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74
Spectral Partitioning Works: Planar graphs and finite element meshes
 In IEEE Symposium on Foundations of Computer Science
, 1996
"... Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extr ..."
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Cited by 144 (8 self)
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Spectral partitioning methods use the Fiedler vectorthe eigenvector of the secondsmallest eigenvalue of the Laplacian matrixto find a small separator of a graph. These methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well. In this paper, we show that spectral partitioning methods work well on boundeddegree planar graphs and finite element meshes the classes of graphs to which they are usually applied. While naive spectral bisection does not necessarily work, we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O( p n) for boundeddegree planar graphs and twodimensional meshes and O i n 1=d j for wellshaped ddimensional meshes. The heart of our analysis is an upper bound on the secondsmallest eigenvalues of the Laplacian matrices of these graphs. 1. Introduction Spectral partitioning has become one of the mos...
On the complexity of solving Markov decision problems
 IN PROC. OF THE ELEVENTH INTERNATIONAL CONFERENCE ON UNCERTAINTY IN ARTIFICIAL INTELLIGENCE
, 1995
"... Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argu ..."
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Cited by 131 (10 self)
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Markov decision problems (MDPs) provide the foundations for a number of problems of interest to AI researchers studying automated planning and reinforcement learning. In this paper, we summarize results regarding the complexity of solving MDPs and the running time of MDP solution algorithms. We argue that, although MDPs can be solved efficiently in theory, more study is needed to reveal practical algorithms for solving large problems quickly. To encourage future research, we sketch some alternative methods of analysis that rely on the structure of MDPs.
Balanced Matroids
"... We introduce the notion of "balance", and say that a matroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen basis, the presence of an element can only make any other element less likely. We establish strong expansion properties for the base ..."
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Cited by 76 (3 self)
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We introduce the notion of "balance", and say that a matroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen basis, the presence of an element can only make any other element less likely. We establish strong expansion properties for the basesexchange graph of balanced matroids; consequently, the set of bases of a balanced matroid can be sampled and approximately counted using rapidly mixing Markov chains. Thus, the general problem of approximately counting bases (known to be #Pcomplete) is reduced to that of showing balance. Specific classes for which balance is known to hold include graphic and regular matroids.
Global clock synchronization in sensor networks
 IEEE Transactions on Computers
"... Abstract—Global synchronization is important for many sensor network applications that require precise mapping of collected sensor data with the time of the events, for example, in tracking and surveillance. It also plays an important role in energy conservation in MAC layer protocols. This paper de ..."
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Cited by 76 (1 self)
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Abstract—Global synchronization is important for many sensor network applications that require precise mapping of collected sensor data with the time of the events, for example, in tracking and surveillance. It also plays an important role in energy conservation in MAC layer protocols. This paper describes four methods to achieve global synchronization in a sensor network: a nodebased approach, a hierarchical clusterbased method, a diffusionbased method, and a faulttolerant diffusionbased method. The diffusionbased protocol is fully localized. We present two implementations of the diffusionbased protocol for synchronous and asynchronous systems and prove its convergence. Finally, we show that, by imposing some constraints on the sensor network, global clock synchronization can be achieved in the presence of malicious nodes that exhibit Byzantine failures. Index Terms—Sensor networks, fault tolerance. æ
Random Walks And An O*(n 5 ) Volume Algorithm For Convex Bodies
, 1996
"... Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n 5 ) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a mul ..."
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Cited by 75 (8 self)
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Given a high dimensional convex body K ` IR n by a separation oracle, we can approximate its volume with relative error ", using O (n 5 ) oracle calls. Our algorithm also brings the body into isotropic position. As all previous randomized volume algorithms, we use "rounding" followed by a multiphase MonteCarlo (product estimator) technique. Both parts rely on sampling (generating random points in K), which is done by random walk. Our algorithm introduces three new ideas: ffl the use of the isotropic position (or at least an approximation of it) for rounding, ffl the separation of global obstructions (diameter) and local obstructions (boundary problems) for fast mixing, and ffl a stepwise interlacing of rounding and sampling. 1 . Introduction For a variety of geometric objects, classical results characterize various geometric parameters. Many of these results are useful even in practical situations: they can easily be transformed into efficient algorithms. Some other theorem...
