7 research outputs found
Deterministic Replacement Path Covering
In this article, we provide a unified and simplified approach to derandomize
central results in the area of fault-tolerant graph algorithms. Given a graph
, a vertex pair , and a set of edge faults , a replacement path is an - shortest path in
. For integer parameters , a replacement path covering
(RPC) is a collection of subgraphs of , denoted by
, such that for every set of at
most faults (i.e., ) and every replacement path of at
most edges, there exists a subgraph that
contains all the edges of and does not contain any of the edges of . The
covering value of the RPC is then defined to be the number
of subgraphs in .
We present efficient deterministic constructions of -RPCs whose
covering values almost match the randomized ones, for a wide range of
parameters. Our time and value bounds improve considerably over the previous
construction of Parter (DISC 2019). We also provide an almost matching lower
bound for the value of these coverings. A key application of our above
deterministic constructions is the derandomization of the algebraic
construction of the distance sensitivity oracle by Weimann and Yuster (FOCS
2010). The preprocessing and query time of the our deterministic algorithm
nearly match the randomized bounds. This resolves the open problem of Alon,
Chechik and Cohen (ICALP 2019)
Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers
Fault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied thoroughly mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distributed networks, not much is known on the distributed computational aspects of these structures.
In this paper, we present distributed algorithms for constructing fault tolerant distance preservers and +2 additive spanners that are resilient to at most two edge faults. Prior to our work, the only non-trivial constructions known were for the single fault and single source setting by [Ghaffari and Parter SPAA\u2716].
Our key technical contribution is a distributed algorithm for computing distance preservers w.r.t. a subset S of source vertices, resilient to two edge faults. The output structure contains a BFS tree BFS(s,G ? {e?,e?}) for every s ? S and every e?,e? ? G. The distributed construction of this structure is based on a delicate balance between the edge congestion (formed by running multiple BFS trees simultaneously) and the sparsity of the output subgraph. No sublinear-round algorithms for constructing these structures have been known before
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
Fixed-Parameter Sensitivity Oracles
We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter
tractability (FPT) to design sensitivity oracles for FPT graph problems. An
oracle with sensitivity for an FPT problem on a graph with
parameter preprocesses in time . When
queried with a set of at most edges of , the oracle reports the
answer to the -with the same parameter -on the graph , i.e.,
deprived of . The oracle should answer queries in a time that is
significantly faster than merely running the best-known FPT algorithm on
from scratch. We mainly design sensitivity oracles for the -Path and the
-Vertex Cover problem. Following our line of research connecting
fault-tolerant FPT and shortest paths problems, we also introduce
parameterization to the computation of distance preservers. We study the
problem, given a directed unweighted graph with a fixed source and
parameters and , to construct a polynomial-sized oracle that efficiently
reports, for any target vertex and set of at most edges, whether
the distance from to increases at most by an additive term of in
.Comment: 19 pages, 1 figure, abstract shortened to meet ArXiv requirements;
accepted at ITCS'2