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291
The noncrossing graph Nathan Linial \Lambda
, 2004
"... Abstract Two sets are noncrossing if they are disjoint or one contains the other. The noncrossing graph N Cn is the graph whose vertex set is the set of nonempty subsets of [n] = f1; : : : ; ng with an edge between any two noncrossing sets. Various facts, some new and some already known, concern ..."
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Abstract Two sets are noncrossing if they are disjoint or one contains the other. The noncrossing graph N Cn is the graph whose vertex set is the set of nonempty subsets of [n] = f1; : : : ; ng with an edge between any two noncrossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show:
An optimal online algorithm for metrical task systems
 Journal of the ACM
, 1992
"... Abstract. In practice, almost all dynamic systems require decisions to be made online, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general online decnion algorithm is developed. It is shown that, for an ..."
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Cited by 213 (8 self)
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Abstract. In practice, almost all dynamic systems require decisions to be made online, without full knowledge of their future impact on the system. A general model for the processing of sequences of tasks is introduced, and a general online decnion algorithm is developed. It is shown that, for an important algorithms. class of special cases, this algorithm is optimal among all online Specifically, a task system (S. d) for processing sequences of tasks consists of a set S of states and a cost matrix d where d(i, j) is the cost of changing from state i to state j (we assume that d satisfies the triangle inequality and all diagonal entries are f)). The cost of processing a given task depends on the state of the system. A schedule for a sequence T1, T2,..., Tk of tasks is a ‘equence sl,s~,..., Sk of states where s ~ is the state in which T ’ is processed; the cost of a schedule is the sum of all task processing costs and state transition costs incurred. An online scheduling algorithm is one that chooses s, only knowing T1 Tz ~.. T’. Such an algorithm is wcompetitive if, on any input task sequence, its cost is within an additive constant of w times the optimal offline schedule cost. The competitive ratio w(S, d) is the infimum w for which there is a wcompetitive online scheduling algorithm for (S, d). It is shown that w(S, d) = 2 ISI – 1 for eoery task system in which d is symmetric, and w(S, d) = 0(1 S]2) for every task system. Finally, randomized online scheduling algorithms are introduced. It is shown that for the uniform task system (in which d(i, j) = 1 for all i, j), the expected competitive ratio w(S, d) =
2 Local and Almost LinearTime Clustering and Partitioning
, 2009
"... You should probably know that • the first problem set (due October 15) is posted on the class website, and • its hints are also posted there. Also, today in class there was a majority vote for posting problem sets earlier. Professor Kelner will post the problem sets from two years ago, but he reserv ..."
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on applications of expander graphs? A course taught by Nathan Linial and Avi Wigderson [3]. Plan for today. We use what we proved last time to obtain a local clustering algorithm from a random walk scheme. Then, noting that similar results to the ones proved last time also hold for PageRank, we obtain a second
ProtoMap: Automatic classification of protein sequences, a hierarchy of protein families, and local maps of the protein space
 PROTEINS
, 1999
"... We investigate the space of all protein sequences in search of clusters of related proteins. Our aim is to automatically detect these sets, and thus obtain a classification of all protein sequences. Our analysis, which uses standard measures of sequence similarity as applied to an allvs.all compar ..."
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Cited by 119 (15 self)
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We investigate the space of all protein sequences in search of clusters of related proteins. Our aim is to automatically detect these sets, and thus obtain a classification of all protein sequences. Our analysis, which uses standard measures of sequence similarity as applied to an allvs.all comparison of SWISSPROT, gives a very conservative initial classification based on the highest scoring pairs. The many classes in this classification correspond to protein subfamilies. Subsequently we merge the subclasses using the weaker pairs in a twophase clustering algorithm. The algorithm makes use of transitivity to identify homologous proteins; however, transitivity is applied restrictively in an attempt to prevent unrelated proteins from clustering together. This process is repeated at varying levels of statistical significance. Consequently, a hierarchical organization of all proteins is obtained. The resulting classification splits the protein space into welldefined groups of proteins, which are closely correlated with natural biological families and superfamilies. Different indices of validity were applied to assess the quality of our classification and compare it with the protein families in the PROSITE and Pfam databases. Our classification agrees with these domainbased classifications for between 64.8 % and 88.5 % of the proteins. It also finds many new clusters of protein sequences which were not classified by these databases. The hierarchical organization suggested by our analysis reveals finer subfamilies in families of known proteins as well as many novel relations between protein families.
