Some Results on Greedy Embeddings in Metric Spaces

Moitra, Ankur; Leighton, Tom

doi:10.1109/focs.2008.18

Cited by 31 publications

(18 citation statements)

References 20 publications

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“…They also showed that any planar 3-connected newtork has a greedy embedding in 3 dimensions, for which an embedding algorithm was described in [10]. The Papadimitriou-Ratajczak conjecture was very recently, settled by Moitra and Leighton [27] who present a polynomial-time algorithm for constructing a greedy embedding of a given 3-connected planar graph. In this paper, we study the embedding of a combinatorial UDG where only the connectivity information but no position information of the vertices is given.…”

Section: Related Workmentioning

confidence: 96%

Greedy Routing with Bounded Stretch

2009

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Greedy routing is a novel routing paradigm where messages are always forwarded to the neighbor that is closest to the destination. Our main result is a polynomial-time algorithm that embeds combinatorial unit disk graphs (CUDGs -a CUDG is a UDG without any geometric information) into O(log 2 n)-dimensional space, permitting greedy routing with constant stretch. To the best of our knowledge, this is the first greedy embedding with stretch guarantees for this class of networks. Our main technical contribution involves extracting, in polynomial time, a constant number of isometric and balanced tree separators from a given CUDG. We do this by extending the celebrated Lipton-Tarjan separator theorem for planar graphs to CUDGs. Our techniques extend to other classes of graphs; for example, for general graphs, we obtain an O(log n)-stretch greedy embedding into O(log 2 n)-dimensional space. The greedy embeddings constructed by our algorithm can also be viewed as a constant-stretch compact routing scheme in which each node is assigned an O(log 3 n)-bit label. To the best of our knowledge, this result yields the best known stretch-space trade-off for compact routing on CUDGs. Extensive simulations on random wireless networks indicate that the average routing overhead is about 10%; only few routes have a stretch above 1.5.

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Section: Related Workmentioning

confidence: 96%

Greedy Routing with Bounded Stretch

2009

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“…Our question is somewhat related to a larger question of embedding metrics into low-dimensional Euclidean spaces. Such questions have motivated a large body of work to aid the understanding of metric/graph embeddings (Kuhn et al 2010;Pemmaraju and Pirwani 2007;Krauthgamer and Lee 2007), various graph classes and special metric spaces as they relate to one another and to the complexity of various optimization problems on these special classes (Raghavan and Spinrad 2001;Nieberg et al 2008;Talwar 2004), and applications in wireless networks such as routing (Papadimitriou and Ratajczak 2005;Dhandapani 2008; Moitra and Leighton 2008;Goodrich and Strash 2009) and localization (Bulusu et al 2000;Bruck et al 2005;Aspnes et al 2004;Nagpal et al 2003). One might think that if a feasible realization of these combinatorial objects can be found, then one can (indirectly) apply the shifting strategy on the objects to obtain good approximation algorithms.…”

Section: Introductionmentioning

confidence: 99%

Shifting strategy for geometric graphs without geometry

Pirwani

2011

J Comb Optim

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We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1): 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization-it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete.Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (lowdimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302-1323, 2005 and Chan (J. Algorithms 46(2): [178][179][180][181][182][183][184][185][186][187][188][189] 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302-1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.

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“…In an upward drawing [6,7], every directed path is monotone with respect to the positive y direction. Even more related to the monotone drawings are the greedy drawings [2,14,16]. In a greedy drawing, for any two vertices u, v, there exists a path P uv from u to v such that the Euclidean distance from an intermediate vertex of P uv to the destination v decreases at each step.…”

Section: Introductionmentioning

confidence: 99%

Optimal Monotone Drawings of Trees

He¹,

He²

2017

SIAM J. Discrete Math.

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A monotone drawing of a graph G is a straight-line drawing of G such that, for every pair of vertices u, w in G, there exists a path P uw in G that is monotone in some direction l uw . (Namely, the order of the orthogonal projections of the vertices of P uw on l uw is the same as the order they appear in P uw .)The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is O(n 1.205 ) × O(n 1.205 ). In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most 12n × 12n. The smaller drawing size is achieved by a new simple Path Draw algorithm, and a procedure that carefully assigns primitive vectors to the paths of the input tree T .We also show that there exists a tree T 0 such that any monotone drawing of T 0 must use a grid of size Ω(n) × Ω(n). So the size of our monotone drawing of trees is asymptotically optimal.

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Some Results on Greedy Embeddings in Metric Spaces

Cited by 31 publications

References 20 publications

Greedy Routing with Bounded Stretch

Greedy Routing with Bounded Stretch

Shifting strategy for geometric graphs without geometry

Optimal Monotone Drawings of Trees

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