17,628 research outputs found
Shortcuts for the Circle
Let C be the unit circle in R^2. We can view C as a plane graph whose vertices are all the points on C, and the distance between any two points on C is the length of the smaller arc between them. We consider a graph augmentation problem on C, where we want to place k >= 1 shortcuts on C such that the diameter of the resulting graph is minimized.
We analyze for each k with 1 <= k <= 7 what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of k. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is 2 + Theta(1/k^(2/3)) for any k
Smallest small-world network
Efficiency in passage times is an important issue in designing networks, such
as transportation or computer networks. The small-world networks have
structures that yield high efficiency, while keeping the network highly
clustered. We show that among all networks with the small-world structure, the
most efficient ones have a single ``center'', from which all shortcuts are
connected to uniformly distributed nodes over the network. The networks with
several centers and a connected subnetwork of shortcuts are shown to be
``almost'' as efficient. Genetic-algorithm simulations further support our
results.Comment: 5 pages, 6 figures, REVTeX
A Quality and Cost Approach for Comparison of Small-World Networks
We propose an approach based on analysis of cost-quality tradeoffs for
comparison of efficiency of various algorithms for small-world network
construction. A number of both known in the literature and original algorithms
for complex small-world networks construction are shortly reviewed and
compared. The networks constructed on the basis of these algorithms have basic
structure of 1D regular lattice with additional shortcuts providing the
small-world properties. It is shown that networks proposed in this work have
the best cost-quality ratio in the considered class.Comment: 27 pages, 16 figures, 1 tabl
Complexity and Algorithms for the Discrete Fr\'echet Distance Upper Bound with Imprecise Input
We study the problem of computing the upper bound of the discrete Fr\'{e}chet
distance for imprecise input, and prove that the problem is NP-hard. This
solves an open problem posed in 2010 by Ahn \emph{et al}. If shortcuts are
allowed, we show that the upper bound of the discrete Fr\'{e}chet distance with
shortcuts for imprecise input can be computed in polynomial time and we present
several efficient algorithms.Comment: 15 pages, 8 figure
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