1,436 research outputs found
New Algorithms and Hardness for Incremental Single-Source Shortest Paths in Directed Graphs
In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a
graph subject to edge insertions and deletions and a source vertex
, and the goal is to maintain the distance for all .
Fine-grained complexity has provided strong lower bounds for exact partially
dynamic SSSP and approximate fully dynamic SSSP [ESA'04, FOCS'14, STOC'15].
Thus much focus has been directed towards finding efficient partially dynamic
-approximate SSSP algorithms [STOC'14, ICALP'15, SODA'14,
FOCS'14, STOC'16, SODA'17, ICALP'17, ICALP'19, STOC'19, SODA'20, SODA'20].
Despite this rich literature, for directed graphs there are no known
deterministic algorithms for -approximate dynamic SSSP that
perform better than the classic ES-tree [JACM'81]. We present the first such
algorithm.
We present a \emph{deterministic} data structure for incremental SSSP in
weighted digraphs with total update time which is
near-optimal for very dense graphs; here is the ratio of the largest weight
in the graph to the smallest. Our algorithm also improves over the best known
partially dynamic \emph{randomized} algorithm for directed SSSP by Henzinger et
al. [STOC'14, ICALP'15] if .
We also provide improved conditional lower bounds. Henzinger et al. [STOC'15]
showed that under the OMv Hypothesis, the partially dynamic exact -
Shortest Path problem in undirected graphs requires amortized update or query
time , given polynomial preprocessing time. Under a hypothesis
about finding Cliques, we improve the update and query lower bound for
algorithms with polynomial preprocessing time to . Further,
under the -Cycle hypothesis, we show that any partially dynamic SSSP
algorithm with preprocessing time requires amortized update
or query time
Algorithms and Hardness for Diameter in Dynamic Graphs
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include:
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP.
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
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
Shortest Paths Avoiding Forbidden Subpaths
In this paper we study a variant of the shortest path problem in graphs:
given a weighted graph G and vertices s and t, and given a set X of forbidden
paths in G, find a shortest s-t path P such that no path in X is a subpath of
P. Path P is allowed to repeat vertices and edges. We call each path in X an
exception, and our desired path a shortest exception-avoiding path. We
formulate a new version of the problem where the algorithm has no a priori
knowledge of X, and finds out about an exception x in X only when a path
containing x fails. This situation arises in computing shortest paths in
optical networks. We give an algorithm that finds a shortest exception avoiding
path in time polynomial in |G| and |X|. The main idea is to run Dijkstra's
algorithm incrementally after replicating vertices when an exception is
discovered.Comment: 12 pages, 2 figures. Fixed a few typos, rephrased a few sentences,
and used the STACS styl
- …