Collisions among Random Walks on a Graph
 SIAM J. on Discrete Mathematics
, 1993
"... A token located at some vertex v of a connected, undirected graph G on n vertices is said to be taking a "random walk" on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of v. We consider the following problem: suppose that two tokens are placed on G, ..."
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Cited by 67 (13 self)
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A token located at some vertex v of a connected, undirected graph G on n vertices is said to be taking a "random walk" on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of v. We consider the following problem: suppose that two tokens are placed on G, and at each tick of the clock a certain demon decides which of them is to make the next move. The demon is trying to keep the tokens apart as long as possible. What is the expected time M before they meet? The problem arises in the study of selfstabilizing systems, a topic of recent interest in distributed computing. Since previous upper bounds for M were exponential in n, the issue was to obtain a polynomial bound. We use a novel potential function argument to show that in the worst case M = \Gamma 4 27 + o(1) \Delta n 3 . 1 Introduction Let G be a connected graph on n vertices, and let v be a fixed vertex of G. A random walk on G, beginning at v, is a stochastic process whose stat...
An Optimal Algorithm for Monte Carlo Estimation
, 1995
"... A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a ..."
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Cited by 53 (4 self)
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A typical approach to estimate an unknown quantity is to design an experiment that produces a random variable Z distributed in [0; 1] with E[Z] = , run this experiment independently a number of times and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0; 1]. We describe an approximation algorithm AA which, given ffl and ffi, when running independent experiments with respect to any Z, produces an estimate that is within a factor 1 + ffl of with probability at least 1 \Gamma ffi. We prove that the expected number of experiments run by AA (which depends on Z) is optimal to within a constant factor for every Z. An announcement of these results appears in P. Dagum, D. Karp, M. Luby, S. Ross, "An optimal algorithm for MonteCarlo Estimation (extended abstract)", Proceedings of the Thirtysixth IEEE Symposium on Foundations of Computer Science, 1995, pp. 142149 [3]. Section ...
TIGHT ANALYSES OF TWO LOCAL LOAD BALANCING ALGORITHMS
 SIAM J. COMPUT.
, 1999
"... This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We ..."
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Cited by 51 (5 self)
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This paper presents an analysis of the following load balancing algorithm. At each step, each node in a network examines the number of tokens at each of its neighbors and sends a token to each neighbor with at least 2d + 1 fewer tokens, where d is the maximum degree of any node in the network. We show that within O(∆/α) steps, the algorithm reduces the maximum difference in tokens between any two nodes to at most O((d 2 log n)/α), where ∆ is the global imbalance in tokens (i.e., the maximum difference between the number of tokens at any node initially and the average number of tokens), n is the number of nodes in the network, and α is the edge expansion of the network. The time bound is tight in the sense that for any graph with edge expansion α, and for any value ∆, there exists an initial distribution of tokens with imbalance ∆ for which the time to reduce the imbalance to even ∆/2 is at least Ω(∆/α). The bound on the final imbalance is tight in the sense that there exists a class of networks that can be locally balanced everywhere (i.e., the maximum difference in tokens between any two neighbors is at most 2d), while the global imbalance remains Ω((d 2 log n)/α). Furthermore, we show that upon reaching a state with a global imbalance of O((d 2 log n)/α), the time for this algorithm to locally balance the network can be as large as Ω(n 1/2). We extend our analysis to a variant of this algorithm for dynamic and asynchronous
Faster Mixing via average Conductance
"... The notion of conductance introduced by Jerrum and Sinclair [JS] has been widely used to prove rapid mixing of Markov chains. Here we introduce a variant of this  instead of measuring the conductance of the worst subset of states, we show that it is enough to bound a certain weighted average conduc ..."
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Cited by 44 (3 self)
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The notion of conductance introduced by Jerrum and Sinclair [JS] has been widely used to prove rapid mixing of Markov chains. Here we introduce a variant of this  instead of measuring the conductance of the worst subset of states, we show that it is enough to bound a certain weighted average conductance (where the average is taken over subsets of states with different sizes.) In the case of convex bodies, we show that this average conductance is better than the known bounds for the worst case; this helps us save a factor of O(n) which is incurred in all proofs as a "penalty" for a "bad start" (i.e., because the starting distribution may be arbitrary).