We consider the problem of dynamic range searching in tree structures that do not replicate data. We propose a new dynamic structure, called the O-tree, that achieves a query time complexity of O(n(d 1)=d) on n d-dimensional points and an amortized insertion/deletion time complexity of O(logn). We show that this structure is optimal when data is not replicated. In addition to optimal query and ...

We call such an algorithm a computation tree. A computation tree for the problem Poly(d) has input the coefficients of a polynomial f (in terms of real and imaginary parts). The output must consist of (z,, . . . , zd) (again given in terms of real and imaginary parts), each zi being within E of {i, the <i being the roots off. The computation nodes do not contribute to the topology of the comput...

Experimental results are shown in Tables 1 and 2. We compare tree length (TL), the sum total path length for all p i ! p j pairs (TPL), and tree diameter D. Also included are the maximum delay between any source-sink pair (MD) and the average of maximum delays for each source (AMD), averaged over all runs. Delays were measured from the input transition to the output reaching 90% of its nal valu...

A tree T is a graph which is connected and doesn’t contain any circuits. Given any two vertices α ̸ = β ∈ T, let αβ be the unique path connecting α and β. Define the graph distance d(α, β) to be the number of edges contained in the path αβ. Let T be an infinite tree with root 0.The set of all vertices with distance n from the root is called the nth generation of T, which is denoted by L n . We d...

In a recently proposed graphical compression algorithm by Choi and Szpankowski (2009), the following tree arose in the course of the analysis. The root contains n balls that are consequently distributed between two subtrees according to a simple rule: In each step, all balls independently move down to the left subtree (say with probability p) or the right subtree (with probability 1− p). A new ...

We study the problem of constructing phylogenetic trees for a given set of species. The problem is formulated as that of finding a minimum Steiner tree on n points over the Boolean hypercube of dimension d. It is known that an optimal tree can be found in linear time [1] if the given dataset has a perfect phylogeny, i.e. cost of the optimal phylogeny is exactly d. Moreover, if the data has a ne...

Given an undirected connected graph G = (V,E) with a set V of vertices, a set E of edges, and costs cij associated to every edge [i, j] ∈ E, with i < j, the Diameter Minimum Spanning Tree Problem (DCMST) consists in finding a minimum spanning tree T = (V,E ), with E ′ ⊆ E, where the diameter required does not exceed a given positive integer value D, where 2 ≤ D ≤ |V | − 1. The diameter of a tre...

We propose an optimal framework for active surface extraction from video sequences. An active surface is a collection of active contours in successive frames such that the active contours are constrained by spatial and temporal energy terms. The spatial energy terms impose constraints on the active contour in a given frame. The temporal energy terms relate the active contours in different frame...

In this paper we discuss an algorithm to perform the conversion from the interior to the boundary of d-dimensional polyhedra, where both the d-polyhedron and its (d 1) boundary faces are represented as BSP trees. This approach allows also to compute the BSP tree of the set intersection between any hyperplane in Ed and the BSP representation of a d-polyhedron. If such section hyperplane is the a...

A minimum Steiner tree for a given set X of points is a network interconnecting the points of X having minimum possible total length. In this note we investigate various properties of minimum Steiner trees in normed planes, i.e., where the "unit disk" is an arbitrary compact convex centrally symmetric domain D having nonempty interior. We show that if the boundary of D is strictly convex and di...