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57
Path Set Selection in Mobile Ad Hoc Networks
, 2002
"... Topological changes in mobile ad hoc networks frequently render routing paths unusable. Such recurrent path failures have detrimental effects on the network ability to support QoSdriven services. A promising technique for addressing this problem is to use multiple redundant paths between the source ..."
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Cited by 40 (6 self)
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Topological changes in mobile ad hoc networks frequently render routing paths unusable. Such recurrent path failures have detrimental effects on the network ability to support QoSdriven services. A promising technique for addressing this problem is to use multiple redundant paths between the source and the destination. However,while multipath routing algorithms can tolerate network failures well,their failure resilience only holds if the paths are selected judiciously. In particular,the correlation between the failures of the paths in a redundant path set should be as small as possible. However,selecting an optimal path set is an NPcomplete problem. Heuristic solutions proposed in the literature are either too complex to be performed in realtime, or too ineffective,or both. This paper proposes a multipath routing algorithm,called Disjoint Pathset Selection Protocol (DPSP),based on a novel heuristic that,in nearly linear time on average,picks a set of highly reliable paths. The convergence to a highly reliable path set is very fast,and the protocol provides flexibility in path selection and routing algorithm. Furthermore,DPSP is suitable for realtime execution,with nearly no message exchange overhead and with minimal additional storage requirements. This paper presents evidence that multipath routing can mask a substantial number of failures in the network compared to single path routing protocols,and that the selection of paths according to DPSP can be beneficial for mobile ad hoc networks,since it dramatically reduces the rate of route discoveries.
A Survey of Graph Pebbling
 Congr. Numer
, 1999
"... We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic question of Erdös and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a gr ..."
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Cited by 31 (13 self)
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We survey results on the pebbling numbers of graphs as well as their historical connection with a numbertheoretic question of Erdös and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal.
Thresholds for Families of Multisets, With an Application to Graph Pebbling
, 2000
"... In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. In add ..."
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Cited by 22 (17 self)
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In this paper we prove two multiset analogs of classical results. We prove a multiset analog of Lovász's version of the KruskalKatona Theorem and an analog of the Bollob asThomason threshold result. As a corollary we obtain the existence of pebbling thresholds for arbitrary graph sequences. In addition, we improve both the lower and upper bounds for the `random pebbling' threshold of the sequence of paths.
Isoperimetric Problems in Discrete Spaces
 Bolyai Soc. Math. Stud
, 1994
"... This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications o ..."
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Cited by 20 (5 self)
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This paper is a survey on discrete isoperimetric type problems. We present here as some known facts about their solutions as well some new results and demonstrate a general techniques used in this area. The main attention is paid to the unit cube and cube like structures. Besides some applications of the isoperimetric approach are listed too. 1 Introduction This paper is devoted to the discrete isoperimetric problem. This problem may be considered as an analog of the well known continuous problem and has some similar features. The discrete isoperimetric problem began to be studied a very long ago and a lot about it's solutions is known now. If there is the only solution of the continuous version, the discrete one, considered for the unit cube, has generally more solutions with much more rich structure, which have no direct continuous analogs. It is mainly due to the facts that, at first not for all values of cardinality of a subset (which is defined as the number of cube points in the...
Weighted 3Wise 2Intersecting Families
 J. COMBIN. THEORY (A
, 2002
"... Let n and r be positive integers. Suppose that a family satisfies 2 for all F 1 , F 2 , F 3 . We prove that if w < 0.5018 then F#F w F  (1 . 1 ..."
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Cited by 13 (12 self)
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Let n and r be positive integers. Suppose that a family satisfies 2 for all F 1 , F 2 , F 3 . We prove that if w < 0.5018 then F#F w F  (1 . 1
List Decoding of qary ReedMuller Codes
 IEEE Trans. Inform. Theory
, 2004
"... The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in ..."
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Cited by 13 (1 self)
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The qary ReedMuller codes RMq(u, m) of length n = qm are a generalization of ReedSolomon codes, which allow polynomials in m variables to encode the message. Using an idea of reducing the multivariate case to univariate case, randomized listdecoding algorithms for ReedMuller codes were given in [1] and [27]. The algorithm in [27] is an improvement of the algorithm in [1], it works for up to E < n(1 − √ 2u/q) errors but is applicable only to codes RMq(u, m) with u < q/2. In this paper, we will propose some deterministic listdecoding algorithms for qary ReedMuller codes. Viewing qary ReedMuller codes as codes from order domains, we present a listdecoding algorithm for qary ReedMuller codes, which is a straightforward generalization of the listdecoding algorithm of ReedSolomon codes in [9]. The algorithm works for up to n(1 − m+1 √ u/q) m − 1 errors, and it is applicable to codes RMq(u, m) with u < q. The algorithm can be implemented to run in time polynomial in the length of the codes. Following [12], we show that qary ReedMuller codes are subfield subcodes of ReedSolomon codes. We then present a second listdecoding algorithm for qary ReedMuller codes. This algorithm works for codes with any rates, and achieves an errorcorrection bound n(1 − √ (n − d)/n) − 1. So the second algorithm achieves a better errorcorrection bound than the algorithm in [27], since when u is small, n(1 − √ (n − d)/n) = n(1 − √ u/q). The implementation of the second algorithm requires O(n) field operations in Fq and O(n3) field operations in Fqm under some assumption. Also, we prove that qary ReedMuller codes can be described as onepoint AG codes. And using the algorithm of AG codes in [9], we give a third listdecoding
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Random walks and multiply intersecting families
 J. Combin. Theory (A
, 2005
"... Let F ⊂ 2 [n] be a 3wise 2intersecting Sperner family. It is proved that n−2 if n even, (n−2)/2 F  ≤ � � n−2 + 2 if n odd (n−1)/2 holds for n ≥ n0. The unique extremal configuration is determined as well. 1 ..."
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Cited by 11 (10 self)
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Let F ⊂ 2 [n] be a 3wise 2intersecting Sperner family. It is proved that n−2 if n even, (n−2)/2 F  ≤ � � n−2 + 2 if n odd (n−1)/2 holds for n ≥ n0. The unique extremal configuration is determined as well. 1
On an Equivalence in Discrete Extremal Problems
 Discr. Math
, 1999
"... We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edgeisoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quit ..."
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Cited by 10 (9 self)
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We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edgeisoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quite independently. We present close connections between them. In particular we show that a number of known results concerning the edgeisoperimetric problem for concrete families of graphs are direct consequences of the Macauleyness of appropriate posets. Keywords: Isoperimetric problem, Macaulay poset, compression, cartesian product. 1 Introduction Let G = (V G ; EG ) be a graph. We consider the following general problem: given a function F : 2 VG 7! IR and a number m (1 m jV G j), find an m element subset A ` VG with maximum (or minimum) value of F (A) among all the melement subsets of VG . Such subsets are called optimal . Similar problems arise in a number of practical situations. ...