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Fenwick Tree
A Fenwick tree or binary indexed tree (BIT) is a data structure that can efficiently update elements and calculate prefix sums in a table of numbers. This structure was proposed by Boris Ryabko in 1989 with a further modification published in 1992. It has subsequently become known under the name Fenwick tree after Peter Fenwick, who described this structure in his 1994 article. When compared with a flat array of numbers, the Fenwick tree achieves a much better balance between two operations: element update and prefix sum calculation. A flat array of n numbers can either store the elements or the prefix sums. In the first case, computing prefix sums requires linear time; in the second case, updating the array elements requires linear time (in both cases, the other operation can be performed in constant time). Fenwick trees allow both operations to be performed in O(\log n) time. This is achieved by representing the numbers as a tree with n+1 nodes where the value of each node in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix Sum
In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the sums of prefixes ( running totals) of the input sequence: : : : :... For instance, the prefix sums of the natural numbers are the triangular numbers: : Prefix sums are trivial to compute in sequential models of computation, by using the formula to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort,. and they form the basis of the scan higher-order function in functional programming languages. Prefix sums have also been much studied in parallel algorithms, both as a test problem to be solved and as a useful primitive to be used as a subroutine in other parallel algorithms.. Abstractly, a prefix sum requires only a binary associative operator ⊕, making it useful for many applications from calculating well-separated pai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Query (data Structures)
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. Definition A range query q_f(A,i,j) on an array A= _1,a_2,..,a_n/math> of ''n'' elements of some set , denoted A ,n/math>, takes two indices 1\leq i\leq j\leq n, a function defined over arrays of elements of and outputs f(A ,j= f(a_i,\ldots,a_j). For example, for f = \sum and A ,n/math> an array of numbers, the range query \sum_ A computes \sum A ,j= (a_i+\ldots + a_j), for any 1 \leq i \leq j \leq n. These queries may be answered in constant time and using O(n) extra space by calculating the sums of the first elements of and storing them into an auxiliary array , such that B /mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trees (data Structures)
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are usable as lumber or plants above a specified height. In wider definitions, the taller palms, tree ferns, bananas, and bamboos are also trees. Trees are not a taxonomic group but include a variety of plant species that have independently evolved a trunk and branches as a way to tower above other plants to compete for sunlight. The majority of tree species are angiosperms or hardwoods; of the rest, many are gymnosperms or softwoods. Trees tend to be long-lived, some reaching several thousand years old. Trees have been in existence for 370 million years. It is estimated that there are some three trillion mature trees in the world. A tree typically has many secondary branches supported clear of the ground by the trunk. This trunk typically co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segment Tree
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. A segment tree for a set of ''n'' intervals uses ''O''(''n'' log ''n'') storage and can be built in ''O''(''n'' log ''n'') time. Segment trees support searching for all the intervals that contain a query point in time ''O''(log ''n'' + ''k''), ''k'' being the number of retrieved intervals or segments. Applications of the segment tree are in the areas of computational geometry, geographic information systems and machine learning. The segment tree can be generalized to higher dimension spaces. Definition Description Let ''S'' be a set of intervals, or segments. Let ''p''1, ''p''2, ..., ''pm'' be the list of distinct interval endpoin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefix Sums
In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers is a second sequence of numbers , the sums of prefixes (running totals) of the input sequence: : : : :... For instance, the prefix sums of the natural numbers are the triangular numbers: : Prefix sums are trivial to compute in sequential models of computation, by using the formula to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort,. and they form the basis of the scan higher-order function in functional programming languages. Prefix sums have also been much studied in parallel algorithms, both as a test problem to be solved and as a useful primitive to be used as a subroutine in other parallel algorithms.. Abstractly, a prefix sum requires only a binary associative operator ⊕, making it useful for many applications from calculating well-separated pai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Statistic Tree
In computer science, an order statistic tree is a variant of the binary search tree (or more generally, a B-tree) that supports two additional operations beyond insertion, lookup and deletion: * Select(''i'') – find the ''ith smallest element stored in the tree * Rank(''x'') – find the rank of element ''x'' in the tree, i.e. its index in the sorted list of elements of the tree Both operations can be performed in worst case time when a self-balancing tree is used as the base data structure. Augmented search tree implementation To turn a regular search tree into an order statistic tree, the nodes of the tree need to store one additional value, which is the size of the subtree rooted at that node (i.e., the number of nodes below it). All operations that modify the tree must adjust this information to preserve the invariant that size = size .html"_;"title="eft[x">eft[x_+_size[right[x_+_1 where_size[nil.html" ;"title="">eft[x_+_size[right[x.html" ;"title=".html" ;"title="eft[x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C (programming Language)
C (''pronounced like the letter c'') is a General-purpose language, general-purpose computer programming language. It was created in the 1970s by Dennis Ritchie, and remains very widely used and influential. By design, C's features cleanly reflect the capabilities of the targeted CPUs. It has found lasting use in operating systems, device drivers, protocol stacks, though decreasingly for application software. C is commonly used on computer architectures that range from the largest supercomputers to the smallest microcontrollers and embedded systems. A successor to the programming language B (programming language), B, C was originally developed at Bell Labs by Ritchie between 1972 and 1973 to construct utilities running on Unix. It was applied to re-implementing the kernel of the Unix operating system. During the 1980s, C gradually gained popularity. It has become one of the measuring programming language popularity, most widely used programming languages, with C compilers avail ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have ''optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; Rives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Heap
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. 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 (logarithmic time) algorithms are known for the two ope ... [...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 defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Fenwick (computer Scientist) (1932–2002), who wrote as Peter Fenwick
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Peter Fenwick may refer to: * Peter Fenwick (politician) (born 1944), Canadian politician * Peter Fenwick (neuropsychologist) (born 1935), neuropsychiatrist and neurophysiologist * Peter Holmes (businessman) Sir Peter Fenwick Holmes (27 September 1932 – 8 March 2002) was a British businessman who was the chairman of Shell Transport and Trading from 1985 to 1993 and chairman of the Royal Dutch/Shell Group from 1992 to 1993. Early life Holmes w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