Naïve Learning in Social Networks and the Wisdom of Crowds
, 2010
"... We study learning in a setting where agents receive independent noisy signals about the true value of a variable and then communicate in a network. They naïvely update beliefs by repeatedly taking weighted averages of neighbors’ opinions. We show that all opinions in a large society converge to the ..."
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Cited by 97 (1 self)
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We study learning in a setting where agents receive independent noisy signals about the true value of a variable and then communicate in a network. They naïvely update beliefs by repeatedly taking weighted averages of neighbors’ opinions. We show that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows. We also identify obstructions to this, including prominent groups, and provide structural conditions on the network ensuring efficient learning. Whether agents converge to the truth is unrelated to how quickly consensus is approached. (JEL D83, D85, Z13)
Compact Distributed Data Structures for Adaptive Routing
 In Proc. 21st ACM Symp. on Theory of Computing
, 1989
"... In designing a routing scheme for a communication network it is desirable to use as short as possible paths for routing messages, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its ..."
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Cited by 76 (9 self)
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In designing a routing scheme for a communication network it is desirable to use as short as possible paths for routing messages, while keeping the routing information stored in the processors' local memory as succinct as possible. The efficiency of a routing scheme is measured in terms of its stretch factor  the maximum ratio between the cost of a route computed by the scheme and that of a cheapest path connecting the same pair of vertices. This paper presents a family of adaptive routing schemes for general networks. The hierarchical schemes HS k (for every fixed k 1) guarantee a stretch factor of O(k 2 \Delta 3 k ) and require storing at most O \Gamma kn 2 k log n \Delta bits of routing information per vertex. The new important features, that make the schemes appropriate for adaptive use, are ffl applicability to networks with arbitrary edge costs; ffl nameindependence, i.e., usage of original names; ffl a balanced distribution of the memory; ffl an efficient onli...
The Moore bound for irregular graphs
 Graphs Combin
, 2001
"... What is the largest number of edges in a graph of order n and girth g? For dregular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an armative answer to an old open problem ([4] p.163, problem 10). ..."
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Cited by 60 (7 self)
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What is the largest number of edges in a graph of order n and girth g? For dregular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an armative answer to an old open problem ([4] p.163, problem 10).
Faulttolerant Computation in the Full Information Model
 SIAM J. Comput
, 1995
"... We initiate an investigation of general faulttolerant distributed computation in the fullinformation model. In the full information model no restrictions are made on the computational power of the faulty parties or the information available to them. (Namely, the faulty players may be infinitely po ..."
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Cited by 33 (4 self)
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powerful and there are no private channels connecting pairs of honest players). Previous work, in this model, has concentrated on the particular problem of simulating a single boundedbias global coin flip (e.g. BenOr and Linial [4] and Alon and Naor [1]). We widen the scope of investigation
Approximate Protein Structural Alignment in Polynomial Time
 Proc. Natl Acad. Sci. USA
, 2004
"... Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of ..."
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Cited by 61 (1 self)
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Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n ) time, for globular protein of length n, when we wish to detect all scores that are at most # distance away from the optimum. We argue that such approximate solutions are, in fact, of greater interest than exact ones, due to the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by the algorithm involves the Euclidean distance between the structures, after rigidly transforming them. We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients in order to both guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield new insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of new scoring functions for which the optimum can be approximated e#ciently, and perhaps in the development of e#cient algorithms for the multiple structural alignment problem.
Results 1  10
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291