Fully dynamic maintenance of k-connectivity in parallel

©2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.Given a graph G=(V, E) with n vertices and m edges, the k-connectivity of G denotes either the k-edge connectivity or the k-vertex connectivity of G. In this paper, we deal with the fully dynamic maintenance of k-connectivity of G in the parallel setting for k=2, 3. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such as edge insertions and deletions, and answering the query of whether two vertices are included in the same k-edge/vertex connected component. Our major results are the following: (1) An NC algorithm for the 2-edge connectivity problem is proposed, which runs in O(log n log(m/n)) time using O(n3/4) processors per update and query. (2) It is shown that the biconnectivity problem can be solved in O(log2 n ) time using O(nα(2n, n)/logn) processors per update and O(1) time with a single processor per query or in O(log n logn/m) time using O(nα(2n, n)/log n) processors per update and O(logn) time using O(nα(2n, n)/logn) processors per query, where α(.,.) is the inverse of Ackermann's function. (3) An NC algorithm for the triconnectivity problem is also derived, which takes O(log n logn/m+logn log log n/α(3n, n)) time using O(nα(3n, n)/log n) processors per update and O(1) time with a single processor per query. (4) An NC algorithm for the 3-edge connectivity problem is obtained, which has the same time and processor complexities as the algorithm for the triconnectivity problem. To the best of our knowledge, the proposed algorithms are the first NC algorithms for the problems using O(n) processors in contrast to Ω(m) processors for solving them from scratch. In particular, the proposed NC algorithm for the 2-edge connectivity problem uses only O(n3/4) processors. All the proposed algorithms run on a CRCW PRAMWeifa Liang, Brent, R.P., Hong She

Optimal decremental connectivity in planar graphs

We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in $O(n)$ time. This results improves over previously known $O(n \log n)$ time algorithm

Parallel Batch-Dynamic Graph Connectivity

In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The most well known sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves $O(\log^2 n)$ amortized time per edge insertion or deletion, and $O(\log n / \log\log n)$ time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where $\Delta$ is the average batch size of all deletions, our algorithm achieves $O(\log n \log(1 + n / \Delta))$ expected amortized work per edge insertion and deletion and $O(\log^3 n)$ depth w.h.p. Our algorithm answers a batch of $k$ connectivity queries in $O(k \log(1 + n/k))$ expected work and $O(\log n)$ depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.Comment: This is the full version of the paper appearing in the ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 201

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

Maintenance of Strongly Connected Component in Shared-memory Graph

In this paper, we present an on-line fully dynamic algorithm for maintaining strongly connected component of a directed graph in a shared memory architecture. The edges and vertices are added or deleted concurrently by fixed number of threads. To the best of our knowledge, this is the first work to propose using linearizable concurrent directed graph and is build using both ordered and unordered list-based set. We provide an empirical comparison against sequential and coarse-grained. The results show our algorithm's throughput is increased between 3 to 6x depending on different workload distributions and applications. We believe that there are huge applications in the on-line graph. Finally, we show how the algorithm can be extended to community detection in on-line graph.Comment: 29 pages, 4 figures, Accepted in the Conference NETYS-201