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### Approximating Shortest Paths in Graphs [chapter]

Sandeep Sen
2009 Lecture Notes in Computer Science
In contrast, there has been a growing interest in the area of algorithms for approximate shortest paths leading to many interesting variations of the original problem.  ...  In this article, we trace some of the fundamental developments like spanners and distance oracles, their underlying constructions, as well as their applications to the approximate all-pairs shortest paths  ...  Compute shortest paths from a large set of vertices in a sparse sub graph 2. Compute shortest paths from a small set of vertices in a dense sub-graph.  ...

### Approximate shortest paths in weighted graphs

Raphael Yuster
2012 Journal of computer and system sciences (Print)
We present an approximation algorithm for the all pairs shortest paths (APSP) problem in weighed graphs.  ...  Hence, if ω = 2+o(1), this algorithm is as fast as any algorithm known for integer APSP in directed graphs, and is more general.  ...  In the shortest paths problem we are given a (possibly weighted, possibly directed) graph G = (V , E) and a set S ⊂ V × V of pairs of vertices, and are required to find distances and shortest paths connecting  ...

### Counting approximately-shortest paths in directed acyclic graphs [article]

Matúš Mihalák and Rastislav Šrámek and Peter Widmayer
2013 arXiv   pre-print
path in a purpose-built auxiliary graph.  ...  That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L_1 and L_2, we show how to approximately count the s-t paths that have  ...  Concluding Remarks We have shown that there is an efficient algorithm to approximate the number of approximately shortest paths in a directed acyclic graph.  ...

### Counting Approximately-Shortest Paths in Directed Acyclic Graphs [chapter]

Matúš Mihalák, Rastislav Šrámek, Peter Widmayer
2014 Lecture Notes in Computer Science
path in a purpose-built auxiliary graph.  ...  That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights L1 and L2, we show how to approximately count the s-t paths that have length  ...  Concluding Remarks We have shown that there is an efficient algorithm to approximate the number of approximately shortest paths in a directed acyclic graph.  ...

### Fuzzy Shortest Paths Approximation For Solving The Fuzzy Steiner Tree Problem In Graphs

Miloš Šeda
2008 Zenodo
We propose a modification of the shortest paths approximation based on the fuzzy shortest paths (FSP) evaluations.  ...  In this paper, we deal with the Steiner tree problem (STP) on a graph in which a fuzzy number, instead of a real number, is assigned to each edge.  ...  SHORTEST PATHS APPROXIMATION The shortest paths approximation  for solving the Steiner tree problem in graphs is an analogue of Jarník's algorithm for finding a minimum spanning tree.  ...

### Approximate Shortest Path Queries in Graphs Using Voronoi Duals

Shinichi Honiden, Michael E. Houle, Christian Sommer, Martin Wolff
2009 2009 Sixth International Symposium on Voronoi Diagrams
We propose an approximation method to answer point-to-point shortest path queries in undirected graphs, based on random sampling and Voronoi duals.  ...  This overlay graph allows for fast computation of approximate shortest paths for general, undirected graphs. The time-quality tradeoff decision can be made at query time.  ...  ACKNOWLEDGMENTS We thank Georg Troxler (staila technologies) for providing us with a machine for our experiments, and Daniel Delling for supplying the public transportation graphs.  ...

### A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs

P. N. Klein, S. Subramanian
1998 Algorithmica
In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks.  ...  Given an error parameter ε such that 0 < ε, our algorithm maintains approximate all-pairs shortest paths in an undirected planar graph G with nonnegative edge lengths.  ...  We are grateful to the referees for their comments and for pointing out an error in an earlier version of the document.  ...

### Approximation Algorithms and Hardness Results for Shortest Path Based Graph Orientations [chapter]

Dima Blokh, Danny Segev, Roded Sharan
2012 Lecture Notes in Computer Science
be approximately shortest.  ...  vertex pairs, assign directions to the edges so as to maximize the number of pairs that are connected by a shortest (in the original graph) directed path.  ...  Exact shortest paths To tackle MSPO, we adapt the approximation algorithm of Elberfeld et al.  , which was initially suggested for MGO in mixed graphs.  ...

### Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs

Surender Baswana, Telikepalli Kavitha
2010 SIAM journal on computing (Print)
We present a simple, novel and generic scheme for all-pairs approximate shortest paths.  ...  Computing all-pairs small stretch distances efficiently (both in terms of time and space) is a well-studied problem in graph algorithms.  ...  In this paper, we presented faster algorithms for all-pairs approximate shortest paths in weighted undirected graphs.  ...

### Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs

Timothy M. Chan, Dimitrios Skrepetos, Marc Herbstritt
2018 International Symposium on Computational Geometry
All the above approximation results for planar graphs rely heavily on the concept of shortest-path separator: a set of shortest paths with common root, such that the removal of their vertices decomposes  ...  In a seminal paper, Gao and Zhang  obtained the first nontrivial set of results on shortest-path problems in weighted unit-disk graphs, by adapting a familiar technique in computational geometry -namely  ...  S o C G 2 0 1 8 24:12 Approximate Shortest Paths and Distance Oracles in Weighted Unit-Disk Graphs Theorem 15 (Approximate apBLSP).  ...

### Approximate shortest paths and distance oracles in weighted unit-disk graphs

Timothy M. Chan, Dimitrios Skrepetos
2018 Journal of Computational Geometry
As a further application we employ our distance oracle, along with additional ideas, to solve the $(1+\eps)$-approximate \emph{all-pairs bounded-leg shortest paths\/} (apBLSP) problem for a set of $n$  ...  We also obtain similar new results for a number of related problems in the weighted unit-disk graph metric such as the radius and the bichromatic closest pair.  ...  We now discuss the technical challenges that arise when trying to address (1 + ε)approximate shortest-path problems in weighted unit-disk graphs.  ...

### A Nearly Optimal Algorithm for Approximating Replacement Paths and k Shortest Simple Paths in General Graphs [chapter]

Aaron Bernstein
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
We also present an even faster (1 + ) approximate algorithm for the simpler problem of approximating the k shortest simple s − t paths in a directed graph with positive edge weights.  ...  Let G = (V, E) be a directed graph with positive edge weights, let s, t be two specified vertices in this graph, and let π(s, t) be the shortest path between them.  ...   showed that approximation cannot help us beat the o(mn) bound for all pairs shortest paths in weighted, directed graphs.  ...

### A Logspace Approximation Scheme for the Shortest Path Problem for Graphs with Bounded Independence Number [chapter]

Till Tantau
2004 Lecture Notes in Computer Science
In contrast, for directed graphs, undirected graphs, and dags we cannot construct paths in logarithmic space (let alone approximate the shortest one), unless complexity class collapses occur.  ...  Concerning the second question, in this paper it is shown that not only can we construct paths in logarithmic space, but there even exists a logspace approximation scheme for this problem.  ...  In Section 3 a logspace approximation scheme for the shortest path problem for graphs with bounded independence number is presented.  ...

### Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

Siamak Tazari, Matthias Müller-Hannemann
2009 Discrete Applied Mathematics
We generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs.  ...  Afterwards, we show that using this algorithm, one can implement Mehlhorn's (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.  ...  Single-source shortest paths on minor-closed graph classes In this section, we prove our main theorem about shortest paths: Theorem 3.1.  ...

### The Borel Distribution: Approximation and Concentration [article]

Fraser Daly, Seva Shneer
2021 arXiv   pre-print
We develop the tools necessary to use Stein's method for approximation by a Borel distribution, which we illustrate by considering the approximation of the number of customers served in the busy period  ...  including some explicit error bounds in the corresponding approximation.  ...  The first is to develop the tools necessary in order to apply Stein's method for probability approximation to problems in which the Borel distribution is a natural limiting or approximating object.  ...
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