽ O n 1/k log D , and stretch (4 − α)k − β for some absolute constants α, β > 0, where D is the normalized diameter. This provides a positive answer to a main open question in this area as to the existence of a routing scheme with stretch c · k for some constant c < 4.