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RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS
, 1999
"... We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for pro ..."
Abstract

Cited by 72 (11 self)
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We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cutapproximation algorithms extend unchanged to weighted graphs while our weightedgraph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a nearlinear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cutbased problems, including approximating the best balanced cut of a graph, finding a kconnected orientation of a 2kconnected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum kconnected subgraph problem from 1.85 to 1 � O(�log n)/k).
SelfAdjusting Computation with Delta ML
"... Abstract. In selfadjusting computation, programs respond automatically and efficiently to modifications to their data by tracking the dynamic data dependences of the computation and incrementally updating the output as needed. In this tutorial, we describe the selfadjustingcomputation model and pr ..."
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Cited by 2 (0 self)
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Abstract. In selfadjusting computation, programs respond automatically and efficiently to modifications to their data by tracking the dynamic data dependences of the computation and incrementally updating the output as needed. In this tutorial, we describe the selfadjustingcomputation model and present the language ∆ML (Delta ML) for writing selfadjusting programs. 1
Dynamic algorithms: Introduction
"... Most algorithms that we have seen so far have been static in that the input was given initially and our task has been to compute some function f of the input while keeping resources at a minimum. This note considers dynamic algorithms, where we in addition to computing ..."
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Most algorithms that we have seen so far have been static in that the input was given initially and our task has been to compute some function f of the input while keeping resources at a minimum. This note considers dynamic algorithms, where we in addition to computing