SI 486H, Unit 1: Discrete Math

Further analysis of the number of spanning trees in circulant graphs

Let 1 s1<s2< · · ·<sk n/2 be given integers. An undirected even-valent circulant graph, C12k n , has n vertices 0, 1, 2, . . ., n− 1, and for each si (1 i k) and j (0 j n− 1) there is an edge between j and j + si (mod n). Let T (C12k n ) stand for the number of spanning trees of C12k n . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X.Yong...

An Improved Line-Separable Algorithm for Discrete Unit Disk Cover

Given a set D of m unit disks and a set P of n points in the plane, the discrete unit disk cover problem is to select a minimum cardinality subset D′ ⊆ D to cover P. This problem is NP-hard [14] and the best ∗[email protected] †[email protected] ‡[email protected] §[email protected] ¶[email protected] ‖[email protected] ∗∗[email protected] ††[email protected]

Bounds on the Expected Size of the Maximum Agreement Subtree

We prove polynomial upper and lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and Yule-Harding distribution. This positively answers a question posed in earlier work. Determining tight upper and lower bounds remains an open problem.

Improved Approximation Algorithms for the Demand Routing and Slotting Problem with Unit Demands on Rings

Canonical Trees, Compact Prefix-Free Codes, and Sums of Unit Fractions: A Probabilistic Analysis

For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane “canonical” trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the...