46 research outputs found
Deterministic Combinatorial Replacement Paths and Distance Sensitivity Oracles
In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms.
For the replacement paths problem, let G = (V,E) be a directed unweighted graph with n vertices and m edges and let P be a shortest path from s to t in G. The replacement paths problem is to find for every edge e in P the shortest path from s to t avoiding e. Roditty and Zwick [ICALP 2005] obtained a randomized algorithm with running time of O~(m sqrt{n}). Here we provide the first deterministic algorithm for this problem, with the same O~(m sqrt{n}) time. Due to matching conditional lower bounds of Williams et al. [FOCS 2010], our deterministic combinatorial algorithm for the replacement paths problem is optimal up to polylogarithmic factors (unless the long standing bound of O~(mn) for the combinatorial boolean matrix multiplication can be improved). This also implies a deterministic algorithm for the second simple shortest path problem in O~(m sqrt{n}) time, and a deterministic algorithm for the k-simple shortest paths problem in O~(k m sqrt{n}) time (for any integer constant k > 0).
For the problem of distance sensitivity oracles, let G = (V,E) be a directed graph with real-edge weights. An f-Sensitivity Distance Oracle (f-DSO) gets as input the graph G=(V,E) and a parameter f, preprocesses it into a data-structure, such that given a query (s,t,F) with s,t in V and F subseteq E cup V, |F| <=f being a set of at most f edges or vertices (failures), the query algorithm efficiently computes the distance from s to t in the graph G F (i.e., the distance from s to t in the graph G after removing from it the failing edges and vertices F).
For weighted graphs with real edge weights, Weimann and Yuster [FOCS 2010] presented several randomized f-DSOs. In particular, they presented a combinatorial f-DSO with O~(mn^{4-alpha}) preprocessing time and subquadratic O~(n^{2-2(1-alpha)/f}) query time, giving a tradeoff between preprocessing and query time for every value of 0 < alpha < 1. We derandomize this result and present a combinatorial deterministic f-DSO with the same asymptotic preprocessing and query time
Space-Efficient Fault-Tolerant Diameter Oracles
We design -edge fault-tolerant diameter oracles (-FDOs). We preprocess
a given graph on vertices and edges, and a positive integer , to
construct a data structure that, when queried with a set of
edges, returns the diameter of .
For a single failure () in an unweighted directed graph of diameter ,
there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch
, constant query time, space , and a combinatorial
preprocessing time of .We
present an FDO for directed graphs with the same stretch, query time, and
space. It has a preprocessing time of .
The preprocessing time nearly matches a conditional lower bound for
combinatorial algorithms, also by Henzinger et al. With fast matrix
multiplication, we achieve a preprocessing time of . We further prove an information-theoretic lower bound
showing that any FDO with stretch better than requires bits
of space.
For multiple failures () in undirected graphs with non-negative edge
weights, we give an -FDO with stretch , query time ,
space, and preprocessing time . We
complement this with a lower bound excluding any finite stretch in
space. We show that for unweighted graphs with polylogarithmic diameter and up
to failures, one can swap approximation for query
time and space. We present an exact combinatorial -FDO with preprocessing
time , query time , and space . When using
fast matrix multiplication instead, the preprocessing time can be improved to
, where is the matrix multiplication
exponent.Comment: Full version of a paper to appear at MFCS'21. Abstract shortened to
meet ArXiv requirement
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
Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms
We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic "approximately-maximal" fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic (2+epsilon)-approximate integral matching algorithm with small polylog worst-case update time. For the (2+epsilon)-approximation regime only a fractional fully-dynamic (2+epsilon)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm
Molecular Detection of Bladder Cancer by Fluorescence Microsatellite Analysis and an Automated Genetic Analyzing System
To investigate the ability of an automated fluorescent analyzing system to detect microsatellite alterations, in patients with bladder cancer. We investigated 11 with pathology proven bladder Transitional Cell Carcinoma (TCC) for microsatellite alterations in blood, urine, and tumor biopsies. DNA was prepared by standard methods from blood, urine and resected tumor specimens, and was used for microsatellite analysis. After the primers were fluorescent labeled, amplification of the DNA was performed with PCR. The PCR products were placed into the automated genetic analyser (ABI Prism 310, Perkin Elmer, USA) and were subjected to fluorescent scanning with argon ion laser beams. The fluorescent signal intensity measured by the genetic analyzer measured the product size in terms of base pairs. We found loss of heterozygocity (LOH) or microsatellite alterations (a loss or gain of nucleotides, which alter the original normal locus size) in all the patients by using fluorescent microsatellite analysis and an automated analyzing system. In each case the genetic changes found in urine samples were identical to those found in the resected tumor sample. The studies demonstrated the ability to detect bladder tumor non-invasively by fluorescent microsatellite analysis of urine samples. Our study supports the worldwide trend for the search of non-invasive methods to detect bladder cancer. We have overcome major obstacles that prevented the clinical use of an experimental system. With our new tested system microsatellite analysis can be done cheaper, faster, easier and with higher scientific accuracy
Improved Approximate Distance Oracles: Bypassing the Thorup-Zwick Bound in Dense Graphs
Despite extensive research on distance oracles, there are still large gaps
between the best constructions for spanners and distance oracles. Notably,
there exist sparse spanners with a multiplicative stretch of
plus some additive stretch. A fundamental open problem is whether such a bound
is achievable for distance oracles as well. Specifically, can we construct a
distance oracle with multiplicative stretch better than 2, along with some
additive stretch, while maintaining subquadratic space complexity? This
question remains a crucial area of investigation, and finding a positive answer
would be a significant step forward for distance oracles. Indeed, such oracles
have been constructed for sparse graphs. However, in the more general case of
dense graphs, it is currently unknown whether such oracles exist.
In this paper, we contribute to the field by presenting the first distance
oracles that achieve a multiplicative stretch of along with a
small additive stretch while maintaining subquadratic space complexity. Our
results represent an advancement particularly for constructing efficient
distance oracles for dense graphs. In addition, we present a whole family of
oracles that, for any positive integer , achieve a multiplicative stretch of
using space
Deep Distance Sensitivity Oracles
One of the most fundamental graph problems is finding a shortest path from a
source to a target node. While in its basic forms the problem has been studied
extensively and efficient algorithms are known, it becomes significantly harder
as soon as parts of the graph are susceptible to failure. Although one can
recompute a shortest replacement path after every outage, this is rather
inefficient both in time and/or storage. One way to overcome this problem is to
shift computational burden from the queries into a pre-processing step, where a
data structure is computed that allows for fast querying of replacement paths,
typically referred to as a Distance Sensitivity Oracle (DSO). While DSOs have
been extensively studied in the theoretical computer science community, to the
best of our knowledge this is the first work to construct DSOs using deep
learning techniques. We show how to use deep learning to utilize a
combinatorial structure of replacement paths. More specifically, we utilize the
combinatorial structure of replacement paths as a concatenation of shortest
paths and use deep learning to find the pivot nodes for stitching shortest
paths into replacement paths.Comment: arXiv admin note: text overlap with arXiv:2007.11495 by other author