and the roundtrip routing schemes for directed graphs that we describe are only slightly worse than the best available spanners and routing schemes for undirected graphs. Our roundtrip routing schemes substantially improve previous results of Cowen and Wagner. Our results are obtained by combining ideas of Cohen

This paper presents compact roundtrip routing schemes with local tables of size Õ( √ n) and stretch 6 for any directed network with arbitrary edge weights; and with local tables of size Õ(ǫ−1 n 2/k) and stretch min((2 k/2 − 1)(k + ǫ),16k 2 + 8k − 8), for any directed network with polynomially

This paper concerns graph spanners that approximate multipaths between pair of vertices of an undirected graphs with n vertices. Classically, a spanner H of stretch s for a graph G is a spanning subgraph such that the distance in H between any two vertices is at most s times the distance in G. We

An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k − 1, 0)-spanner of size O(n 1+1/k) and an (additive) (1, 2)-spanner of size O(n 3

We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k

The IP multicast model allows scalable and efficient multi-party communication, particularly for groups of large size. However, deployment of IP multicast requires substantial infrastructure modifications and is hampered by a host of unresolved open problems. To circumvent this situation, we have designed and implemented ALMI, an application level group communication middleware, which allows accelerated application deployment and simplified network configuration, without the need of network infrastructure support. ALMI is tailored toward support of multicast groups of relatively small size (several I Os of members) with many to many semantics. Session participants are connected via a vir- tual multicast tree, which consists of unicast connections between end hosts and is formed as a minimum spanning tree (MST) using application-specific performance metric. Using simulation, we show that the performance penalties, introduced by this shift of multicast to end systems, is a relatively small increase in traffic load and that ALMI multicast trees approach the efficiency of IP multicast trees. We have also implemented ALMi as a Java based middleware package and performed experiments over the Internet. Experimental results show that ALMI is able to cope with network dynamics and keep the mul- ticast tree efficient.

We define the notion of a transitive-closure spanner of a directed graph. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k

, and it is necessary to direct clients to the closest mirrors based on some distance metric in order to realize the benefit of mirrors. We suggest a scalable Internet-wide architecture, called IDMaps, which measures and disseminates distance information on the global Internet. Higher-level services can collect

We survey results on transitive-closure spanners and their applications. Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k

(·) is a directed graph I(V,E, r), whose vertices are the points of V and whose edge set includes a directed edge from p to q if δ(p,q) ≤ r(p). In [8] we presented an algorithm for constructing a (1 + ǫ)-spanner of size O(n/ǫ d log M), where M is the maximal radius r(p). The current paper presents two