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Finding shortest non-separating and non-contractible cycles for topologically embedded graphs
- In Proceedings 13th European Symp. Algorithms
, 2005
"... Abstract. We present an algorithm for finding shortest surface nonseparating cycles in graphs with given edge-lengths that are embedded on surfaces. The time complexity is O(g 3/2 V 3/2 log V + g 5/2 V 1/2 ), where V is the number of vertices in the graph and g is the genus of the sur- This result ..."
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Cited by 46 (9 self)
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Abstract. We present an algorithm for finding shortest surface nonseparating cycles in graphs with given edge-lengths that are embedded on surfaces. The time complexity is O(g 3/2 V 3/2 log V + g 5/2 V 1/2 ), where V is the number of vertices in the graph and g is the genus of the sur- This result can be applied for computing the (non-separating) facewidth of embedded graphs. Using similar ideas we provide the first nearlinear running time algorithm for computing the face-width of a graph embedded on the projective plane, and an algorithm to find the facewidth of embedded toroidal graphs in O(V 5/4 log V ) time.
A New Approach to All-Pairs Shortest Paths on Real-Weighted Graphs
- Theoretical Computer Science
, 2003
"... 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 ..."
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Cited by 41 (3 self)
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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.
Transitive-closure spanners
, 2008
"... 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. These spanner ..."
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Cited by 35 (11 self)
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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. These spanners were studied implicitly in access control, property testing, and data structures, and properties of these spanners have been rediscovered over the span of 20 years. We bring these areas under the unifying framework of TC-spanners. We abstract the common task implicitly tackled in these diverse applications as the problem of constructing sparse TCspanners. We study the approximability of the size of the sparsest k-TC-spanner for a given digraph. Our technical contributions fall into three categories: algorithms for general digraphs,
Combining Speed-Up Techniques for Shortest-Path Computations
- In Proc. 3rd Workshop on Experimental and Efficient Algorithms. LNCS
, 2004
"... 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. ..."
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Cited by 29 (7 self)
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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.
Fast and accurate estimation of shortest paths in large graphs
- In Proceedings of Conference on Information and Knowledge Management (CIKM
, 2010
"... Computing shortest paths between two given nodes is a fundamental operation over graphs, but known to be nontrivial over large disk-resident instances of graph data. While a numberoftechniquesexistfor answeringreachabilityqueries and approximating node distances efficiently, determining actual short ..."
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Cited by 28 (1 self)
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Computing shortest paths between two given nodes is a fundamental operation over graphs, but known to be nontrivial over large disk-resident instances of graph data. While a numberoftechniquesexistfor answeringreachabilityqueries and approximating node distances efficiently, determining actual shortest paths (i.e. the sequence of nodes involved) is often neglected. However, in applications arising in massive online social networks, biological networks, and knowledge graphs it is often essential to find out many, if not all, shortest paths between two given nodes. In this paper, we address this problem and present a scalable sketch-based index structure that not only supports estimation of node distances, but also computes corresponding shortest paths themselves. Generating the actual path information allows for further improvements to the estimation accuracy of distances (and paths), leading to near-exact shortest-path approximations in real world graphs. We evaluate our techniques – implemented within a fully functional RDF graph database system – over large realworld social and biological networks of sizes ranging from tens of thousand to millions of nodes and edges. Experiments on several datasets show that we can achieve query response times providing several orders of magnitude speedup over traditional path computations while keeping the estimation errors between 0 % and 1 % on average.
Complex network measurements: Estimating the relevance of observed properties
- In INFOCOM 2008. 27th IEEE International Conference on Computer Communications, Joint Conference of the IEEE Computer and Communications Societies
, 2008
"... Abstract—Complex networks, modeled as large graphs, re-ceived much attention during these last years. However, data on such networks is only available through intricate measurement procedures. Until recently, most studies assumed that these proce-dures eventually lead to samples large enough to be r ..."
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Cited by 25 (3 self)
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Abstract—Complex networks, modeled as large graphs, re-ceived much attention during these last years. However, data on such networks is only available through intricate measurement procedures. Until recently, most studies assumed that these proce-dures eventually lead to samples large enough to be representative of the whole, at least concerning some key properties. This has crucial impact on network modeling and simulation, which rely on these properties. Recent contributions proved that this approach may be mis-leading, but no solution has been proposed. We provide here the first practical way to distinguish between cases where it is indeed misleading, and cases where the observed properties may be trusted. It consists in studying how the properties of interest evolve when the sample grows, and in particular whether they reach a steady state or not.
Efficient search ranking in social networks
- in CIKM, 2007
"... In social networks such as Orkut, www.orkut.com, a large portion of the user queries refer to names of other people. Indeed, more than 50 % of the queries in Orkut are about names of other users, with an average of 1.8 terms per query. Further, the users usually search for people with whom they main ..."
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Cited by 23 (1 self)
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In social networks such as Orkut, www.orkut.com, a large portion of the user queries refer to names of other people. Indeed, more than 50 % of the queries in Orkut are about names of other users, with an average of 1.8 terms per query. Further, the users usually search for people with whom they maintain relationships in the network. These relationships can be modelled as edges in a friendship graph, a graph in which the nodes represent the users. In this context, search ranking can be modelled as a function that depends on the distances among users in the graph, more specifically, of shortest paths in the friendship graph. However, applica-tion of this idea to ranking is not straightforward because the large size of modern social networks (dozens of millions of users) prevents efficient computation of shortest paths at query time. We overcome this by designing a ranking for-mula that strikes a balance between producing good results and reducing query processing time. Using data from the Orkut social network, which includes over 40 million users, we show that our ranking, augmented by this new signal, produces high quality results, while maintaining query pro-cessing time small.
Efficient algorithms for constructing (1 + ɛ, β)-spanners in the distributed and streaming models (Extended Abstract)
- PODC
, 2004
"... 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 ..."
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Cited by 20 (6 self)
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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
