Randomized online algorithms for minimum metric bipartite matching

Abstract

We present the rst poly-logarithmic competitive online algorithm for minimum metric bipartite matching. Via induction and a careful use of potential functions, we show that a simple randomized greedy algorithm is competitive on a hierarchically separated tree. Application of recent results on randomized embedding of metrics into trees yield the poly-logarithmic result for general metrics.

The Online Transportation Problem: On the Exponential Boost of One Extra Server

“…It is easy to see that the randomized greedy algorithm is O(log k)-competitive for online matching in a star. [3,12] extend this analysis to show that randomized greedy is O(log k)-competitive in (log k)-Hierarchically Separated Trees (log k-HST's). [1,4] show that for every metric space, there exists a probability distribution over log k-HST's, for which the expected distance between any pair of points in a randomly drawn log k-HST is at most O(log 2 k) times the distance between this pair of points in the metric space.…”

“…One can also see from the analyses in [8] that the greedy algorithm is 3-server O(1)-competitive for online transportation. [3,12] consider randomized algorithms for the online matching problem against an oblivious adversary that must specify the input a priori. In particular, [3,12] consider the simple randomized greedy algorithm that services each request with an unused server site picked uniformly at random from the closest server sites.…”

“…[3,12] consider randomized algorithms for the online matching problem against an oblivious adversary that must specify the input a priori. In particular, [3,12] consider the simple randomized greedy algorithm that services each request with an unused server site picked uniformly at random from the closest server sites. It is easy to see that the randomized greedy algorithm is O(log k)-competitive for online matching in a star.…”

“…The results in [3,12] can be viewed as a reduction from a general metric space to the star via HST's. So we begin by first considering the star metric space.…”

“…We then generalize our analysis to show that BODS is +1-server O(log k)-competitive for online transportation on a log k-HST. Our proof proceeds by induction on the height of the HST, following the same general structure as the proof in [12]. However the introduction of resource augmentation adds some technical difficulties.…”