In computer science, dynamization is the process of transforming a static data structure into a dynamic one. Although static data structures may provide very good functionality and fast queries, their utility is limited because of their inability to grow/shrink quickly, thus making them inapplicable for the solution of dynamic problems, where the input data changes. Dynamization techniques provide uniform ways of creating dynamic data structures.
We define problem of searching for the predicate match in the set as . Problem is decomposable if the set can be decomposed into subsets and there exists an operation of result unification such that .
Decomposition is a term used in computer science to break static data structures into smaller units of unequal size. The basic principle is the idea that any decimal number can be translated into a representation in any other base. For more details about the topic see Decomposition (computer science). For simplicity, binary system will be used in this article but any other base (as well as other possibilities such as Fibonacci numbers) can also be utilized.
If using the binary system, a set of elements is broken down into subsets of sizes with
elements where is the -th bit of in binary. This means that if has -th bit equal to 0, the corresponding set does not contain any elements. Each of the subset has the same property as the original static data structure. Operations performed on the new dynamic data structure may involve traversing sets formed by decomposition. As a result, this will add factor as opposed to the static data structure operations but will allow insert/delete operation to be added.
Kurt Mehlhorn proved several equations for time complexity of operations on the data structures dynamized according to this idea. Some of these equalities are listed.
If
then
If is at least polynomial, then .
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the middle element of the array. If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found. If the search ends with the remaining half being empty, the target is not in the array.
A splay tree is a binary search tree with the additional property that recently accessed elements are quick to access again. Like self-balancing binary search trees, a splay tree performs basic operations such as insertion, look-up and removal in O(log n) amortized time. For random access patterns drawn from a non-uniform random distribution, their amortized time can be faster than logarithmic, proportional to the entropy of the access pattern. For many patterns of non-random operations, also, splay trees can take better than logarithmic time, without requiring advance knowledge of the pattern. According to the unproven dynamic optimality conjecture, their performance on all access patterns is within a constant factor of the best possible performance that could be achieved by any other self-adjusting binary search tree, even one selected to fit that pattern. The splay tree was invented by Daniel Sleator and Robert Tarjan in 1985.
The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided design (CAD).
In computer science, a fusion tree is a type of tree data structure that implements an associative array on w-bit integers on a finite universe, where each of the input integer has size less than 2w and is non-negative. When operating on a collection of n key–value pairs, it uses O(n) space and performs searches in O(logwn) time, which is asymptotically faster than a traditional self-balancing binary search tree, and also better than the van Emde Boas tree for large values of w. It achieves this speed by using certain constant-time operations that can be done on a machine word. Fusion trees were invented in 1990 by Michael Fredman and Dan Willard.
In computer science, an interval tree is a tree data structure to hold intervals. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computerized map inside a rectangular viewport, or to find all visible elements inside a three-dimensional scene. A similar data structure is the segment tree.
In computer science, a k-d tree is a space-partitioning data structure for organizing points in a k-dimensional space. k-d trees are a useful data structure for several applications, such as searches involving a multidimensional search key and creating point clouds. k-d trees are a special case of binary space partitioning trees.
In graph theory and computer science, the lowest common ancestor (LCA) of two nodes v and w in a tree or directed acyclic graph (DAG) T is the lowest node that has both v and w as descendants, where we define each node to be a descendant of itself.
In computer science, fractional cascading is a technique to speed up a sequence of binary searches for the same value in a sequence of related data structures. The first binary search in the sequence takes a logarithmic amount of time, as is standard for binary searches, but successive searches in the sequence are faster. The original version of fractional cascading, introduced in two papers by Chazelle and Guibas in 1986, combined the idea of cascading, originating in range searching data structures of Lueker (1978) and Willard (1978), with the idea of fractional sampling, which originated in Chazelle (1983). Later authors introduced more complex forms of fractional cascading that allow the data structure to be maintained as the data changes by a sequence of discrete insertion and deletion events.
Dynamic problems in computational complexity theory are problems stated in terms of the changing input data. In the most general form a problem in this category is usually stated as follows:
In computer science, the range searching problem consists of processing a set S of objects, in order to determine which objects from S intersect with a query object, called the range. For example, if S is a set of points corresponding to the coordinates of several cities, find the subset of cities within a given range of latitudes and longitudes.
In computer science, a succinct data structure is a data structure which uses an amount of space that is "close" to the information-theoretic lower bound, but still allows for efficient query operations. The concept was originally introduced by Jacobson to encode bit vectors, (unlabeled) trees, and planar graphs. Unlike general lossless data compression algorithms, succinct data structures retain the ability to use them in-place, without decompressing them first. A related notion is that of a compressed data structure, in which the size of the data structure depends upon the particular data being represented.
In computer science, a segment tree, also known as a statistic tree, is a tree data structure used for storing information about intervals, or segments. It allows querying which of the stored segments contain a given point. It is, in principle, a static structure and cannot be modified once built. A similar data structure is the interval tree.
In computer science, a range tree is an ordered tree data structure to hold a list of points. It allows all points within a given range to be reported efficiently, and is typically used in two or higher dimensions. Range trees were introduced by Jon Louis Bentley in 1979. Similar data structures were discovered independently by Lueker, Lee and Wong, and Willard. The range tree is an alternative to the k-d tree. Compared to k-d trees, range trees offer faster query times of but worse storage of , where n is the number of points stored in the tree, d is the dimension of each point and k is the number of points reported by a given query.
Vector space model or term vector model is an algebraic model for representing text documents as vectors of identifiers. It is used in information filtering, information retrieval, indexing and relevancy rankings. Its first use was in the SMART Information Retrieval System.
Collective operations are building blocks for interaction patterns, that are often used in SPMD algorithms in the parallel programming context. Hence, there is an interest in efficient realizations of these operations.
In data structures, a range query consists of preprocessing some input data into a data structure to efficiently answer any number of queries on any subset of the input. Particularly, there is a group of problems that have been extensively studied where the input is an array of unsorted numbers and a query consists of computing some function, such as the minimum, on a specific range of the array.
The Wavelet Tree is a succinct data structure to store strings in compressed space. It generalizes the and operations defined on bitvectors to arbitrary alphabets.
In computer science, an optimal binary search tree , sometimes called a weight-balanced binary tree, is a binary search tree which provides the smallest possible search time for a given sequence of accesses. Optimal BSTs are generally divided into two types: static and dynamic.
In computer science, the predecessor problem involves maintaining a set of items to, given an element, efficiently query which element precedes or succeeds that element in an order. Data structures used to solve the problem include balanced binary search trees, van Emde Boas trees, and fusion trees. In the static predecessor problem, the set of elements does not change, but in the dynamic predecessor problem, insertions into and deletions from the set are allowed.
Approximate Membership Query Filter (AMQ-Filter) is a group of space-efficient probabilistic data structures that supports approximate membership queries. An approximate membership query answers if an element is in a set or not with a false positive rate of .