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Min-heap
In computer science, a heap is a tree-based data structure that satisfies the heap property: In a ''max heap'', for any given node C, if P is the parent node of C, then the ''key'' (the ''value'') 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 (with no parents) is called the ''root'' node. The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact, priority queues are often referred to as "heaps", regardless of how they may be implemented. In a heap, the highest (or lowest) priority element is always stored at the root. However, a heap is not a sorted structure; it can be regarded as being partially ordered. A heap is a useful data structure when it is necessary to repeatedly remove the object with the highest (or lowest) priority, or when insertions need to be interspersed with removals of the root node. A common implement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Binary Heap
A binary heap is a heap (data structure), 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 implementing heapsort. A binary heap is defined as a binary tree with two additional constraints: *Shape property: a binary heap is a ''complete binary tree''; that is, all levels of the tree, except possibly the last one (deepest) are fully filled, and, if the last level of the tree is not complete, the nodes of that level are filled from left to right. *Heap property: the key stored in each node is either greater than or equal to (≥) or less than or equal to (≤) the keys in the node's children, according to some total order. Heaps where the parent key is greater than or equal to (≥) the child keys are called ''max-heaps''; those where it is less than or equal to (≤) are called ''min-heaps''. Efficient (that is, logarithmic tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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D-ary Heap
The -ary heap or -heap is a priority queue data structure, a generalization of the binary heap in which the nodes have 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., -ary heaps were invented by Donald B. Johnson in 1975.. This data structure allows decrease priority operations to be performed more quickly than binary heaps, at the expense of slower delete minimum operations. This tradeoff leads to better running times for algorithms such as Dijkstra's algorithm in which decrease priority operations are more common than delete min operations. Additionally, -ary heaps have better memory cache behavior than binary heaps, allowing them to run more quickly in practice despite having a theoretically larger worst-case running time. Like binary heaps, -ary heaps are an in-place data structure that use no additional storage beyond that needed to store the array of items in the heap.. Data structure The -ary h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Priority Queue
In computer science, a priority queue is an abstract data type similar to a regular queue (abstract data type), queue or stack (abstract data type), stack abstract data type. In a priority queue, each element has an associated ''priority'', which determines its order of service. Priority queue serves highest priority items first. Priority values have to be instances of an ordered data type, and higher priority can be given either to the lesser or to the greater values with respect to the given order relation. For example, in Java (programming language), Java standard library, ''PriorityQueues the least elements with respect to the order have the highest priority. This implementation detail is without much practical significance, since passing to the converse relation, opposite order relation turns the least values into the greatest, and vice versa. While priority queues are often implemented using Heap (data structure) , heaps, they are conceptually distinct. A priority queue can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Dictionary Of Algorithms And Data Structures
The NIST ''Dictionary of Algorithms and Data Structures'' is a reference work maintained by the U.S. National Institute of Standards and Technology. It defines a large number of terms relating to algorithms and data structures. For algorithms and data structures not necessarily mentioned here, see list of algorithms and list of data structures. This list of terms was originally derived from the index of that document, and is in the public domain, as it was compiled by a Federal Government employee as part of a Federal Government work. Some of the terms defined are: __NOTOC__ A * absolute performance guarantee * abstract data type (ADT) * abstract syntax tree (AST) * (a,b)-tree * accepting state * Ackermann's function * active data structure * acyclic directed graph * adaptive heap sort * adaptive Huffman coding * adaptive k-d tree * adaptive sort * address-calculation sort * adjacency list representation * adjacency matrix representation * adversary * algorithm * a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Binomial Heap
In computer science, a binomial heap is a data structure that acts as a priority queue. It is an example of a mergeable heap (also called meldable heap), as it supports merging two heaps in logarithmic time. It is implemented as a Heap (data structure), 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. Binomial heap A binomial heap is implemented as a set of binomial tree data structure, trees (compare with a binary heap, which has a shape of a single binary tree), which are defined recursively as follows: * A binomial tree of order 0 is a single node * A binomial tree of order k has a root node whose children are roots of binomial trees of orders k-1, k-2, ..., 2, 1, 0 (in this order). A binomial tree of order k has 2^k nodes, and height k. The name comes from the shape: a binomial tree of order k has \tbinom k d nodes at depth d, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Beap
