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A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds su ..."
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Cited by 386 (1 self)
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Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.
Explicit construction of linear sized tolerant networks
, 2006
"... For every ɛ > 0 and every integer m > 0, we construct explicitly graphs with O(m/ɛ) vertices and maximum degree O(1/ɛ²), such that after removing any (1 − ɛ) portion of their vertices or edges, the remaining graph still contains a path of length m. This settles a problem of Rosenberg, which was moti ..."
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Cited by 104 (13 self)
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For every ɛ > 0 and every integer m > 0, we construct explicitly graphs with O(m/ɛ) vertices and maximum degree O(1/ɛ²), such that after removing any (1 − ɛ) portion of their vertices or edges, the remaining graph still contains a path of length m. This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.
How Good is Recursive Bisection?
 SIAM J. Sci. Comput
, 1995
"... . The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the opti ..."
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Cited by 84 (4 self)
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. The most commonly used pway partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem, that partitions a graph into two equal sized subgraphs to minimize the number of edges cut, is NPcomplete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the recursive bisection method, even when an optimal bisection algorithm is assumed, may produce a pway partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: First we show that for some important classes of graphs that occur in practical applications, such as wellshaped finite element and finite difference...
Communication complexity of simultaneous messages
 SIAM Journal on Computing
"... In the multiparty communication game (CFLgame) of Chandra, Furst, and Lipton (Proc. 15th ACM STOC, 1983, 94–99) k players collaboratively evaluate a function f(x0,..., xk−1) in which player i knows all inputs except xi. The players have unlimited computational power. The objective is to minimize co ..."
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Cited by 15 (0 self)
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In the multiparty communication game (CFLgame) of Chandra, Furst, and Lipton (Proc. 15th ACM STOC, 1983, 94–99) k players collaboratively evaluate a function f(x0,..., xk−1) in which player i knows all inputs except xi. The players have unlimited computational power. The objective is to minimize communication. In this paper, we study the Simultaneous Messages (SM) model of multiparty communication complexity. The SM model is a restricted version of the CFLgame in which the players are not allowed to communicate with each other. Instead, each of the k players simultaneously sends a message to a referee, who sees none of the inputs. The referee then announces the function value. We prove lower and upper bounds on the SMcomplexity of several classes of explicit functions. Our lower bounds extend to randomized SM complexity via an entropy argument. A lemma establishing a tradeoff between average Hamming distance and range size for transformations of the Boolean cube might be of independent interest. Our lower bounds on SMcomplexity imply an exponential gap between the SMmodel and
On Separators, Segregators and Time versus Space
"... We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that N T IM E(n ..."
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Cited by 6 (0 self)
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We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that N T IM E(n
Matrix Rigidity
, 1999
"... The rigidity of a matrix M is the function RM (r), which, for a given r, gives the minimum number of entries of M which one has to change in order to reduce its rank to at most r. This notion has been introduced by Valiant in 1977 in connection with the complexity of computing linear forms. Despi ..."
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Cited by 2 (0 self)
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The rigidity of a matrix M is the function RM (r), which, for a given r, gives the minimum number of entries of M which one has to change in order to reduce its rank to at most r. This notion has been introduced by Valiant in 1977 in connection with the complexity of computing linear forms. Despite more than 20 years of research, very little is known about the rigidity of matrices. Nonlinear lower bounds on matrix rigidity would lead to new lower bound techniques for the computation of linear forms, e.g., for the computation of the DFT, as well as to more general advances in complexity theory. We put forward a number of linear algebra research issues arising in the above outlined context.