375 research outputs found
Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity
We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time.
As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
Linear-Time Algorithms for Computing Maximum-Density Sequence Segments with Bioinformatics Applications
We study an abstract optimization problem arising from biomolecular sequence
analysis. For a sequence A of pairs (a_i,w_i) for i = 1,..,n and w_i>0, a
segment A(i,j) is a consecutive subsequence of A starting with index i and
ending with index j. The width of A(i,j) is w(i,j) = sum_{i <= k <= j} w_k, and
the density is (sum_{i<= k <= j} a_k)/ w(i,j). The maximum-density segment
problem takes A and two values L and U as input and asks for a segment of A
with the largest possible density among those of width at least L and at most
U. When U is unbounded, we provide a relatively simple, O(n)-time algorithm,
improving upon the O(n \log L)-time algorithm by Lin, Jiang and Chao. When both
L and U are specified, there are no previous nontrivial results. We solve the
problem in O(n) time if w_i=1 for all i, and more generally in
O(n+n\log(U-L+1)) time when w_i>=1 for all i.Comment: 23 pages, 13 figures. A significant portion of these results appeared
under the title, "Fast Algorithms for Finding Maximum-Density Segments of a
Sequence with Applications to Bioinformatics," in Proceedings of the Second
Workshop on Algorithms in Bioinformatics (WABI), volume 2452 of Lecture Notes
in Computer Science (Springer-Verlag, Berlin), R. Guigo and D. Gusfield
editors, 2002, pp. 157--17
Fully dynamic maintenance of k-connectivity in parallel
Ā©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(nĪ±(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(nĪ±(2n, n)/log n) processors per update and O(logn) time using O(nĪ±(2n, n)/logn) processors per query, where Ī±(.,.) 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/Ī±(3n, n)) time using O(nĪ±(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 Ī©(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
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