1,574 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
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
Simple Local Computation Algorithms for the General Lovasz Local Lemma
We consider the task of designing Local Computation Algorithms (LCA) for
applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear
algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot
of attention in recent years. The LLL is an existential, sufficient condition
for a collection of sets to have non-empty intersection (in applications,
often, each set comprises all objects having a certain property). The
ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully
constructive, following earlier results by Beck~\cite{beck_lll} and
Alon~\cite{alon_lll} giving algorithms under significantly stronger LLL-like
conditions. LCAs under those stronger conditions were given in~\cite{Ronitt},
where it was asked if the Moser-Tardos algorithm can be used to design LCAs
under the standard LLL condition. The main contribution of this paper is to
answer this question affirmatively. In fact, our techniques yield LCAs for
settings beyond the standard LLL condition
Fast Deterministic Consensus in a Noisy Environment
It is well known that the consensus problem cannot be solved
deterministically in an asynchronous environment, but that randomized solutions
are possible. We propose a new model, called noisy scheduling, in which an
adversarial schedule is perturbed randomly, and show that in this model
randomness in the environment can substitute for randomness in the algorithm.
In particular, we show that a simplified, deterministic version of Chandra's
wait-free shared-memory consensus algorithm (PODC, 1996, pp. 166-175) solves
consensus in time at most logarithmic in the number of active processes. The
proof of termination is based on showing that a race between independent
delayed renewal processes produces a winner quickly. In addition, we show that
the protocol finishes in constant time using quantum and priority-based
scheduling on a uniprocessor, suggesting that it is robust against the choice
of model over a wide range.Comment: Typographical errors fixe
On Strong Diameter Padded Decompositions
Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles
Randomized protocols for asynchronous consensus
The famous Fischer, Lynch, and Paterson impossibility proof shows that it is
impossible to solve the consensus problem in a natural model of an asynchronous
distributed system if even a single process can fail. Since its publication,
two decades of work on fault-tolerant asynchronous consensus algorithms have
evaded this impossibility result by using extended models that provide (a)
randomization, (b) additional timing assumptions, (c) failure detectors, or (d)
stronger synchronization mechanisms than are available in the basic model.
Concentrating on the first of these approaches, we illustrate the history and
structure of randomized asynchronous consensus protocols by giving detailed
descriptions of several such protocols.Comment: 29 pages; survey paper written for PODC 20th anniversary issue of
Distributed Computin
Research summary, January 1989 - June 1990
The Research Institute for Advanced Computer Science (RIACS) was established at NASA ARC in June of 1983. RIACS is privately operated by the Universities Space Research Association (USRA), a consortium of 62 universities with graduate programs in the aerospace sciences, under a Cooperative Agreement with NASA. RIACS serves as the representative of the USRA universities at ARC. This document reports our activities and accomplishments for the period 1 Jan. 1989 - 30 Jun. 1990. The following topics are covered: learning systems, networked systems, and parallel systems
Orderly Spanning Trees with Applications
We introduce and study the {\em orderly spanning trees} of plane graphs. This
algorithmic tool generalizes {\em canonical orderings}, which exist only for
triconnected plane graphs. Although not every plane graph admits an orderly
spanning tree, we provide an algorithm to compute an {\em orderly pair} for any
connected planar graph , consisting of a plane graph of , and an
orderly spanning tree of . We also present several applications of orderly
spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem,
(2) the first area-optimal 2-visibility drawing of , and (3) the best known
encodings of with O(1)-time query support. All algorithms in this paper run
in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of
the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001),
Washington D.C., USA, January 7-9, 2001, pp. 506-51
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