We present an algorithm for finding shortest surface non-separating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and Har-Peled. We also give algorithms to find a shortest non-contractible cycle in O(g O(g) V 3/2) time, which improves previous results for fixed genus. This result can be applied for computing the (non-separating) face-width of embedded graphs. Using similar ideas we provide the first near-linear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the face-width of embedded toroidal graphs in O(V 5/4 log V) time. 1

We present a new all-pairs shortest path algorithm that works with real-weighted graphs in the traditional comparison-addition model. It runs in O(mn+n time, improving on the long-standing bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.

Computing a shortest path from one node to another in a directed graph is a very common task in practice. This problem is classically solved by Dijkstra's algorithm. Many techniques are known to speed up this algorithm heuristically, while optimality of the solution can still be guaranteed. In most studies, such techniques are considered individually.

For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)-spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there exists an integer β = β(ɛ, κ) such that for every n-vertex graph G there exists a (1 + ɛ, β)-spanner G ′ with O(n 1+1/κ) edges. An efficient distributed protocol for constructing (1+ ɛ, β)-spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n 1+ρ) and O(|E|n ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n ρ) as opposed to O(n 1+ρ)) for constructing (1 + ɛ, β)-spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18]. We also show that our protocol for constructing (1+ɛ, β)spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O(n 1+1/κ · log n) bits of space for computing allpairs-almost-shortest-paths of length at most by a multiplicative factor (1 + ɛ) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n ρ), for an arbitrarily small ρ> 0. The only

Abstract. We construct a readable, compact and efficient implementation of Dijkstra’s shortest path algorithm and Fibonacci heaps using Constraint Handling Rules (CHR), which is increasingly used as a high-level rule-based general-purpose programming language. We measure its performance in different CHR systems, investigating both the theoretical asymptotic complexity and the constant factors realized in practice. 1

We evaluate the practical eciency of a new shortest path algorithm for undirected graphs which was developed by the rst two authors. This algorithm works on the fundamental comparison-addition model.

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Abstract. Let G(V, E) be an unweighted undirected graph on |V | = n vertices. Let δ(u, v) denote the shortest distance between vertices u, v ∈ V. An algorithm is said to compute all-pairs t-approximate shortestpaths/distances, for some t ≥ 1, if for each pair of vertices u, v ∈ V, the path/distance reported by the algorithm is not longer/greater than t · δ(u, v). This paper presents two randomized algorithms for computing allpairs nearly 2-approximate distances. The first algorithm takes expected O(m 2/3 n log n+n 2) time, and for any u, v ∈ V reports distance no greater than 2δ(u, v) + 1. Our second algorithm requires expected O(n 2 log 3/2) time, and for any u, v ∈ V reports distance bounded by 2δ(u, v) + 3. This paper also presents the first expected O(n 2) time algorithm to compute all-pairs 3-approximate distances. 1

We present a faster all-pairs shortest paths algorithm for arbitrary real-weighted directed graphs.

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-sized edges, both in the topology-independent node-name model. 1 These are the first topology-independent results that apply to routing in directed networks.