9 research outputs found
Restorable Shortest Path Tiebreaking for Edge-Faulty Graphs
The restoration lemma by Afek, Bremler-Barr, Kaplan, Cohen, and Merritt
[Dist. Comp. '02] proves that, in an undirected unweighted graph, any
replacement shortest path avoiding a failing edge can be expressed as the
concatenation of two original shortest paths. However, the lemma is
tiebreaking-sensitive: if one selects a particular canonical shortest path for
each node pair, it is no longer guaranteed that one can build replacement paths
by concatenating two selected shortest paths. They left as an open problem
whether a method of shortest path tiebreaking with this desirable property is
generally possible.
We settle this question affirmatively with the first general construction of
restorable tiebreaking schemes. We then show applications to various problems
in fault-tolerant network design. These include a faster algorithm for subset
replacement paths, more efficient fault-tolerant (exact) distance labeling
schemes, fault-tolerant subset distance preservers and additive spanners
with improved sparsity, and fast distributed algorithms that construct these
objects. For example, an almost immediate corollary of our restorable
tiebreaking scheme is the first nontrivial distributed construction of sparse
fault-tolerant distance preservers resilient to three faults
Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles
Given a graph with a source vertex , the Single Source Replacement Paths
(SSRP) problem is to compute, for every vertex and edge , the length
of a shortest path from to that avoids . A Single-Source
Distance Sensitivity Oracle (Single-Source DSO) is a data structure that
answers queries of the form by returning the distance . We
show how to deterministically compress the output of the SSRP problem on
-vertex, -edge graphs with integer edge weights in the range into
a Single-Source DSO of size with query time
. The space requirement is optimal (up to the word size) and
our techniques can also handle vertex failures.
Chechik and Cohen [SODA 2019] presented a combinatorial, randomized
time SSRP algorithm for undirected and
unweighted graphs. Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020]
gave an algebraic, randomized time SSRP algorithm
for graphs with integer edge weights in the range , where
is the matrix multiplication exponent. We derandomize both algorithms for
undirected graphs in the same asymptotic running time and apply our compression
to obtain deterministic Single-Source DSOs. The
and preprocessing times are polynomial improvements
over previous -space oracles.
On sparse graphs with edges, for any
constant , we reduce the preprocessing to randomized
time. This is
the first truly subquadratic time algorithm for building Single-Source DSOs on
sparse graphs.Comment: Full version of a paper to appear at ESA 2021. Abstract shortened to
meet ArXiv requirement
Fault-Tolerant ST-Diameter Oracles
We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D? of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch ? ? 1 if diam(G-F,S,T) ? D? ? ? diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures.
We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature.
We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ? 2 and stretch less than 5/3 requires ?(n^{3/2}) bits of space, regardless of the query time
Network creation games: anarchy and dynamics
En aquest projecte hem dut a terme recerca teĂČrica i empĂrica de propietats topolĂČgiques sobre el model clĂ ssic de Jocs de CreaciĂł de Xarxes introduĂŻt per Fabrikant et al. al voltant de la conjectura de l'arbre i la conjectura del PoA constant.In this project we conduct theoretical and empirical research of topological properties for the classic model of Network Creation Games introduced by Fabrikant et al. around the Tree Conjecture and the Constat PoA Conjecture
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum