The minimum spanning tree problem (MSTP) in the plane requires finding the length of the shortest tree spanning n points of R 2. We are concerned here with stochastic versions of the problem. First let Xi, 1 _ i < o0, be uniformly and independently distributed random variables in [0,1] 2 and let LMsT(n) be the length of the shortest tree spanning {X 1, X 2 . . . . . Xn}. Steele [9] proved that ...

Reconstructing phylogenetic trees using the criterion of minimum evolution requires the use of a formula FT (d) that estimates the total length of a tree T given only the estimated distances d between the leaves of the tree. Let U(T ) be the collection of linear formulas FT (d) that correctly estimate the total length of T whenever d is an additive distance function on T . The current paper cha...

In a seminal paper [10], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (which is the same as approximating the partition function of the hard-core model from statistical physics) in graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently S...

zs denotes the concatenation of a data zz and a string s, l(s) is the length of a string s, 0 < ps < 1 is a constant given to a string s if 0 < l(s) 5 D 1, ,Bs = 0 if 2(s) = D (D > 1: some integer), Q,(x;) is the product of the predicted probabilities over the time instances such that the string defined below is s. The CTW assumes that the source is some tree source: the probability of x1+1 E A...

Multidimensional similarity join finds pairs of multidimensional points that are within some small distance of each other. The -k-d-B tree has been proposed as a data structure that scales better as the number of dimensions increases compared to previous data structures. We present a cost model of the -k-d-B tree and use it to optimize the leaf

Let G = (V, E) be a simple graph. A dominating set D is called a complementary tree dominating set if the induced subgraph is a tree. The minimum cardinality of a complementary tree dominating set is called the complementary tree domination number of G and is denoted by ctd(G). For a graph G, let V(G) = {v : v  V(G)} be a copy of V(G). The splitting graph Sp(G) of G is the graph with ...

Nearest-neighbor search dominates the asymptotic complexity of sampling-based motion planning algorithms and is often addressed with k-d tree data structures. While it is generally believed that the expected complexity of nearest-neighbor queries is O(log(N)) in the size of the tree, this paper reveals that when a classic k-d tree approach is used with sub-Riemannian metrics, the expected query...