In computer science, an AVL tree is a self-balancing binary search tree (BST). It was the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.
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.
In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree.
In computer science, a heap is a specialized tree-based data structure which is essentially an almost complete tree that satisfies the heap property: in a max heap, for any given node C, if P is a parent node of C, then the key of P is greater than or equal to the key of C. In a min heap, the key of P is less than or equal to the key of C. The node at the "top" of the heap is called the root node.
In computer science, a priority queue is an abstract data-type similar to a regular queue or stack data structure in which each element additionally has a priority associated with it. In a priority queue, an element with high priority is served before an element with low priority. In some implementations, if two elements have the same priority, they are served according to the order in which they were enqueued; in other implementations ordering of elements with the same priority remains undefined.
In computer science, a red–black tree is a kind of self-balancing binary search tree. Each node stores an extra bit representing "color", used to ensure that the tree remains balanced during insertions and deletions.
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.
Dijkstra's algorithm is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.
A binary heap is a heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap was introduced by J. W. J. Williams in 1964, as a data structure for heapsort.
In computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform.
In computer science, a binomial heap is a data structure that acts as a priority queue but also allows pairs of heaps to be merged. It is important as an implementation of the mergeable heap abstract data type, which is a priority queue supporting merge operation. It is implemented as a heap similar to a binary heap but using a special tree structure that is different from the complete binary trees used by binary heaps. Binomial heaps were invented in 1978 by Jean Vuillemin.
In computer science, a Fibonacci heap is a data structure for priority queue operations, consisting of a collection of heap-ordered trees. It has a better amortized running time than many other priority queue data structures including the binary heap and binomial heap. Michael L. Fredman and Robert E. Tarjan developed Fibonacci heaps in 1984 and published them in a scientific journal in 1987. Fibonacci heaps are named after the Fibonacci numbers, which are used in their running time analysis.
A quadtree is a tree data structure in which each internal node has exactly four children. Quadtrees are the two-dimensional analog of octrees and are most often used to partition a two-dimensional space by recursively subdividing it into four quadrants or regions. The data associated with a leaf cell varies by application, but the leaf cell represents a "unit of interesting spatial information".
In computer science, a scapegoat tree is a self-balancing binary search tree, invented by Arne Andersson in 1989 and again by Igal Galperin and Ronald L. Rivest in 1993. It provides worst-case lookup time and amortized insertion and deletion time.
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 leftist tree or leftist heap is a priority queue implemented with a variant of a binary heap. Every node x has an s-value which is the distance to the nearest leaf in subtree rooted at x. In contrast to a binary heap, a leftist tree attempts to be very unbalanced. In addition to the heap property, leftist trees are maintained so the right descendant of each node has the lower s-value.
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.
The d-ary heap or d-heap is a priority queue data structure, a generalization of the binary heap in which the nodes have d children instead of 2. Thus, a binary heap is a 2-heap, and a ternary heap is a 3-heap. According to Tarjan and Jensen et al., d-ary heaps were invented by Donald B. Johnson in 1975.
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; that is, it's a structure that cannot be modified once it's built. A similar data structure is the interval tree.
In computer science, k-way merge algorithms or multiway merges are a specific type of sequence merge algorithms that specialize in taking in k sorted lists and merging them into a single sorted list. These merge algorithms generally refer to merge algorithms that take in a number of sorted lists greater than two. Two-way merges are also referred to as binary merges.
