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### All-pairs shortest paths with a sublinear additive error

Liam Roditty, Asaf Shapira
2011 ACM Transactions on Algorithms
Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error.  ...  pair of vertices u, v in the graph to within an additive error δ p (u, v), where δ(u, v) is the exact length of the shortest path between u and v.  ...  all we can hope for is to obtain a polynomially small additive error.  ...

### All-Pairs Shortest Paths with a Sublinear Additive Error [chapter]

Liam Roditty, Asaf Shapira
2008 Lecture Notes in Computer Science
Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error.  ...  pair of vertices u, v in the graph to within an additive error δ p (u, v), where δ(u, v) is the exact length of the shortest path between u and v.  ...  all we can hope for is to obtain a polynomially small additive error.  ...

### Breaking the Linear Error Barrier in Differentially Private Graph Distance Release [article]

Chenglin Fan, Ping Li, Xiaoyun Li
2022 arXiv   pre-print
By adding a set of shortcut edges to the original graph, we show that any node pair has a shortest path with link length Õ(n^1/2).  ...  Computing all pairwise distances on the constructed graph only introduces Õ(n^1/2) error in answering all pairwise shortest path distances for fixed privacy parameter.  ...  Is this linear dependence on n inherent, or is it possible to release all-pairs distances with error sublinear in n ?"  ...

### A Hierarchy of Lower Bounds for Sublinear Additive Spanners [article]

Amir Abboud, Greg Bodwin, Seth Pettie
2017 arXiv   pre-print
Specifically, for any integer k> 2, any compression scheme with size O(n^1+1/2^k-1 - ϵ) has a sublinear additive stretch function f: f(d) = d + Ω(d^1-1/k).  ...  This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators.  ...  We thank Michael Elkin for proposing the question of finding a lower bound hierarchy for mixed spanners (as shown here), as well as observing the corresponding upper bounds.  ...

### Sublinear-Space Distance Labeling using Hubs [article]

Paweł Gawrychowski, Adrian Kosowski, Przemysław Uznański
2016 arXiv   pre-print
For a queried pair of nodes (u,v), the length of a shortest u-v-path passing through a hub node from S(u)∩ S(v) is then used as an upper bound on the distance between u and v.  ...  To our knowledge, this is the first additive scheme with constant absolute error to use labels of sublinear size. The corresponding decoding time is then small (any T=ω(1) is sufficient).  ...  provide a shortest path cover with sufficiently high probability.  ...

### Sublinear Distance Labeling [article]

Stephen Alstrup and Søren Dahlgaard and Mathias Bæk Tejs Knudsen and Ely Porat
2016 arXiv   pre-print
For approximate r-additive labeling schemes, that return distances within an additive error of r we show a scheme of size O ( n/r·polylog (r n)/ n ) for r > 2.  ...  A D-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least D from each other.  ...  Consider a pair of nodes u, v P V with dist G pu, vq ě D. Let p be a shortest path between u and v, then |p| ě D.  ...

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

P. N. Klein, S. Subramanian
1998 Algorithmica
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.  ...  In this paper we give a fully dynamic approximation scheme for maintaining all-pairs shortest paths in planar networks.  ...  We are grateful to the referees for their comments and for pointing out an error in an earlier version of the document.  ...

### Graph Spanners: A Tutorial Review [article]

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Richard Spence
2020 arXiv   pre-print
As an additional contribution, we offer a list of open problems on graph spanners.  ...  This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners.  ...  In fact, the latter construction has a stronger property: for each node pair u, v, the additive error scales with W = W (u, v) the maximum edge weight along the true u v shortest path in the input graph  ...

### Analysis and Experimental Evaluation of Time-Dependent Distance Oracles [chapter]

Spyros Kontogiannis, George Michalopoulos, Georgia Papastavrou, Andreas Paraskevopoulos, Dorothea Wagner, Christos Zaroliagis
2014 2015 Proceedings of the Seventeenth Workshop on Algorithm Engineering and Experiments (ALENEX)
The core of this preprocessing phase is based on a novel, quite efficient and simple oneto-all approximation method for creating approximations of shortest travel-time functions.  ...  We conduct an extensive, comparative experimental study with all query algorithms and six landmark sets.  ...  The relative error for a given od−path p is the percentage of surplus from the exact shortest travel-time (as computed by TDD), i.e.: With respect to the observed query times, in all cases FCA is the fastest  ...

### Spanners and emulators with sublinear distance errors

Mikkel Thorup, Uri Zwick
2006 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06
These are the first such results with additive error terms of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated.  ...  We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G = (V, E ) with O(kn 1+1/k ) edges such that for any two vertices .  ...  distances with a sublinear error term in the distance.  ...

### Distance Oracles for Time-Dependent Networks [article]

Spyros Kontogiannis, Christos Zaroliagis
2015 arXiv   pre-print
Our oracle uses subquadratic space and time preprocessing, and provides two sublinear-time query algorithms that deliver constant and (1+σ)-approximate shortest-travel-times, respectively, for arbitrary  ...  origin-destination pairs in the network, for any constant σ > ϵ.  ...  Introduction Distance oracles are succinct data structures encoding shortest path information among a carefully selected subset of pairs of vertices in a graph.  ...

### Approximate Online Pattern Matching in Sublinear Time

Diptarka Chakraborty, Debarati Das, Michal Koucký, Michael Wagner
2019 Foundations of Software Technology and Theoretical Computer Science
We design an algorithm that upon arrival of the t-th symbol of T computes kt approximately within O(1)multiplicative factor and m 8/9 -additive error.  ...  The goal is to find all the positions j in T such that there is a substring of T ending at j which has edit distance at most k from the pattern P .  ...  Acknowledgements Authors would like to thank anonymous reviewers for many helpful suggestions and comments on an earlier version of this paper. 10:14 Approximate Online Pattern Matching in Sublinear  ...

### The 4/3 additive spanner exponent is tight

Amir Abboud, Greg Bodwin
2016 Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2016
Intuitively, it takes two graphs G, H and produces a new graph G ⊗ H whose shortest paths structure looks locally like H but globally like G.  ...  of additive error is allowed.  ...  G is the union of many shortest paths between a set of node pairs P , with the following properties: • Each pair p ∈ P has distance k • Each pair p ∈ P has a unique shortest path between its endpoints  ...

### The 4/3 Additive Spanner Exponent is Tight [article]

Amir Abboud, Greg Bodwin
2020 arXiv   pre-print
of additive error is allowed.  ...  A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error.  ...  G is the union of many shortest paths between a set of node pairs P , with the following properties: • Each pair p ∈ P has distance k • Each pair p ∈ P has a unique shortest path between its endpoints  ...

### Preserving Distances in Very Faulty Graphs [article]

Greg Bodwin, Fabrizio Grandoni, Merav Parter, Virginia Vassilevska Williams
2017 arXiv   pre-print
Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively.  ...  We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^2-ϵ) for ϵ>0, then the additive error has to be Ω( f).  ...  additive error n δ .  ...
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