We consider the problem of name-independent routing in doubling metrics. A doubling metric is a metric space whose doubling dimension is a constant, where the doubling dimension of a metric space is the least value α such that any ball of radius r can be covered by at most 2 α balls of radius r/2. Given any δ> 0 and a weighted undirected network G whose shortest path metric d is a doubling metric with doubling dimension α, we present a name-independent routing scheme for G with (9+δ)-stretch, (2+ 1 δ)O(α) (log ∆) 2 (log n)bit routing information at each node, and packet headers of size O(log n), where ∆ is the ratio of the largest to the smallest shortest path distance in G. In addition, we prove that for any ǫ ∈ (0, 8), there is a doubling metric network G with n nodes, doubling dimension α ≤ 6 − log ǫ, and ∆ = O(2 1/ǫ n) such that any name-independent routing scheme on G with routing information at each node of size o(n (ǫ/60)2)-bits has stretch larger than 9 − ǫ. Therefore assuming that ∆ is bounded by a polynomial on n, our algorithm basically achieves optimal stretch for name-independent routing in doubling metrics with packet header size and routing information at each node both bounded by a polylogarithmic function of n.

Let G = (V (G), E(G)) be a weighted directed graph and let P be a shortest path from s to t in G. In the replacement paths problem we are required to compute for every edge e in P, the length of a shortest path from s to t that avoids e. The fastest known algorithm for solving the problem in weighted directed graphs is the trivial one: each edge in P is removed from the graph in its turn and the distance from s to t in the modified graph is computed. The running time of this algorithm is O � mn + n2 log n � , where n = |V (G) | and m = |E(G)|. The replacement paths problem is strongly motivated by two different applications. First, the fastest algorithm to compute the k simple shortest paths from s to t in directed graphs [21, 13] repeatedly computes the replacement paths from s to t. Its running time is O(kn(m + n log n)). Second, the computation of Vickrey pricing of edges in distributed networks can be reduced to the replacement paths problem. An open question raised by Nisan and Ronen [16] asks whether it is possible to compute the Vickrey pricing faster than the trivial algorithm described in the previous paragraph. In this paper we present a near-linear time algorithm for computing replacement paths in

One of the fundamental trade-offs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the maximum ratio over all pairs between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. All previous routing schemes required storage that is dependent on the diameter of the network. We present a new scale-free routing scheme, whose storage and header sizes are independent of the aspect ratio of the network. Our scheme is based on a decomposition into sparse and dense neighborhoods. Given an undirected network with arbitrary weights and n arbitrary node names, for any integer k ≥ 1 we present the first scale-free routing scheme with asymptotically optimal space-stretch tradeoff that does not require edge weights to be polynomially bounded. The scheme uses � O(n 1/k) space routing table at each node, and routes along paths of asymptotically optimal linear stretch O(k).

Algorithms for route planning in transportation networks have recently undergone a rapid development, leading to methods that are up to three million times faster than Dijkstra’s algorithm. We give an overview of the techniques enabling this development and point out frontiers of ongoing research on more challenging variants of the problem that include dynamically changing networks, time-dependent routing, and flexible objective functions.

We consider compact routing schemes in networks of low doubling dimension, where the doubling dimension is the least value α such that any ball in the network can be covered by at most 2 α balls of half radius. There are two variants of routing scheme design: (i) labeled (name-dependent) routing, where the designer is allowed to rename the nodes so that the names (labels) can contain additional routing information, e.g. topological information; and (ii) name-independent routing, which works on top of the arbitrary original node names in the network, i.e. the node names are independent of the routing scheme. In this paper, given any constant ǫ ∈ (0, 1), and an n-node weighted network of low doubling dimension α ∈ O(loglog n), we present • A (1+ǫ)-stretch labeled compact routing scheme with ⌈log n⌉-bit routing labels, O(log 2 � n/log log n)bit packet headers, and-bit routing information at each node; ( 1 ǫ)O(α) log 3 n • A (9 + ǫ)-stretch name-independent compact routing scheme with O(log 2 � n/log log n)-bit packet headers, and-bit routing information at each node. ( 1 ǫ)O(α) log 3 n In addition, we also prove a lower bound: any name-independent routing scheme with o(n (ǫ/60)2) bits of storage at each node has stretch no less than 9 −ǫ, for any ǫ ∈ (0, 8). Therefore our name-independent routing scheme achieves asymptotically optimal stretch with polylogarithmic storage at each node and packet headers. Note that both schemes are scale-free in the sense that their space requirements do not depend on the normalized diameter ∆ of the network. We also present a simpler non-scale-free (9 + ǫ)-stretch name-independent compact routing scheme with improved space requirements if ∆ is polynomial in n. 1

