Consider an $n$-vertex, $m$-edge, undirected graph with integral capacities and max-flow value $v$. We give a new $\tilde{O}(m + nv)$-time maximum flow algorithm. After assigning certain special sampling probabilities to edges in $\tilde{O}(m)$ time, our algorithm is very simple: repeatedly find an augmenting path in a random sample of edges from the residual graph. Breaking from past work, we demonstrate that we can benefit by random sampling from directed (residual) graphs. We also slightly improve an algorithm for approximating flows of arbitrary value, finding a flow of value $(1-\epsilon)$ times the maximum in $\tilde{O}(m\sqrt{n/\epsilon})$ time.

ABSTRACT Consider an n-vertex, m-edge, undirected graph with maximumflow value v. We give a new ~O (m + nv)-time maximum flow algo-rithm based on finding augmenting paths in random samples of the edges of residual graphs. After assigning certain special samplingprobabilities to edges in ~O (m) time, our algorithm is very simple:repeatedly find an augmenting path in a random sample of edges from the residual graph.

Consider an rr-vertex, m-edge, undirected graph with maximum llow value v.We give a method to find augmenting paths in such a graph in amortized sub-linear (O(n@) time per path.This lets us improve the time bound of the classic augmenting path algorithm to O(m + nvsi2) on simple graphs.The addition of a blocking flow subroutine gives a simple, deterministic O(nm2/3v1/6)-time algorithm, We also use our technique to improve known randomized algorithms, giving @rtr+nv5/4)-time and d(m+-nt'~gv)-time algorithms for capacitated undirected graphs.-Forsimple graphs, in which v s II, the last bound is a(n2s2), improving on the best previous bound of O(n2*5), which is also the best known time bound for bipartite matching.

We describe a family of caching protocols for distrib-uted networks that can be used to decrease or eliminate the occurrence of hot spots in the network. Our protocols are particularly designed for use with very large networks such as the Internet, where delays caused by hot spots can be severe, and where it is not feasible for every server to have complete information about the current state of the entire network. The protocols are easy to implement using existing network protocols such as TCP/IP, and require very little overhead. The protocols work with local control, make efficient use of existing resources, and scale gracefully as the network grows. Our caching protocols are based on a special kind of hashing that we call consistent hashing. Roughly speaking, a consistent hash function is one which changes minimally as the range of the function changes. Through the development of good consistent hash functions, we are able to develop caching protocols which do not require users to have a current or even consistent view of the network. We believe that consistent hash functions may eventually prove to be useful in other applications such as distributed name servers and/or quorum systems.