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ConstantLevel Greedy Triangulations Approximate the MWT Well
, 1995
"... The wellknown greedy triangulation GT #S# of a #nite point set S is obtained by inserting compatible edges in increasing length order, where an edge is compatible if it does not cross previously inserted ones. Exploiting the concept of socalled light edges, we introduce a new way of de#ning ..."
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Cited by 5 (2 self)
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The wellknown greedy triangulation GT #S# of a #nite point set S is obtained by inserting compatible edges in increasing length order, where an edge is compatible if it does not cross previously inserted ones. Exploiting the concept of socalled light edges, we introduce a new way of de#ning GT #S#. The new de#nition does not rely on the length ordering of the edges. It provides a decomposition of GT #S#into levels, and the number of levels allows us to bound the total edge length of GT #S#. In particular, we show jGT #S#j#3#2 k+1 jMWT#S#j, where k is the number of levels and MWT#S# is the minimumweight triangulation of S. This constitutes the #rst nontrivial upper bound on jGT #S#j for general points sets S. 1 Introduction A triangulation of a given set S of n points in the plane is a maximal set of noncrossing line segments #called edges# whichhave both endpoints in S. Besides the Delaunay triangulation and the minimumweight triangulation, the greedy triang...
Oracles for bounded length shortest paths in planar graphs
 ACM Trans. Algorithms
"... We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so ..."
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Cited by 4 (0 self)
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We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed. Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service. Our results can be easily generalized to other wide classes of graphs – for instance we can take any minorclosed family of graphs.
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
An equivalent version of the CaccettaHäggkvist conjecture in an online load balancing problem
 In Proceedings of the 33rd Workshop on Graphs (WG), LNCS
, 2007
"... Abstract. We study the competitive ratio of certain online algorithms for a wellstudied class of load balancing problems. These algorithms are obtained and analyzed according to a method by Crescenzi et al (2004). We show that an exact analysis of their competitive ratio on certain “uniform ” insta ..."
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Abstract. We study the competitive ratio of certain online algorithms for a wellstudied class of load balancing problems. These algorithms are obtained and analyzed according to a method by Crescenzi et al (2004). We show that an exact analysis of their competitive ratio on certain “uniform ” instances would resolve a fundamental conjecture by Caccetta and Häggkvist (1978). The conjecture is that any digraph on n nodes and minimum outdegree d must contain a directed cycle involving at most ⌈n/d ⌉ nodes. Our results are the first relating this conjecture to the competitive analysis of certain algorithms, thus suggesting a new approach to the conjecture itself. We also prove that, on “uniform ” instances, the analysis by Crescenzi et al (2004) gives only trivial upper bounds, unless we find a counterexample to the conjecture. This is in contrast with other (notable) examples where the same analysis yields optimal (nontrivial) bounds. Key words: CaccettaHäggkvist conjecture, online load balancing, competitive analysis 1
On the korientability of random graphs
, 2009
"... Let G(n, m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n, m) is said to be korientable, wher ..."
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Let G(n, m) be an undirected random graph with n vertices and m multiedges that may include loops, where each edge is realized by choosing its two vertices independently and uniformly at random with replacement from the set of all n vertices. The random graph G(n, m) is said to be korientable, where k ≥ 2 is an integer, if there exists an orientation of the edges such that the maximum outdegree is at most k. Let ck = sup {c: G(n, cn) is korientable w.h.p.}. We prove that for k large enough, 1 − 2 k exp −k + 1 + e −k/4) < ck/k < 1 − exp