61,918 research outputs found

    Finding tight Hamilton cycles in random hypergraphs faster

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    In an rr-uniform hypergraph on nn vertices a tight Hamilton cycle consists of nn edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of rr vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random rr-uniform hypergraphs with edge probability at least Clog3n/nC \log^3n/n. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for p=ω(1/n)p=\omega(1/n) for r=3r=3 and p=(e+o(1))/np=(e + o(1))/n for r4r\ge 4 using a second moment argument. Moreover our algorithm is superior to previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities pn1+εp\ge n^{-1+\varepsilon}, while the algorithm of Nenadov and \v{S}kori\'c is a randomised quasipolynomial time algorithm working for edge probabilities pClog8n/np\ge C\log^8n/n.Comment: 17 page

    Robust geometric forest routing with tunable load balancing

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    Although geometric routing is proposed as a memory-efficient alternative to traditional lookup-based routing and forwarding algorithms, it still lacks: i) adequate mechanisms to trade stretch against load balancing, and ii) robustness to cope with network topology change. The main contribution of this paper involves the proposal of a family of routing schemes, called Forest Routing. These are based on the principles of geometric routing, adding flexibility in its load balancing characteristics. This is achieved by using an aggregation of greedy embeddings along with a configurable distance function. Incorporating link load information in the forwarding layer enables load balancing behavior while still attaining low path stretch. In addition, the proposed schemes are validated regarding their resilience towards network failures
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