The (maximum receiver-centric) interference of a geometric graph (von (2005)) is studied. It is shown that, with high probability, the following results hold for a set, V, of n points independently and uniformly distributed in the unit d-cube, for constant dimension d: (1) there exists a connected graph with vertex set V that has interference O((log n)^ 1 3 ); (2) no connected graph with vertex set V has interference o((log n)^ 1 4 ); and (3) the minimum spanning tree of @math has interference Theta(( n)^ 1 2 ).
In higher dimensions, Khabbazian, Durocher, and Haghnegahdar @cite_0 use their version of log to show that minimum spanning trees of random point sets have at most logarithmic interference. Part 1 of main improves the upper bound in durocher to @math and Part 1 of lower-bound gives a matching lower bound.
@cite_0
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We consider distance labeling schemes for trees: given a tree with n nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by [, J. Alg., 2004] and an upper bound by Peleg [Peleg, J. Graph Theory, 2000] establish that labels must use Theta(log^2(n)) bits. [, ESA, 2001] show that for very small approximate stretch, labels use Theta(log(n) log(log(n))) bits. Several other papers investigate various variants such as, for example, small distances in trees [, SODA, 2003]. We improve the known upper and lower bounds of exact distance labeling by showing that 1 4*log^2(n) bits are needed and that 1 2*log^2(n) bits are sufficient. We also give (1 + epsilon)-stretch labeling schemes using Theta(log(n)) bits for constant epsilon > 0. (1 + epsilon)-stretch labeling schemes with polylogarithmic label size have previously been established for doubling dimension graphs by Talwar [Talwar, STOC, 2004]. In addition, we present matching upper and lower bounds for distance labeling for caterpillars, showing that labels must have size 2*log(n) - Theta(log(log(n))). For simple paths with k nodes and edge weights in [1,n], we show that labels must have size (k - 1) k*log(n) + Theta(log(k)).