167 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Incremental Dead State Detection in Logarithmic Time
Identifying live and dead states in an abstract transition system is a
recurring problem in formal verification; for example, it arises in our recent
work on efficiently deciding regex constraints in SMT. However,
state-of-the-art graph algorithms for maintaining reachability information
incrementally (that is, as states are visited and before the entire state space
is explored) assume that new edges can be added from any state at any time,
whereas in many applications, outgoing edges are added from each state as it is
explored. To formalize the latter situation, we propose guided incremental
digraphs (GIDs), incremental graphs which support labeling closed states
(states which will not receive further outgoing edges). Our main result is that
dead state detection in GIDs is solvable in amortized time per edge
for edges, improving upon per edge due to Bender, Fineman,
Gilbert, and Tarjan (BFGT) for general incremental directed graphs.
We introduce two algorithms for GIDs: one establishing the logarithmic time
bound, and a second algorithm to explore a lazy heuristics-based approach. To
enable an apples-to-apples experimental comparison, we implemented both
algorithms, two simpler baselines, and the state-of-the-art BFGT baseline using
a common directed graph interface in Rust. Our evaluation shows -x
speedups over BFGT for the largest input graphs over a range of graph classes,
random graphs, and graphs arising from regex benchmarks.Comment: 22 pages + reference
Dynamic -Approximate Matching Size in Truly Sublinear Update Time
We show a fully dynamic algorithm for maintaining -approximate
\emph{size} of maximum matching of the graph with vertices and edges
using update time. This is the first polynomial
improvement over the long-standing update time, which can be trivially
obtained by periodic recomputation. Thus, we resolve the value version of a
major open question of the dynamic graph algorithms literature (see, e.g.,
[Gupta and Peng FOCS'13], [Bernstein and Stein SODA'16],[Behnezhad and Khanna
SODA'22]).
Our key technical component is the first sublinear algorithm for -approximate maximum matching with sublinear running time on dense graphs.
All previous algorithms suffered a multiplicative approximation factor of at
least or assumed that the graph has a very small maximum degree
Maintaining Expander Decompositions via Sparse Cuts
In this article, we show that the algorithm of maintaining expander
decompositions in graphs undergoing edge deletions directly by removing sparse
cuts repeatedly can be made efficient. Formally, for an -edge undirected
graph , we say a cut is -sparse if . A
-expander decomposition of is a partition of into sets such that each cluster contains no -sparse cut
(meaning it is a -expander) with edges crossing
between clusters. A natural way to compute a -expander decomposition is
to decompose clusters by -sparse cuts until no such cut is contained in
any cluster. We show that even in graphs undergoing edge deletions, a slight
relaxation of this meta-algorithm can be implemented efficiently with amortized
update time . Our approach naturally extends to maintaining
directed -expander decompositions and -expander hierarchies and
thus gives a unifying framework while having simpler proofs than previous
state-of-the-art work. In all settings, our algorithm matches the run-times of
previous algorithms up to subpolynomial factors. Moreover, our algorithm
provides stronger guarantees for -expander decompositions. For example,
for graphs undergoing edge deletions, our approach is the first to maintain a
dynamic expander decomposition where each updated decomposition is a refinement
of the previous decomposition, and our approach is the first to guarantee a
sublinear bound on the total number of edges that cross
between clusters across the entire sequence of dynamic updates
Approximating Edit Distance in the Fully Dynamic Model
The edit distance is a fundamental measure of sequence similarity, defined as
the minimum number of character insertions, deletions, and substitutions needed
to transform one string into the other. Given two strings of length at most
, simple dynamic programming computes their edit distance exactly in
time, which is also the best possible (up to subpolynomial factors)
assuming the Strong Exponential Time Hypothesis (SETH). The last few decades
have seen tremendous progress in edit distance approximation, where the runtime
has been brought down to subquadratic, near-linear, and even sublinear at the
cost of approximation.
In this paper, we study the dynamic edit distance problem, where the strings
change dynamically as the characters are substituted, inserted, or deleted over
time. Each change may happen at any location of either of the two strings. The
goal is to maintain the (exact or approximate) edit distance of such dynamic
strings while minimizing the update time. The exact edit distance can be
maintained in time per update (Charalampopoulos, Kociumaka,
Mozes; 2020), which is again tight assuming SETH. Unfortunately, even with the
unprecedented progress in edit distance approximation in the static setting,
strikingly little is known regarding dynamic edit distance approximation.
