Results 1 
3 of
3
An almostlineartime algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations
"... In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
In this paper we present an almost linear time algorithm for solving approximate maximum flow in undirected graphs. In particular, given a graph with m edges we show how to produce a 1−ε approximate maximum flow in time O(m 1+o(1) · ε −2). Furthermore, we present this algorithm as part of a general framework that also allows us to achieve a running time of O(m 1+o(1) ε −2 k 2) for the maximum concurrent kcommodity flow problem, the first such algorithm with an almost linear dependence on m. We also note that independently Jonah Sherman has produced an almost linear time algorithm for maximum flow and we thank him for coordinating submissions.
Oblivious Routing for the Lpnorm
"... Abstract — Gupta et al. [13] introduced a very general multicommodity flow problem in which the cost of a given flow solution on a graph G = (V, E) is calculated by first computing the link loads via a loadfunction ℓ, that describes the load of a link as a function of the flow traversing the link, ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract — Gupta et al. [13] introduced a very general multicommodity flow problem in which the cost of a given flow solution on a graph G = (V, E) is calculated by first computing the link loads via a loadfunction ℓ, that describes the load of a link as a function of the flow traversing the link, and then aggregating the individual link loads into a single number via an aggregation function agg: R E  → R. In this paper we show the existence of an oblivious routing scheme with competitive ratio O(log n) and a lower bound of Ω(log n / log log n) for this model when the aggregation function agg is an Lpnorm. Our results can also be viewed as a generalization of the work on approximating metrics by a distribution over dominating tree metrics (see e.g. [4], [5], [8]) and the work on minimum congestion oblivious routing [20], [14], [21]. We provide a convex combination of trees such that routing according to the tree distribution approximately minimizes the Lpnorm of the link loads. The embedding techniques of Bartal [4], [5] and Fakcharoenphol et al. [8] can be viewed as solving this problem in the L1norm while the result of Räcke [21] solves it for L∞. We give a single proof that shows the existence of a good treebased oblivious routing for any Lpnorm. For the Euclidean norm, we also show that it is possible to compute a treebased oblivious routing scheme in polynomial time. Keywordsoblivious routing; metric embeddings; norm; 1.
On a New Competitive Measure for Oblivious Routing
"... Abstract—Oblivious routing algorithms use only locally available information at network nodes to forward traffic, and as such, a plausible choice for distributed implementations. It is a natural desire to quantify the performance penalty we pay for this distributedness. Recently, Räcke has shown tha ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract—Oblivious routing algorithms use only locally available information at network nodes to forward traffic, and as such, a plausible choice for distributed implementations. It is a natural desire to quantify the performance penalty we pay for this distributedness. Recently, Räcke has shown that for general undirected graphs the competitive ratio is only, that is, the maximum congestion caused by the oblivious algorithm is within a logarithmic factor of the best possible congestion. And while the performance penalty is larger for directed networks (Azar gives a lower bound), experiments on many realworld topologies show that it usually remains under 2. These competitive measures, however, are of worstcase type, and therefore do not always give adequate characterization. The more different combinations of demands a routing algorithm can accommodate in the network without congestion, the better. Driven by this observation, in this paper we introduce a new competitive measure, the volumetric competitive ratio, as the measure of all admissible demands compared to the measure of demands routed without congestion. The main result of the paper is a general lower bound on the volumetric ratio; and we also show a directed graph with competitive ratio that exhibits volumetric ratio. Our numerical evaluations show that the competitivity of oblivious routing in terms of the new measure quickly vanishes even in relatively small commonplace topologies. Keywords—competitive ratio; oblivious routing; norm; norm; throughput polytope; feasible region; probability of congestion; hyperspherical coordinates I.