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Establishing strong connectivity using optimal radius half-disk antennas

Given a set S of points in the plane representing wireless devices, each point equipped with a directional antenna of radius r and aperture angle @a>=180^o, our goal is to find orientations and a minimum r for these antennas such that the induced communication graph is strongly connected. We show that r=3 if @a@?[180^o,240^o), r=2 if @a@?[240^o,270^o), r=2sin(36^o) if @a@?[270^o,288^o), and r=1 if @a>=288^o suffices to establish strong connectivity, assuming that the longest edge in the Euclidean minimum spanning tree of S is 1. These results are worst-case optimal and match the lower bounds presented in [I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc, A. Wiese, Communication in wireless networks with directional antennae, in: Proc. of the 20th Symp. on Parallelism in Algorithms and Architectures, 2008, pp. 344-351]. In contrast, r=2 is sometimes necessary when @a<180^o.

Why Some Heaps Support Constant-Amortized-Time Decrease-Key Operations, and Others Do Not

A lower bound is presented which shows that a class of heap algorithms in the pointer model with only heap pointers must spend \(\Omega \left( \frac{\log \log n}{\log \log \log n} \right)\) amortized time on the Decrease-Key operation (given O(logn) amortized-time Extract-Min). Intuitively, this bound shows the key to having O(1)-time Decrease-Key is the ability to sort O(logn) items in O(logn) time; Fibonacci heaps [M. .L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this through the use of bucket sort. Our lower bound also holds no matter how much data is augmented; this is in contrast to the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] who showed a tradeoff between the number of augmented bits and the amortized cost of Decrease-Key. A new heap data structure, the sort heap, is presented. This heap is a simplification of the heap of Elmasry [SODA 2009: 471-476] and shares with it a O(loglogn) amortized-time Decrease-Key, but with a straightforward implementation such that our lower bound holds. Thus a natural model is presented for a pointer-based heap such that the amortized runtime of a self-adjusting structure and amortized lower asymptotic bounds for Decrease-Key differ by but a O(logloglogn) factor.

Max-Throughput for (Conservative) k-of-n Testing

We define a variant of

Cache-Oblivious Persistence

k

The complexity of order type isomorphism

k-of-

Establishing Strong Connectivity using Optimal Radius Half-Disk Antennas.

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