This paper studies labeling schemes for the vertex connectivity function on general graphs. We consider the problem of labeling the nodes of any n-node graph is such a way that given the labels of two nodes u and v, one can decide whether u and v are k-vertex connected in G, i.e., whether there exist k vertex disjoint paths connecting u and v. The paper establishes an upper bound of k 2 log n on the number of bits used in a label. The best previous upper bound for the label size of such labeling scheme is 2 k log n.

Distance labeling schemes are schemes that label the vertices of a graph with short labels in such a way that the distance between any two vertices u and v can be determined efficiently (e.g., in constant or logarithmic time) by merely inspecting the labels of u and v, without using any other information. Similarly, routing labeling schemes are schemes that label the vertices of a graph with short labels in such a way that given the label of a source vertex and the label of a destination, it is possible to compute efficiently (e.g., in constant or logarithmic time) the port number of the edge from the source that heads in the direction of the destination. In this paper we show that the three major classes of nonpositively curved plane graphs enjoy such distance and routing labeling schemes using O(log n) bit labels on n-vertex graphs. In constructing these labeling schemes interesting metric properties of those graphs are employed.

When you drive to somewhere ‘far away’, you will leave your current location via one of only a few ‘important’ traffic junctions. Starting from this informal observation, we develop an algorithmic approach—transit node routing— that allows us to reduce quickest-path queries in road networks to a small number of table lookups. We present two implementations of this idea, one based on a simple grid data structure and one based on highway hierarchies. For the road map of the United States, our best query times improve over the best previously published figures by two orders of magnitude. Our results exhibit various trade-offs between average query time (5 µs to 63 µs), preprocessing time (59 min to 1200 min), and storage overhead (21 bytes/node to 244 bytes/node).

We define a vertex labelling for every planar 3-connected graph with n vertices from which one can answer connectivity queries. A connectivity query asks whether there exists in the given graph a path linking u and v that avoids a set F of edges and a set X of vertices. The vertices u,v and those of X are given by their labels. The edges of F are given by the labels of their ends. Each label has a size of O(log(n)) bits. Our construction makes an essential use of straight-line embeddings on n × n grids of simple loop-free planar graphs. Such embeddings can be constructed in linear time by Schnyder’s algorithm [7].

Given an arbitrary real constant ε> 0, and a geometric graph G in d-dimensional Euclidean space with n points, O(n) edges, and constant dilation, our main result is a data structure that answers (1 + ε)-approximate shortest path length queries in constant time. The data structure can be constructed in O(n log n) time using O(n log n) space. This represents the first data structure that answers (1 + ε)-approximate shortest path queries in constant time, and hence functions as an approximate distance oracle. The data structure is also applied to several other problems. In particular, we also show that approximate shortest path queries between vertices in a planar polygonal domain with “rounded ” obstacles can be answered in constant time. Other applications include query versions of closest pair problems, and the efficient computation of the approximate dilations of geometric graphs. Finally, we show how to extend the main result to answer (1 + ε)approximate shortest path length queries in constant time for geometric spanner graphs with m = ω(n) edges. The resulting data structure can be constructed in O(m + n log n) time using O(n log n) space.