28 research outputs found
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
\~{O}ptimal Vertex Fault-Tolerant Spanners in \~{O}ptimal Time: Sequential, Distributed and Parallel
We (nearly) settle the time complexity for computing vertex fault-tolerant
(VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT
spanners are sparse subgraphs that preserve distance information, up to a small
multiplicative stretch, in the presence of vertex failures. These structures
were introduced by [Chechik et al., STOC 2009] and have received a lot of
attention since then. We provide algorithms for computing nearly optimal
-VFT spanners for any -vertex -edge graph, with near optimal running
time in several computational models:
- A randomized sequential algorithm with a runtime of
(i.e., independent in the number of faults ). The state-of-the-art time
bound is by [Bodwin, Dinitz and
Robelle, SODA 2021].
- A distributed congest algorithm of rounds. Improving
upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with
near-optimal sparsity in rounds.
- A PRAM (CRCW) algorithm with work and
depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained
sub-optimal FT spanners using work and
depth.
An immediate corollary provides the first nearly-optimal PRAM algorithm for
computing nearly optimal -\emph{vertex} connectivity certificates
using polylogarithmic depth and near-linear work. This improves the
state-of-the-art parallel bounds of depth and
work, by [Karger and Motwani, STOC'93].Comment: STOC 202
Dynamic Algorithms for the Massively Parallel Computation Model
The Massive Parallel Computing (MPC) model gained popularity during the last
decade and it is now seen as the standard model for processing large scale
data. One significant shortcoming of the model is that it assumes to work on
static datasets while, in practice, real-world datasets evolve continuously. To
overcome this issue, in this paper we initiate the study of dynamic algorithms
in the MPC model.
We first discuss the main requirements for a dynamic parallel model and we
show how to adapt the classic MPC model to capture them. Then we analyze the
connection between classic dynamic algorithms and dynamic algorithms in the MPC
model. Finally, we provide new efficient dynamic MPC algorithms for a variety
of fundamental graph problems, including connectivity, minimum spanning tree
and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and
Architectures (SPAA 2019