Utilizing the off-the-shelf tools, it is possible to achieve an
-approximation in update time for any constant . Improving upon this trade-off remains open.
The contribution of this work is a dynamic -approximation algorithm
with amortized expected update time of . In other words, we bring the
approximation-ratio and update-time product down to . Our solution
utilizes an elegant framework of precision sampling tree for edit distance
approximation (Andoni, Krauthgamer, Onak; 2010).Comment: Accepted to FOCS 202
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Processing massive graphs under limited visibility
Graphs are one of the most important and widely used combinatorial structures in mathematics. Their ability to model many real world scenarios which involve a large network of related entities make them useful across disciplines. They are useful as an abstraction in the analysis of networked structures such as the Internet, social networks, road networks, biological networks and many more. The graphs arising out of many real world phenomenon can be very large and they keep evolving over time. For example, Facebook reported over 2:9 billion monthly active users in 2022. Another very large and dynamic network is the human brain consisting of around 1011 nodes and many more edges. These large and evolving graphs present new challenges for algorithm designers. Traditional graph algorithms designed to work with centralised and sequential computing models are rendered useless due to their prohibitively high resource usage. In fact one needs huge amounts of resources just to read the entire graph. A number of new theoretical models have been devised over the years to keep up with the trends in the modern computing systems capable of handing massive input datasets. Some of these models such as streaming model and the query model allow the algorithm to view the graph piecemeal. In some cases, the model allows the graph to be processed by a set of interconnected computing elements such as in distributed computing. In this thesis we address some graph problems in these non-centralised, non-sequential models of computing with a limited access to the input graph. Specifically, we address three different graph problems, each in a different computing model. The first problem we look at is the computation of approximate shortest paths in dynamic streams. The second problem deals with finding kings in tournament graphs, given query access to the arcs of the tournament. The third and the final problem we investigate is a local test criteria for testing the expansion of a graph in the distributed CONGEST model
Fully Dynamic Shortest Paths and Reachability in Sparse Digraphs
We study the exact fully dynamic shortest paths problem. For real-weighted directed graphs, we show a deterministic fully dynamic data structure with O?(mn^{4/5}) worst-case update time processing arbitrary s,t-distance queries in O?(n^{4/5}) time. This constitutes the first non-trivial update/query tradeoff for this problem in the regime of sparse weighted directed graphs.
Moreover, we give a Monte Carlo randomized fully dynamic reachability data structure processing single-edge updates in O?(n?m) worst-case time and queries in O(?m) time. For sparse digraphs, such a tradeoff has only been previously described with amortized update time [Roditty and Zwick, SIAM J. Comp. 2008]
Max -Flow Oracles and Negative Cycle Detection in Planar Digraphs
We study the maximum -flow oracle problem on planar directed graphs
where the goal is to design a data structure answering max -flow value (or
equivalently, min -cut value) queries for arbitrary source-target pairs
. For the case of polynomially bounded integer edge capacities, we
describe an exact max -flow oracle with truly subquadratic space and
preprocessing, and sublinear query time. Moreover, if
-approximate answers are acceptable, we obtain a static oracle
with near-linear preprocessing and query time and a
dynamic oracle supporting edge capacity updates and queries in
worst-case time.
To the best of our knowledge, for directed planar graphs, no (approximate)
max -flow oracles have been described even in the unweighted case, and
only trivial tradeoffs involving either no preprocessing or precomputing all
the possible answers have been known.
One key technical tool we develop on the way is a sublinear (in the number of
edges) algorithm for finding a negative cycle in so-called dense distance
graphs. By plugging it in earlier frameworks, we obtain improved bounds for
other fundamental problems on planar digraphs. In particular, we show: (1) a
deterministic time algorithm for negatively-weighted SSSP in
planar digraphs with integer edge weights at least . This improves upon the
previously known bounds in the important case of weights polynomial in , and
(2) an improved bound on finding a perfect matching in a
bipartite planar graph.Comment: Extended abstract to appear in SODA 202
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