In this paper we study local routing algorithms on geometric networks. Formally speaking, suppose that we want to travel from a vertex s to a vertex t of a geometric network. A routing algorithm is called a local routing algorithm if it satisfies the following conditions:

We present the first universal compact routing algorithm with maximum stretch bounded by 3 that uses sublinear space at every vertex. The algorithm uses local routing tables of size O(n 2=3 log 4=3 n) and achieves paths that are most 3 times the length of the shortest path distances for all nodes in an arbitrary weighted undirected network. This answers an open question of Gavoille and Gengler who showed that any universal compact routing algorithm with maximum stretch strictly less than 3 must use\Omega\Gamma n) local space at some vertex. 1 Introduction Let G = (V; E) with jV j = n be a labeled undirected network. Assuming that a positive cost, or distance is assigned with each edge, the stretch of path p(u; v) from node u to node v is defined as jp(u;v)j jd(u;v)j , where jd(u; v)j is the length of the shortest u \Gamma v path. The approximate all-pairs shortest path problem involves a tradeoff of stretch against time-- short paths with stretch bounded by a constant are com...

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

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.

We study the problem of routing in doubling metrics, and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(X) at most α if every set of diameter D can be covered by 2 α sets of diameter D/2. (A doubling metric is one whose doubling dimension dim(X) is a constant.) We show how to perform (1 + τ)-stretch routing on metrics for any 0 < τ ≤ 1 with routing tables of size at most (α/τ) O(α) log 2 ∆ bits with only (α/τ) O(α) log ∆ entries, where ∆ is the diameter of the graph; hence the number of routing table entries is just τ −O(1) log ∆ for doubling metrics. These results extend and improve on those of Talwar (2004). We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables above; for τ> 0, we give algorithms to construct (1 + τ)stretch spanners for a metric (X, d) with maximum degree at most (2 + 1/τ) O(dim(X)) , matching the results of Das et al. for Euclidean metrics.

We consider the problem of labeling the nodes of an n-node graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and AT-free graphs). We also show lower bounds, and prove that most of our schemes are optimal in length of labels generated and in the quality of the approximation, leaving some open problems.

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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.

In this paper, we deal with the compact routing problem, that is implementing routing schemes that use a minimum memory size on each router. A universal routing scheme is a scheme that applies to all n-node networks. In [31], Peleg and Upfal showed that one can not implement a universal routing scheme with less than a total of \Omega\Gamma n 1+1=(2s+4) ) memory bits for any given stretch factor s 1. We improve this bound for stretch factors s, 1 s ! 2, by proving that any near-shortest path universal routing scheme uses a total of \Omega\Gamma n 2 ) memory bits in the worst-case. This result is obtained by counting the minimum number of routing functions necessary to route on all n-node networks. Moreover, and more fundamentally, we give a tight bound of \Theta(n log n) bits for the local minimum memory requirement of universal routing scheme of stretch factors s, 1 s ! 2. More precisely, for any fixed constant ", 0 ! " ! 1, there exists an n-node network G on which at least \O...

This paper deals with compact shortest path routing tables on weighted graphs with n nodes. For planar graphs we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log 2+ n) bit-operations per node to extract the route, for any constant > 0. We obtain the same bounds for graphs of crossing-edge number bounded by o(n= log n), and we generalize for graphs of genus bounded by > 0 yielding a size of n log +O(n) bits per node. Actually we prove a sharp upper bound of 2n log k +O(n) for graphs of pagenumber k, and a lower bound of n log k o(n log k) bits. These results are obtained by the use of dominating sets, compact coding of non-crossing partitions, and k-page representation of graphs.