641 research outputs found
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
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
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
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
Conditional Hardness for Sensitivity Problems
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity.
In this paper we prove conditional lower bounds for these and additional problems in a sensitivity setting. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3-varepsilon)-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We extend our reduction from BMM to Diameter to give a reduction from All Pairs Shortest Paths to Diameter under one deletion in weighted graphs. This is intriguing, as in the static setting it is a big open problem whether Diameter is as hard as APSP. We further get a nearly tight lower bound for shortest paths after two edge deletions based on the APSP conjecture. We give more lower bounds under the Strong Exponential Time Hypothesis. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required.
Finally, we give the first algorithm for the (1+varepsilon)-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain
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
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