107 research outputs found
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
Deterministic Fully Dynamic SSSP and More
We present the first non-trivial fully dynamic algorithm maintaining exact
single-source distances in unweighted graphs. This resolves an open problem
stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019].
Previous fully dynamic single-source distances data structures were all
approximate, but so far, non-trivial dynamic algorithms for the exact setting
could only be ruled out for polynomially weighted graphs (Abboud and
Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main
case for which neither a subquadratic dynamic algorithm nor a quadratic lower
bound was known.
Our dynamic algorithm works on directed graphs, is deterministic, and can
report a single-source shortest paths tree in subquadratic time as well. Thus
we also obtain the first deterministic fully dynamic data structure for
reachability (transitive closure) with subquadratic update and query time. This
answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019].
Finally, using the same framework we obtain the first fully dynamic data
structure maintaining all-pairs -approximate distances within
non-trivial sub- worst-case update time while supporting optimal-time
approximate shortest path reporting at the same time. This data structure is
also deterministic and therefore implies the first known non-trivial
deterministic worst-case bound for recomputing the transitive closure of a
digraph.Comment: Extended abstract to appear in FOCS 202
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
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
Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form
Algebraic techniques have had an important impact on graph algorithms so far.
Porting them, e.g., the matrix inverse, into the dynamic regime improved
best-known bounds for various dynamic graph problems. In this paper, we develop
new algorithms for another cornerstone algebraic primitive, the Frobenius
normal form (FNF). We apply our developments to dynamic and fault-tolerant
exact distance oracle problems on directed graphs.
For generic matrices over a finite field accompanied by an FNF, we show
(1) an efficient data structure for querying submatrices of the first
powers of , and (2) a near-optimal algorithm updating the FNF explicitly
under rank-1 updates.
By representing an unweighted digraph using a generic matrix over a
sufficiently large field (obtained by random sampling) and leveraging the
developed FNF toolbox, we obtain: (a) a conditionally optimal distance
sensitivity oracle (DSO) in the case of single-edge or single-vertex failures,
providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a
multiple-failures DSO improving upon the state of the art (vd. Brand and
Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved
dynamic distance oracles in the case of single-edge updates, and (d) a dynamic
distance oracle supporting vertex updates, i.e., changing all edges incident to
a single vertex, in worst-case time and distance queries in
time.Comment: To appear at FOCS 202
Planar Reachability Under Single Vertex or Edge Failures
International audienceIn this paper we present an efficient reachability oracle under single-edge or single-vertex failures for planar directed graphs. Specifically, we show that a planar digraph G can be preprocessed in O(n log 2 n/log log n) time, producing an O(n log n)-space data structure that can answer in O(log n) time whether u can reach v in G if the vertex x (the edge f) is removed from G, for any query vertices u, v and failed vertex x (failed edge f). To the best of our knowledge, this is the first data structure for planar directed graphs with nearly optimal preprocessing time that answers all-pairs queries under any kind of failures in polylogarithmic time. We also consider 2-reachability problems, where we are given a planar digraph G and we wish to determine if there are two vertex-disjoint (edge-disjoint) paths from u to v, for query vertices u, v. In this setting we provide a nearly optimal 2-reachability oracle, which is the existential variant of the reachability oracle under single failures, with the following bounds. We can construct in O(n polylog n) time an O(n log 3+o(1) n)-space data structure that can check in O(log 2+o(1) n) time for any query vertices u, v whether v is 2-reachable from u, or otherwise find some separating vertex (edge) x lying on all paths from u to v in G. To obtain our results, we follow the general recursive approach of Thorup for reachability in planar graphs [J. ACM '04] and we present new data structures which generalize dominator trees and previous data structures for strong-connectivity under failures [Georgiadis et al., SODA '17]. Our new data structures work also for general digraphs and may be of independent interest
New Extremal Bounds for Reachability and Strong-Connectivity Preservers Under Failures
In this paper, we consider the question of computing sparse subgraphs for any
input directed graph on vertices and edges, that preserves
reachability and/or strong connectivity structures.
We show bound on a
subgraph that is an -fault-tolerant reachability preserver for a given
vertex-pair set , i.e., it preserves reachability
between any pair of vertices in under single edge (or vertex)
failure. Our result is a significant improvement over the previous best bound obtained as a corollary of single-source reachability
preserver construction. We prove our upper bound by exploiting the special
structure of single fault-tolerant reachability preserver for any pair, and
then considering the interaction among such structures for different pairs.
In the lower bound side, we show that a 2-fault-tolerant reachability
preserver for a vertex-pair set of size
, for even any arbitrarily small , requires at
least edges. This refutes the existence of
linear-sized dual fault-tolerant preservers for reachability for any polynomial
sized vertex-pair set.
We also present the first sub-quadratic bound of at most size, for strong-connectivity preservers of directed graphs under
failures. To the best of our knowledge no non-trivial bound for this
problem was known before, for a general . We get our result by adopting the
color-coding technique of Alon, Yuster, and Zwick [JACM'95]
Fast Deterministic Fully Dynamic Distance Approximation
In this paper, we develop deterministic fully dynamic algorithms for
computing approximate distances in a graph with worst-case update time
guarantees. In particular, we obtain improved dynamic algorithms that, given an
unweighted and undirected graph undergoing edge insertions and
deletions, and a parameter , maintain
-approximations of the -distance between a given pair of
nodes and , the distances from a single source to all nodes
("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the
distances between all nodes ("APSP").
Our main result is a deterministic algorithm for maintaining
-approximate -distance with worst-case update time
(for the current best known bound on the matrix multiplication
exponent ). This even improves upon the fastest known randomized
algorithm for this problem. Similar to several other well-studied dynamic
problems whose state-of-the-art worst-case update time is , this
matches a conditional lower bound [BNS, FOCS 2019]. We further give a
deterministic algorithm for maintaining -approximate
single-source distances with worst-case update time , which also
matches a conditional lower bound.
At the core, our approach is to combine algebraic distance maintenance data
structures with near-additive emulator constructions. This also leads to novel
dynamic algorithms for maintaining -emulators that improve
upon the state of the art, which might be of independent interest. Our
techniques also lead to improved randomized algorithms for several problems
such as exact -distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st
distances using new algebraic data structure
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