A beap, or bi-parental heap, is a data structure for a set (or map, or multiset or multimap) that enables elements (or mappings) to be located, inserted, or deleted in sublinear time. In a beap, each element is stored in a node with up to two parents and up to two children, with the property that the value of a parent node is never greater than the value of either of its children. Beaps are implemented using an array containing only the values to be stored, with the parent-child relationships being determined implicitly by the array indices. (That is: beaps are an implicit data structure.) In that respect they are similar to binary heaps, which are usually implemented that way as well. However, their performance characteristics are different from heaps; in particular, a beap enables sublinear retrieval of arbitrary elements. The beap was introduced by Ian Munro and Hendra Suwanda. A related data structure is the Young tableau. Performance The height of the structure is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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B-heap
A B-heap is a binary heap implemented to keep subtrees in a single Page (computer memory) , page. This reduces the number of pages accessed by up to a factor of ten for big heaps when using virtual memory, compared with the traditional implementation. The traditional mapping of elements to locations in an Array data structure , array puts almost every level in a different page. There are other heap variants which are efficient in computers using virtual memory or caches, such as cache-oblivious algorithms, k-heaps, and van Emde Boas tree , van Emde Boas layouts. Motivation Traditionally, binary trees are laid out in consecutive memory according to a n -> rule, meaning that if a node is at position n, its left and right child are taken to be at positions 2n and 2n+1 in the array. The root is at position 1. A common operation on binary trees is the vertical traversal; stepping down through the levels of a tree in order to arrive at a searched node. However, because of the way memor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Peek (data Type Operation)
In computer science, peek is an operation on certain abstract data types, specifically sequential collection (abstract data type), collections such as Stack (abstract data type), stacks and Queue (abstract data type), queues, which returns the value of the top ("front") of the collection without removing the element from the collection. It thus returns the same value as operations such as "pop" or "dequeue", but does not modify the data. The name "peek" is similar to the basic "push" and "pop" operations on a stack, but the name for this operation varies depending on data type and language. Peek is generally considered an inessential operation, compared with the more basic operations of adding and removing data, and as such is not included in the basic definition of these data types. However, since it is a useful operation and generally easily implemented, it is frequently included in practices, and in some definitions peek is included as basic, with pop (or analog) defined in terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Brodal Queue
In computer science, the Brodal queue is a heap/priority queue structure with very low worst case time bounds: O(1) for insertion, find-minimum, meld (merge two queues) and decrease-key and O(\mathrm(n)) for delete-minimum and general deletion. They are the first heap variant to achieve these bounds without resorting to amortization of operational costs. Brodal queues are named after their inventor Gerth Stølting Brodal.Gerth Stølting Brodal (1996). Worst-case efficient priority queues. Proc. 7th ACM-SIAM Symposium on Discrete Algorithms, pp. 52–58 While having better asymptotic bounds than other priority queue structures, they are, in the words of Brodal himself, "quite complicated" and " otapplicable in practice." Brodal and Okasaki describe a persistent ( purely functional) version of Brodal queues.Gerth Stølting Brodal and Chris Okasaki (1996)Optimal purely functional priority queues Journal of Functional Programming. Summary of running times Gerth Stølting Brodal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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2–3 Heap
In computer science, a 2–3 heap is a data structure that implements a priority queue. It is a variation on the heap (data structure), heap, designed by Tadao Takaoka in 1999. The structure is similar to a Fibonacci heap, and borrows ideas from the 2–3 tree. The time needed for some common heap operations are as follows. * ''Delete-min'' takes O(\log(n)) amortized time and in the worst case. * ''Decrease-key'' takes constant amortized time. * ''Insertion'' takes constant amortized time and O(\log(n)) time in the worst case. Polynomial of trees Source: A linear tree of size r is a sequential path of r nodes with the first node as a root of the tree and it is represented by a bold \mathbf (e.g. \mathbf is a linear tree of a single node). Product P = ST of two trees S and T, is a tree produced by replacing every node of S by a copy of T; for each edge of S, there is an edge in ST connecting the roots of the trees that replaced the endpoints of the edge. This definition of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Stirling's Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm of n. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to instead use the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |