In today's lecture, we will present a randomized algorithm that embeds any arbitrary metric into a dominating tree metric such that each edge is distorted, in expectation, in the worst case by a factor of O(log n) and such that each edge is not contracted. Recall that a metric on graph G = (V, E) is a function d : E → R. One can define a metric M = (X, d) on an arbitrary set of vertices X with ...

In a recently proposed graphical compression algorithm [1], 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 node is created as long as t...

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 boundary faces are represented as BSP trees This approach allows also to compute the BSP tree of the set intersection between any hyperplane in E and the BSP representation of a d polyhedron If such section hyperplane is the a ne supp...

Efficient indexing support is essential to allow content-based image and video databases using similaritybased retrieval to scale to large databases (tens of thousands up to millions of images). In this paper, we take an in depth look at this problem. One of the major difficulties in solving this problem is the high dimension (6-100) of the feature vectors that are used to represent objects. We...

Relaxed balancing of search trees was introduced with the aim of speeding up the updates and allowing a high degree of concurrency. In a relaxed version of a search tree the rebalancing operations are uncoupled from the updates and may be delayed. Using local transformations the rebalancing can be performed concurrently with updates and search operations. In this paper we revisit the rebalancin...

One can prove the following propositions: (1) For every set D and for every finite sequence p and for every natural number n such that p ∈ D holds p↾ Segn ∈ D. (2) For every binary tree T holds every element of T is a finite sequence of elements of Boolean. Let T be a binary tree. We see that the element of T is a finite sequence of elements of Boolean. Next we state several propositions: (3) F...

Let P : Rd → A be a query problem over Rd for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q ∈ Rd. Let D be a probability measure over Rd representing a distribution of queries. We describe a data structure T = TP,D, called the odds-on tree, of size O(n ) that can be used as a filter that quickly computes P(q) for some query values in Rd and r...

An important feature to be considered in the design of multimedia DBMSs is content-based retrieval of images. Most work in this area has focused on feature-based retrieval; we focus on retrieval based on spatial relationship, which include directional and topological relationships. The most common data structure that is used for representing directional relations is the 2-D string. The search p...

Let G=(V,E) be a finite, simple and undirected graph. For S ⊆ V , let δ(S, G) = {(u, v) ∈ E : u ∈ S and v ∈ V − S} be the edge boundary of S. Given an integer i, 1 ≤ i ≤ |V |, let the edge isoperimetric value of G at i be defined as be(i, G) = minS⊆V ;|S|=i |δ(S, G)|. The edge isoperimetric peak of G is defined as be(G) = max1≤j≤|V | be(j, G). The depth of a tree d is the number of nodes on the...

The k-D tree is a well-studied acceleration data structure for ray tracing. It is used to organize primitives in a scene to allow efficient execution of intersection operations between rays and the primitives. The highest quality k-D tree can be obtained using greedy cost optimization based on a surface area heuristc (SAH). While the high quality enables very fast ray tracing times, a key drawb...