155 research outputs found
Minimum-Weight Edge Discriminator in Hypergraphs
In this paper we introduce the concept of minimum-weight edge-discriminators
in hypergraphs, and study its various properties. For a hypergraph , a function is said to be an {\it edge-discriminator} on if
, for all hyperedges , and
, for every two
distinct hyperedges . An {\it optimal
edge-discriminator} on , to be denoted by , is
an edge-discriminator on satisfying , where
the minimum is taken over all edge-discriminators on . We prove
that any hypergraph , with , satisfies ,
and equality holds if and only if the elements of are mutually
disjoint. For -uniform hypergraphs , it
follows from results on Sidon sequences that , and
the bound is attained up to a constant factor by the complete -uniform
hypergraph. Next, we construct optimal edge-discriminators for some special
hypergraphs, which include paths, cycles, and complete -partite hypergraphs.
Finally, we show that no optimal edge-discriminator on any hypergraph , with , satisfies
, which, in turn,
raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure
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
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
Online horizontal partitioning of heterogeneous data
In an increasing number of use cases, databases face the challenge of managing heterogeneous data. Heterogeneous data is characterized by a quickly evolving variety of entities without a common set of attributes. These entities do not show enough regularity to be captured in a traditional database schema. A common solution is to centralize the diverse entities in a universal table. Usually, this leads to a very sparse table. Although today’s techniques allow efficient storage of sparse universal tables, query efficiency is still a problem. Queries that address only a subset of attributes have to read the whole universal table includingmany irrelevant entities. Asolution is to use a partitioning of the table, which allows pruning partitions of irrelevant entities before they are touched. Creating and maintaining such a partitioning manually is very laborious or even infeasible, due to the enormous complexity. Thus an autonomous solution is desirable. In this article, we define the Online Partitioning Problem for heterogeneous data. We sketch how an optimal solution for this problem can be determined based on hypergraph partitioning. Although it leads to the optimal partitioning, the hypergraph approach is inappropriate for an implementation in a database system. We present Cinderella, an autonomous online algorithm for horizontal partitioning of heterogeneous entities in universal tables. Cinderella is designed to keep its overhead low by operating online; it incrementally assigns entities to partition while they are touched anyway duringmodifications. This enables a reasonable physical database design at runtime instead of static modeling
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