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Approximating Shortest Paths in Graphs
[chapter]

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

2012
*
Journal of computer and system sciences (Print)
*

We present an

doi:10.1016/j.jcss.2011.09.001
fatcat:6lcrvt5ytrfndbhfdq4q2g7rg4
*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]

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]

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

2008
*
Zenodo
*

We propose a modification of the

doi:10.5281/zenodo.1081662
fatcat:zixyalispfg7tjxpm2xyptvahy
*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*[12] 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

2009
*
2009 Sixth International Symposium on Voronoi Diagrams
*

We propose an

doi:10.1109/isvd.2009.30
dblp:conf/isvd/HonidenHSW09
fatcat:7n5gvkparvfm3mhr4xfinm5vea
*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

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]

2012
*
Lecture Notes in Computer Science
*

be

doi:10.1007/978-3-642-31265-6_6
fatcat:3kkfozydnjhqdcicqypsaelvd4
*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. [3] , which was initially suggested for MGO*in*mixed*graphs*. ...##
###
Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs

2010
*
SIAM journal on computing (Print)
*

We present a simple, novel and generic scheme for all-pairs

doi:10.1137/080737174
fatcat:d5ofwojusjeibda5nlatzhebde
*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

2018
*
International Symposium on Computational Geometry
*

All the above

doi:10.4230/lipics.socg.2018.24
dblp:conf/compgeom/ChanS18
fatcat:acijbbi4zve4tmyk25mdoqbzzq
*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 [13] 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

2018
*
Journal of Computational Geometry
*

As a further application we employ our distance oracle, along with additional ideas, to solve the $(1+\eps)$-

doi:10.20382/jocg.v10i2a2
dblp:journals/jocg/ChanS19a
fatcat:sl5fvcsbxfdlhodb5qclagjnle
*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]

2010
*
Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
*

We also present an even faster (1 + )

doi:10.1137/1.9781611973075.61
dblp:conf/soda/Bernstein10
fatcat:rttztkh7azf6ja3qhge7czk524
*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. ... [3] 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]

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

2009
*
Discrete Applied Mathematics
*

We generalize the linear-time

doi:10.1016/j.dam.2008.08.002
fatcat:u3a6452dkjgrrelg7nknyvb6ay
*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]

2021
*
arXiv
*
pre-print

We develop the tools necessary to use Stein's method for

arXiv:1912.03219v3
fatcat:w26qmwoyq5dwfhcbtrrwmox4lu
*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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