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Heap (computer Science)
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] |
Radix Tree
In computer science, a radix tree (also radix trie or compact prefix tree or compressed trie) is a data structure that represents a space-optimized trie (prefix tree) in which each node that is the only child is merged with its parent. The result is that the number of children of every internal node is at most the radix of the radix tree, where = 2 for some integer ≥ 1. Unlike regular trees, edges can be labeled with sequences of elements as well as single elements. This makes radix trees much more efficient for small sets (especially if the strings are long) and for sets of strings that share long prefixes. Unlike regular trees (where whole keys are compared ''en masse'' from their beginning up to the point of inequality), the key at each node is compared chunk-of-bits by chunk-of-bits, where the quantity of bits in that chunk at that node is the radix of the radix trie. When is 2, the radix trie is binary (i.e., compare that node's 1-bit portion of the key), which mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
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] |
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] |
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] |
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] |
Heapsort
In computer science, heapsort is an efficient, comparison-based sorting algorithm that reorganizes an input array into a heap (a data structure where each node is greater than its children) and then repeatedly removes the largest node from that heap, placing it at the end of the array in a similar manner to Selection sort. Although somewhat slower in practice on most machines than a well-implemented quicksort, it has the advantages of very simple implementation and a more favorable worst-case runtime. Most real-world quicksort variants include an implementation of heapsort as a fallback should they detect that quicksort is becoming degenerate. Heapsort is an in-place algorithm, but it is not a stable sort. Heapsort was invented by J. W. J. Williams in 1964. The paper also introduced the binary heap as a useful data structure in its own right. In the same year, Robert W. Floyd published an improved version that could sort an array in-place, continuing his earlier research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Data Structure
In computer science, an implicit data structure or space-efficient data structure is a data structure that stores very little information other than the main or required data: a data structure that requires low overhead. They are called "implicit" because the position of the elements carries meaning and relationship between elements; this is contrasted with the use of pointers to give an ''explicit'' relationship between elements. Definitions of "low overhead" vary, but generally means constant overhead; in big O notation, ''O''(1) overhead. A less restrictive definition is a succinct data structure, which allows greater overhead. Definition An implicit data structure is one with constant space overhead (above the information-theoretic lower bound). Historically, defined an implicit data structure (and algorithms acting on one) as one "in which structural information is implicit in the way data are stored, rather than explicit in pointers." They are somewhat vague in the def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
