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78
Polynomial time algorithms for multicast network code construction
 IEEE Trans. on Info. Thy
"... Abstract—The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the int ..."
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Cited by 183 (15 self)
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Abstract—The famous maxflow mincut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the mincut separating and. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to reencode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures. Index Terms—Communication networks, efficient algorithms, linear coding, multicasting rate maximization. I.
Scheduling Algorithms for Inputqueued Cell Switches
, 1995
"... The algorithms described in this thesis are designed to schedule cells in a very highspeed, parallel, inputqueued crossbar switch. We present several novel scheduling algorithms that we have devised, each aims to match the set of inputs of an inputqueued switch to the set of outputs more effici ..."
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Cited by 138 (4 self)
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The algorithms described in this thesis are designed to schedule cells in a very highspeed, parallel, inputqueued crossbar switch. We present several novel scheduling algorithms that we have devised, each aims to match the set of inputs of an inputqueued switch to the set of outputs more efficiently, fairly and quickly than existing techniques. In Chapter 2 we present the simplest and fastest of these algorithms: SLIP  a parallel algorithm that uses rotating priority ("roundrobin") arbitration. SLIP is simple: it is readily implemented in hardware and can operate at high speed. SLIP has high performance: for uniform i.i.d. Bernoulli arrivals, SLIP is stable for any admissible load, because the arbiters tend to desynchronize. We present analytical results to model this behavior. However, SLIP is not always stable and is not always monotonic: adding more traffic can actually make the algorithm operate more efficiently. We present an approximate analytical model of this behavior. SLIP prevents starvation: all contending inputs are eventually served. We present simulation results, indicating SLIP's performance. We argue that SLIP can be readily implemented for a 32x32 switch on a single chip. In Chapter 3 we present iSLIP, an iterative algorithm that improves upon SLIP by converging on a maximal size match. The performance of iSLIP improves with up to log 2 N iterations. We show that although it has a longer running time than SLIP, an iSLIP scheduler is little more complex to implement. In Chapter 4 we describe maximum or maximal weight matching algorithms based on the occupancy of queues, or waiting times of cells. These algorithms are stabl...
Faster scaling algorithms for network problems
 SIAM J. COMPUT
, 1989
"... This paper presents algorithms for the assignment problem, the transportation problem, and the minimumcost flow problem of operations research. The algorithms find a minimumcost solution, yet run in time close to the bestknown bounds for the corresponding problems without costs. For example, the ..."
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Cited by 126 (4 self)
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This paper presents algorithms for the assignment problem, the transportation problem, and the minimumcost flow problem of operations research. The algorithms find a minimumcost solution, yet run in time close to the bestknown bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimumcost matching in a bipartite graph) can be solved in O(v/’rn log(nN)) time, where n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.
Polynomial Time Algorithms for Network Information Flow
 in 15th ACM Symposium on Parallel Algorithms and Architectures
, 2003
"... The famous maxflow mincut theorem states that a source node s can send information through a network (V; E) to a sink node t at a data rate determined by the mincut separating s and t. Recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that t ..."
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Cited by 96 (1 self)
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The famous maxflow mincut theorem states that a source node s can send information through a network (V; E) to a sink node t at a data rate determined by the mincut separating s and t. Recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to reencode the information they receive. In contrast, we present graphs where without coding the rate must be a factor jV j) smaller. However, so far no fast algorithms for constructing appropriate coding schemes were known. Our main result are polynomial time algorithms for constructing coding schemes for multicasting at the maximal data rate.
Scaling Algorithms for Network Problems
, 1985
"... This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a b ..."
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Cited by 59 (2 self)
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This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n3 % log N) time. For appropriate N this improves the traditional Hungarian method, whose most efftcient implementation is O(n(m + n log n)). The speedup results from finding augmenting paths in batches. The matching algorithm gives similar improvements for the following problems: singlesource shortest paths for arbitrary edge lengths (Bellman’s algorithm); maximum weight degreeconstrained subgraph; minimum cost flow on a cl network. Scaling gives a simple maximum value flow algorithm that matches the best known bound (Sleator and Tarjan’s algorithm) when log N = O(log n). Scaling also gives a good algorithm for shortest paths on a directed graph with nonnegative edge lengths (Dijkstra’s algorithm).
Structural Cohesion and Embeddedness: A hierarchical conception of social groups.
 American Sociological Review
, 2000
"... While questions about social cohesion lie at the core of our discipline, definitions are often vague and difficult to operationalize. We link research on social cohesion and social embeddedness by developing a conception of structural cohesion based on network nodeconnectivity. Structural cohesion i ..."
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Cited by 55 (11 self)
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While questions about social cohesion lie at the core of our discipline, definitions are often vague and difficult to operationalize. We link research on social cohesion and social embeddedness by developing a conception of structural cohesion based on network nodeconnectivity. Structural cohesion is defined as the minimum number of actors who, if removed from a group, would disconnect the group. A structural dimension of embeddedness can then be defined through the hierarchical nesting of these cohesive structures. We demonstrate the empirical applicability of our conception of nestedness in two dramatically different substantive settings and discuss additional theoretical implications with reference to a wide array of substantive fields. "...social solidarity is a wholly moral phenomenon which by itself is not amenable to exact observation and especially not to measurement." (Durkheim, (1893 [1984], p.24) "The social structure [of the dyad] rests immediately on the one and on the other of the two, and the secession of either would destroy the whole. ... As soon, however, as there is a sociation of three, a group continues to exist even in case one of the members drops out." (Simmel (1908 [1950], p. 123)
New scaling algorithms for the assignment and minimum mean cycle problems
, 1992
"... In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing th ..."
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Cited by 50 (4 self)
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In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes, m arcs, and integer arc costs bounded by C, the algorithm runs in O(,/n m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under the similarity assumption, i.e., C = O(n k) for some k. We next consider the minimum mean cycle problem. The mean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. The minimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(~/n m log nC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.
Edge Disjoint Paths Revisited
 In Proceedings of the 14th ACMSIAM Symposium on Discrete Algorithms
, 2003
"... The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the numb ..."
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Cited by 39 (5 self)
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The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )hardness follows. On the other hand, the O( m)approximation algorithms do not guarantee a sublinear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sublinear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.