We study a variant of Interval Routing [SK85, LT86] where the routing range associated with every link is represented by a linear (i.e., contiguous) interval with no wrap around. This kind of routing schemes arises naturally in the study of dynamic Prefix Routing [BLT90]. Linear Interval Routing schemes are precisely the Prefix Routing schemes that use an alphabet of one symbol. We characterize the type of networks that admit optimum Linear Interval Routing schemes. It is shown that several well-known interconnection networks such as hypercubes, certain n-tori, and n-dimensional grids all with unit-cost links, have optimum Linear Interval Routing schemes. We also introduce the multi-label Linear Interval Routing schemes where each link may contain more than one label, and we prove several characterization results for these schemes. 1 Introduction In communication networks routing algorithms are employed for selecting suitable routes from origin to destination nodes and ensuring that m...

This survey concerns the role of data structures for compactly storing and representing various types of information in a localized and distributed fashion. Traditional approaches to data representation are based on global data structures, which require access to the entire structure even if the sought information involves only a small and local set of entities. In contrast, localized data representation schemes are based on breaking the information into small local pieces, or labels, selected in a way that allows one to infer information regarding a small set of entities directly from their labels, without using any additional (global) information. The survey focuses on combinatorial and algorithmic techniques, and covers complexity results on various applications, including compact localized schemes for message routing in communication networks, and adjacency and distance labeling schemes.

This article focuses on routing messages in distributed networks with efficient data structures. After an overview of the various results of the literature, we point some interestingly open problems.

Several methods exist for routing messages in a network without using complete routing tables (compact routing). In k-interval routing schemes (k-IR.S), links carry up to k intervals each. A message is routed over certain link if its destination belongs to one of the intervals of the link. We give some results for the necessary value of k in order to achieve shortest path routing. Even though for very structured networks low values of suce, we show that for 'general graphs' interval routing cannot significantly reduce the space-requirements for shortest path routing. In particular we show that for suitably large n, there are suitable values of p such that for randomly chosen graphs G 6 ,p the following holds, with high probability: if G admits an optimal k-IIS, then k = The result is obtained by means of a novel matrix representation for the shortest paths in a network.

This paper presents the first dynamic routing scheme for high-speed networks. The scheme is based on a hierarchical bubbles partition of the underlying communication graph.

We study a variant of Interval Routing [SK85, LT86] where the routing range associated with every link is represented by a linear (i.e., contiguous) interval with no wrap around. This kind of routing schemes arises naturally in the study of dynamic Prefix Routing [BLT90]. Linear Interval Routing schemes are precisely the Prefix Routing schemes that use an alphabet of one symbol. We characterize the type of networks that admit optimum Linear Interval Routing schemes. It is shown that several well-known interconnection networks such as hypercubes, certain n-tori, and n-dimensional grids all with unit-cost links, have optimum Linear Interval Routing schemes. We also introduce the multi-label Linear Interval Routing schemes where each link may contain more than one label, and we prove several characterization results for these schemes.

We consider the problem of searching for a piece of information in a fully interconnected computer network (also called complete network or clique) by exploiting advice about its location from the network nodes. Each node contains a database that "knows" what kind of documents or information are stored in other nodes (e.g. a node could be a Web server that answers queries about documents stored on the Web). The databases in each node, when queried, provide a pointer that leads to the node that contains the information. However, this information is up-to-date (or correct) with some bounded probability. While, in principle, one may always locate the information by simply visiting the network nodes in some prescribed ordering, this requires a time complexity in the order of the number of nodes of the network. In this paper, we provide algorithms for locating an information node in the complete communication network, that take advantage of advice given from network nodes. The nodes may either give correct advice, by pointing directly to the information node, or give wrong advice by pointing elsewhere. On the lower bounds side, we show that no fixed memory (i.e. with memory independent on the network size) deterministic algorithm may locate the information node in a constant (independent on the network size) expected number of steps. Moreover, if p = #(1/n) is the probability that a node of an n-node clique gives correct advice, we show that no algorithm may locate the information node in expected number of steps less than 1/p-o(1). In order to study how the expected number of steps is a#ected by the amount of memory allowed to the algorithms, we give a memoryless randomized algorithm with expected number of steps 4/p + o(1/p) + o(1) and a 1-bit randomize...

We present deterministic algorithms to search for an item s contained in a node of a network, without prior knowledge of its exact location. Each node of the network has a database that will answer queries of the form \how do I get to s?" by responding with the rst edge on a shortest path to the node containing s. It may happen that some nodes, called liars, give bad advice. If the number of liars k is bounded, we show dierent strategies to nd the item depending on the topology of the network. In particular we consider the complete graph, ring, torus, hypercube and bounded degree trees.

Awerbuch, Afek, Plotkin, and Saks identified an important fundamental problem inherent to distributed networks, which they called the Resource Controller problem. Consider, first, the problem in which one node (called the ‘root’) is required to estimate the number of events that occurred all over the network. The counting can be viewed as a useful variant of the heavily studied and used task of topology update (that deals with collecting all remote information). The Resource Controller problem generalizes the counting problem further: such remote events are considered as requests, and the counting node, i.e., the ‘root’, also issues permits for the requests. That way, the number of request granted can be controlled (bounded). In the paper by Awerbuch et al., it was assumed that the network is spanned by a tree that may only grow, and only by allowing leaves to join the tree (after receiving a permit). In contrast, the Resource Controller presented here can operate under a more general dynamic model. Specifically, the dynamic model considered in this paper allows both controlled insertions and deletions of leaves as well as controlled insertions and deletions of internal nodes. Despite the more dynamic network model we allow, the message complexity of our controller is always at most the message complexity of the

Currently, there are no known explicit algorithms for the great majority of graph problems in the dynamic distributed message-passing model. Instead, most state-of-the-art dynamic distributed algorithms are constructed by composing a static algorithm for the problem at hand with a simulation technique that converts static algorithms to dynamic ones. We argue that this powerful methodology does not provide satisfactory solutions for many important dynamic distributed problems, and this necessitates developing algorithms for these problems from scratch. In this paper we develop a fully dynamic distributed algorithm for maintaining sparse spanners. Our algorithm improves drastically the quiescence time of the state-of-the-art algorithm for the problem. Moreover, we show that the quiescence time of our algorithm is optimal up to a small constant factor. In addition, our algorithm improves significantly upon the state-of-the-art algorithm in all efficiency parameters, specifically, it has smaller quiescence message and space complexities, and smaller local processing time. Finally, our algorithm is self-contained and fairly simple, and is, consequently, amenable to implementation on unsophisticated network devices.