The paper addresses the region search problem in three-dimensional (3D) space. The data used is a dynamically growing point cloud as it is typically gathered with a 3D-sensing device like a laser range-scanner. An encoding of space in combination with a new region search algorithm is introduced. The algorithm allows for fast access to spherical subsets of variable size. An octree based and a ba...

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let hmin(n) be the minimal height of a binary search tree for n keys. We consider the problem to construct an optimal binary search tree with near minimal height, i.e. with height h ≤ hmin(n) +∆ for some fixed ∆. It is shown, that for any fixed∆ optimal binary search trees wi...

We give linear-time algorithms for re-ordering and heightrestricting a binary search tree with only a small increase in cost, constructing a nearly optimal binary search tree given the rank by probability of each possible outcome, and height-restricting an optimal binary search tree when the increase in cost is restricted. Whereas most algorithms for constructing good binary search trees need t...

II Binary search 4 II-A Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 II-B Binary search with stretch-intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 II-B.1 Within a stretch-interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 II-B.2 Binary search among stretch intervals . . . . . . . . . . . ...

We consider random binary search trees when the input consists of a multiset, i.e. a set with multiple occurrences of equal elements, and prove that the randomized insertion and deletion algorithms produce random search trees regardless of multiplicities; even if all the elements are equal during the tree updates, a search tree will maintain its randomness. Thus, equal elements do not degenerat...

Binary Search Trees are a frequently used data structure for rapid access to the stored data. Data structures like arrays, vectors and linked lists are limited by the trade-off between the ability to perform a fast search and resize easily. They are an alternative that is both dynamic in size and easily searchable. Due to efficiency reason, complete and nearly complete binary search trees are o...

In the next two lectures we study the question of dynamic optimality, or whether there exists a binary search tree algorithm that performs ”as well” as all other algorithms on any input string. In this lecture we will define a binary search tree as a formal model of computation, show some analytic bounds that a dynamically optimal binary search tree needs to satisfy, and show two search trees t...

In 1985, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence—any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any othe...

In the next two lectures we study the question of dynamic optimality, or whether there exists a binary search tree algorithm that performs ”as well” as all other algorithms on any input string. In this lecture we will define a binary search tree as a formal model of computation, show some analytic bounds that a dynamically optimal binary search tree needs to satisfy, and show two search trees t...