1,321 research outputs found
Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs
We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation).
Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions
Fully dynamic all-pairs shortest paths with worst-case update-time revisited
We revisit the classic problem of dynamically maintaining shortest paths
between all pairs of nodes of a directed weighted graph. The allowed updates
are insertions and deletions of nodes and their incident edges. We give
worst-case guarantees on the time needed to process a single update (in
contrast to related results, the update time is not amortized over a sequence
of updates).
Our main result is a simple randomized algorithm that for any parameter
has a worst-case update time of and answers
distance queries correctly with probability , against an adaptive
online adversary if the graph contains no negative cycle. The best
deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time
of and assumes non-negative weights. This is the first
improvement for this problem for more than a decade. Conceptually, our
algorithm shows that randomization along with a more direct approach can
provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201
Decremental Single-Source Shortest Paths on Undirected Graphs in Near-Linear Total Update Time
In the decremental single-source shortest paths (SSSP) problem we want to
maintain the distances between a given source node and every other node in
an -node -edge graph undergoing edge deletions. While its static
counterpart can be solved in near-linear time, this decremental problem is much
more challenging even in the undirected unweighted case. In this case, the
classic total update time of Even and Shiloach [JACM 1981] has been the
fastest known algorithm for three decades. At the cost of a
-approximation factor, the running time was recently improved to
by Bernstein and Roditty [SODA 2011]. In this paper, we bring the
running time down to near-linear: We give a -approximation
algorithm with expected total update time, thus obtaining
near-linear time. Moreover, we obtain time for the weighted
case, where the edge weights are integers from to . The only prior work
on weighted graphs in time is the -time algorithm by
Henzinger et al. [STOC 2014, ICALP 2015] which works for directed graphs with
quasi-polynomial edge weights. The expected running time bound of our algorithm
holds against an oblivious adversary.
In contrast to the previous results which rely on maintaining a sparse
emulator, our algorithm relies on maintaining a so-called sparse -hop set introduced by Cohen [JACM 2000] in the PRAM literature. An
-hop set of a graph is a set of weighted edges
such that the distance between any pair of nodes in can be
-approximated by their -hop distance (given by a path
containing at most edges) on . Our algorithm can maintain
an -hop set of near-linear size in near-linear time under
edge deletions.Comment: Accepted to Journal of the ACM. A preliminary version of this paper
was presented at the 55th IEEE Symposium on Foundations of Computer Science
(FOCS 2014). Abstract shortened to respect the arXiv limit of 1920 character
Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization
We study dynamic -approximation algorithms for the all-pairs
shortest paths problem in unweighted undirected -node -edge graphs under
edge deletions. The fastest algorithm for this problem is a randomized
algorithm with a total update time of and constant
query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic
algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total
update time of and constant query time. We improve these results as
follows: (1) We present an algorithm with a total update time of and constant query time that has an additive error of
in addition to the multiplicative error. This beats the previous
time when . Note that the additive
error is unavoidable since, even in the static case, an -time
(a so-called truly subcubic) combinatorial algorithm with
multiplicative error cannot have an additive error less than ,
unless we make a major breakthrough for Boolean matrix multiplication [Dor et
al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and
Williams FOCS 2010]. The algorithm can also be turned into a
-approximation algorithm (without an additive error) with the
same time guarantees, improving the recent -approximation
algorithm with running
time of Bernstein and Roditty [SODA 2011] in terms of both approximation and
time guarantees. (2) We present a deterministic algorithm with a total update
time of and a query time of . The
algorithm has a multiplicative error of and gives the first
improved deterministic algorithm since 1981. It also answers an open question
raised by Bernstein [STOC 2013].Comment: A preliminary version was presented at the 2013 IEEE 54th Annual
Symposium on Foundations of Computer Science (FOCS 2013
Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees
Let be an -node and -edge positively real-weighted undirected
graph. For any given integer , we study the problem of designing a
sparse \emph{f-edge-fault-tolerant} (-EFT) {\em -approximate
single-source shortest-path tree} (-ASPT), namely a subgraph of
having as few edges as possible and which, following the failure of a set
of at most edges in , contains paths from a fixed source that are
stretched at most by a factor of . To this respect, we provide an
algorithm that efficiently computes an -EFT -ASPT of size . Our structure improves on a previous related construction designed for
\emph{unweighted} graphs, having the same size but guaranteeing a larger
stretch factor of , plus an additive term of .
Then, we show how to convert our structure into an efficient -EFT
\emph{single-source distance oracle} (SSDO), that can be built in
time, has size , and is able to report,
after the failure of the edge set , in time a
-approximate distance from the source to any node, and a
corresponding approximate path in the same amount of time plus the path's size.
Such an oracle is obtained by handling another fundamental problem, namely that
of updating a \emph{minimum spanning forest} (MSF) of after that a
\emph{batch} of simultaneous edge modifications (i.e., edge insertions,
deletions and weight changes) is performed. For this problem, we build in time a \emph{sensitivity} oracle of size , that
reports in time the (at most ) edges either exiting from
or entering into the MSF. [...]Comment: 16 pages, 4 figure
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