23,848 research outputs found
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Heuristics for Network Coding in Wireless Networks
Multicast is a central challenge for emerging multi-hop wireless
architectures such as wireless mesh networks, because of its substantial cost
in terms of bandwidth. In this report, we study one specific case of multicast:
broadcasting, sending data from one source to all nodes, in a multi-hop
wireless network. The broadcast we focus on is based on network coding, a
promising avenue for reducing cost; previous work of ours showed that the
performance of network coding with simple heuristics is asymptotically optimal:
each transmission is beneficial to nearly every receiver. This is for
homogenous and large networks of the plan. But for small, sparse or for
inhomogeneous networks, some additional heuristics are required. This report
proposes such additional new heuristics (for selecting rates) for broadcasting
with network coding. Our heuristics are intended to use only simple local
topology information. We detail the logic of the heuristics, and with
experimental results, we illustrate the behavior of the heuristics, and
demonstrate their excellent performance
The Binary Space Partitioning-Tree Process
The Mondrian process represents an elegant and powerful approach for space
partition modelling. However, as it restricts the partitions to be
axis-aligned, its modelling flexibility is limited. In this work, we propose a
self-consistent Binary Space Partitioning (BSP)-Tree process to generalize the
Mondrian process. The BSP-Tree process is an almost surely right continuous
Markov jump process that allows uniformly distributed oblique cuts in a
two-dimensional convex polygon. The BSP-Tree process can also be extended using
a non-uniform probability measure to generate direction differentiated cuts.
The process is also self-consistent, maintaining distributional invariance
under a restricted subdomain. We use Conditional-Sequential Monte Carlo for
inference using the tree structure as the high-dimensional variable. The
BSP-Tree process's performance on synthetic data partitioning and relational
modelling demonstrates clear inferential improvements over the standard
Mondrian process and other related methods
Small Cuts and Connectivity Certificates: A Fault Tolerant Approach
We revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds).
Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS \u2711], Ghaffari and Kuhn [DISC \u2713].
- Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG \u2711]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC \u2719] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design.
- Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest.
- Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds.
Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. \u2797] and Dori [PODC \u2718] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC \u2718] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds
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