Max-Balancing Weighted Directed Graphs and Matrix Scaling

Schneider, Hans; Schneider, Michael

doi:10.1287/moor.16.1.208

Cited by 61 publications

(61 citation statements)

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“…In [3] and [17], generalizing the early work of Schneider and Schneider [40] and Young, Tarjan and Orlin [48], we have developed slack balancing algorithms for very general situations. The most general problem can be formulated as follows.…”

Section: B Slack Balancing Models and Algorithmsmentioning

confidence: 99%

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

2007

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Abstract-The BonnTools provide innvovative solutions for layout and timing closure that are used for many of the most complex integrated circuits. During 20 years of cooperation between the University of Bonn and IBM, new mathematical foundations and algorithms have been developed for the need of new technologies and leading-edge designs. In this paper we present the main ideas for placement, routing, timing optimization, and clock tree synthesis, which are the foundation of a continuing success story.

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Section: B Slack Balancing Models and Algorithmsmentioning

confidence: 99%

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

2007

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“…The average edge cost of such a cycle is called the minimum cycle mean. Solutions to this problem are needed in a minimum-cost circulation algorithm of Goldberg and Tarjan [3] and in a graph minimum-balancing algorithm of Schneider and Schneider [11]. The problem has been studied by Karp [6], who gave an O(nm)-time dynamic programming algorithm, and by Ahuja and Orlin [1], who gave an O( √ nm log nC)-time scaling algorithm.…”

Section: The Minimum Mean Cycle Problemmentioning

confidence: 99%

“…Schneider and Schneider, who gave the above method in [11], also noted that Karp's O(nm)-time algorithm for finding minimum mean cycles can be extended to yield shortest path potentials, thus obtaining an O(n 2 m)-time algorithm.…”

Section: The Minimum Balance Problemmentioning

confidence: 99%

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Faster parametric shortest path and minimum‐balance algorithms

1991

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We use Fibonacci heaps to improve a parametric shortest path algorithm of Karp and Orlin, and we combine our algorithm and the method of Schneider and Schneider's minimum-balance algorithm to obtain a faster minimum-balance algorithm.For a graph with n vertices and m edges, our parametric shortest path algorithm and our minimum-balance algorithm both run in O(nm + n 2 log n) time, improved from O(nm log n) for the parametric shortest path algorithm of Karp and Orlin and O(n 2 m) for the minimum-balance algorithm of Schneider and Schneider.An important application of the parametric shortest path algorithm is in finding a minimum mean cycle. Experiments on random graphs suggest that the expected time for finding a minimum mean cycle with our algorithm is O(n log n + m).

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“…For matrix balancing in the L ∞ norm, Schneider and Schneider [11] gave an O(n 4 )-time non-iterative algorithm. This running time was improved to O(mn + n 2 log n) by Young, Tarjan, and Orlin [14].…”

Section: Introductionmentioning

confidence: 99%

Matrix Balancing in L_p Norms: Bounding the Convergence Rate of Osborne's Iteration

Ostrovsky¹,

Rabani²,

Yousefi³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study an iterative matrix conditioning algorithm due to Osborne (1960). The goal of the algorithm is to convert a square matrix into a balanced matrix where every row and corresponding column have the same norm. The original algorithm was proposed for balancing rows and columns in the L2 norm, and it works by iterating over balancing a row-column pair in fixed round-robin order. Variants of the algorithm for other norms have been heavily studied and are implemented as standard preconditioners in many numerical linear algebra packages. Recently, Schulman and Sinclair (2015), in a first result of its kind for any norm, analyzed the rate of convergence of a variant of Osborne's algorithm that uses the L∞ norm and a different order of choosing row-column pairs. In this paper we study matrix balancing in the L1 norm and other Lp norms. We show the following results for any matrix A = (aij) 1. We analyze the iteration for the L1 norm under a greedy order of balancing. We show that it converges to an -balanced matrix in K = O(min{ −2 log w, −1 n 3/2 log(w/ )}) iterations that cost a total of O(m + Kn log n) arithmetic operations over O(n log(w/ ))-bit numbers. Here m is the number of non-zero entries of A, and w = i,j |aij|/amin with amin = min{|aij| : aij = 0}. 2. We show that the original round-robin implementation converges to an -balanced matrix in O( −2 n 2 log w) iterations totaling O( −2 mn log w) arithmetic operations over O(n log(w/ ))-bit numbers. 3. We show that a random implementation of the iteration converges to an -balanced matrix in O( −2 log w) iterations using O(m + −2 n log w) arithmetic operations over O(log(wn/ ))-bit numbers.4. We demonstrate a lower bound of Ω(1/ √ ) on the convergence rate of any implementation of the iteration. 5. We observe, through a known trivial reduction, that our results for L1 balancing apply to any Lp norm for all finite p, at the cost of increasing the number of iterations by only a factor of p.

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Max-Balancing Weighted Directed Graphs and Matrix Scaling

Cited by 61 publications

References 10 publications

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

Faster parametric shortest path and minimum‐balance algorithms

Matrix Balancing in L_p Norms: Bounding the Convergence Rate of Osborne's Iteration

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Max-Balancing Weighted Directed Graphs and Matrix Scaling

Cited by 61 publications

References 10 publications

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

Faster parametric shortest path and minimum‐balance algorithms

Matrix Balancing in Lp Norms: Bounding the Convergence Rate of Osborne's Iteration

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Product

Resources

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Matrix Balancing in L_p Norms: Bounding the Convergence Rate of Osborne's Iteration