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Balanced Merge Sort
In computer science, merge sort (also commonly spelled as mergesort) is an efficient, general-purpose, and comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the order of equal elements is the same in the input and output. Merge sort is a divide-and-conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up merge sort appeared in a report by Goldstine and von Neumann as early as 1948. Algorithm Conceptually, a merge sort works as follows: #Divide the unsorted list into ''n'' sublists, each containing one element (a list of one element is considered sorted). #Repeatedly merge sublists to produce new sorted sublists until there is only one sublist remaining. This will be the sorted list. Top-down implementation Example C-like code using indices for top-down merge sort algorithm that recursively splits the list (called ''runs'' in this example) into sublists until sublist size i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sorting Algorithm
In computer science, a sorting algorithm is an algorithm that puts elements of a List (computing), list into an Total order, order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the Algorithmic efficiency, efficiency of other algorithms (such as search algorithm, search and merge algorithm, merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for Canonicalization, canonicalizing data and for producing human-readable output. Formally, the output of any sorting algorithm must satisfy two conditions: # The output is in monotonic order (each element is no smaller/larger than the previous element, according to the required order). # The output is a permutation (a reordering, yet retaining all of the original elements) of the input. For optimum efficiency, the input data should be stored in a data structure which allows random access ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Theorem (analysis Of Algorithms)
In the analysis of algorithms, the master theorem for divide-and-conquer recurrences provides an asymptotic analysis (using Big O notation) for recurrence relations of types that occur in the analysis of many divide and conquer algorithms. The approach was first presented by Jon Bentley, Dorothea Blostein (née Haken), and James B. Saxe in 1980, where it was described as a "unifying method" for solving such recurrences. The name "master theorem" was popularized by the widely used algorithms textbook ''Introduction to Algorithms'' by Cormen, Leiserson, Rivest, and Stein. Not all recurrence relations can be solved with the use of this theorem; its generalizations include the Akra–Bazzi method. Introduction Consider a problem that can be solved using a recursive algorithm such as the following: procedure p(input ''x'' of size ''n''): if ''n'' 1). Crucially, a and b must not depend on n. The theorem below also assumes that, as a base case for the recurrence, T(n)=\Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patience Sorting
In computer science, patience sorting is a sorting algorithm inspired by, and named after, the card game patience (or forbearance) is the ability to endure difficult circumstances. Patience may involve perseverance in the face of delay; tolerance of provocation without responding in disrespect/anger; or forbearance when under strain, especially when faced .... A variant of the algorithm efficiently computes the length of a longest increasing subsequence in a given Array data structure, array. Overview The algorithm's name derives from a simplified variant of the patience card game. The game begins with a shuffled deck of cards. The cards are dealt one by one into a sequence of piles on the table, according to the following rules. # Initially, there are no piles. The first card dealt forms a new pile consisting of the single card. # Each subsequent card is placed on the leftmost existing pile whose top card has a value greater than or equal to the new card's value, or to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tournament Sort
Tournament sort is a sorting algorithm. It improves upon the naive selection sort by using a priority queue to find the next element in the sort. In the naive selection sort, it takes O(''n'') operations to select the next element of ''n'' elements; in a tournament sort, it takes O(log ''n'') operations (after building the initial tournament in O(''n'')). Tournament sort is a variation of heapsort. Common application Tournament replacement selection sorts are used to gather the initial runs for external sorting algorithms. Conceptually, an external file is read and its elements are pushed into the priority queue until the queue is full. Then the minimum element is pulled from the queue and written as part of the first run. The next input element is read and pushed into the queue, and the min is selected again and added to the run. There's a small trick that if the new element being pushed into the queue is less than the last element added to the run, then the element's s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timsort
Timsort is a hybrid, stable sorting algorithm, derived from merge sort and insertion sort, designed to perform well on many kinds of real-world data. It was implemented by Tim Peters in 2002 for use in the Python programming language. The algorithm finds subsequences of the data that are already ordered (runs) and uses them to sort the remainder more efficiently. This is done by merging runs until certain criteria are fulfilled. Timsort has been Python's standard sorting algorithm since version 2.3. It is also used to sort arrays of non-primitive type in Java SE 7, on the Android platform, in GNU Octave, on V8, Swift, and Rust. It uses techniques from Peter McIlroy's 1993 paper "Optimistic Sorting and Information Theoretic Complexity". Operation Timsort was designed to take advantage of ''runs'' of consecutive ordered elements that already exist in most real-world data, ''natural runs''. It iterates over the data collecting elements into runs and simultaneously putting those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stack (abstract Data Type)
In computer science, a stack is an abstract data type that serves as a collection of elements, with two main operations: * Push, which adds an element to the collection, and * Pop, which removes the most recently added element that was not yet removed. Additionally, a peek operation can, without modifying the stack, return the value of the last element added. Calling this structure a ''stack'' is by analogy to a set of physical items stacked one atop another, such as a stack of plates. The order in which an element added to or removed from a stack is described as last in, first out, referred to by the acronym LIFO. As with a stack of physical objects, this structure makes it easy to take an item off the top of the stack, but accessing a datum deeper in the stack may require taking off multiple other items first. Considered as a linear data structure, or more abstractly a sequential collection, the push and pop operations occur only at one end of the structure, referred to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Queue (abstract Data Type)
In computer science, a queue is a collection of entities that are maintained in a sequence and can be modified by the addition of entities at one end of the sequence and the removal of entities from the other end of the sequence. By convention, the end of the sequence at which elements are added is called the back, tail, or rear of the queue, and the end at which elements are removed is called the head or front of the queue, analogously to the words used when people line up to wait for goods or services. The operation of adding an element to the rear of the queue is known as ''enqueue'', and the operation of removing an element from the front is known as ''dequeue''. Other operations may also be allowed, often including a ''peek'' or ''front'' operation that returns the value of the next element to be dequeued without dequeuing it. The operations of a queue make it a first-in-first-out (FIFO) data structure. In a FIFO data structure, the first element added to the queue will b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Run Of A Sequence
In computer science, a run of a sequence is a non-decreasing range of the sequence that cannot be extended. The ''number of runs'' of a sequence is the number of increasing subsequences of the sequence. This is a measure of presortedness, and in particular measures how many subsequences must be merged to sort a sequence. Definition Let X=\langle x_1,\dots,x_n\rangle be a sequence of elements from a totally ordered set. A run of X is a maximal increasing sequence \langle x_i,x_,\dots, x_,x_j \rangle. That is, x_>x_i and x_>x_ assuming that x_ and x_ exists. For example, if n is a natural number, the sequence \langle n+1,n+2,\dots, 2n, 1,2,\dots, n\rangle has the two runs \langle n+1,\dots, 2n \rangle and \langle 1,\dots,n \rangle. Let \mathtt(X) be defined as the number of positions i such that 1\le i |
Lisp Programming Language
Lisp (historically LISP) is a family of programming languages with a long history and a distinctive, fully parenthesized prefix notation. Originally specified in 1960, Lisp is the second-oldest high-level programming language still in common use, after Fortran. Lisp has changed since its early days, and many dialects have existed over its history. Today, the best-known general-purpose Lisp dialects are Common Lisp, Scheme, Racket and Clojure. Lisp was originally created as a practical mathematical notation for computer programs, influenced by (though not originally derived from) the notation of Alonzo Church's lambda calculus. It quickly became a favored programming language for artificial intelligence (AI) research. As one of the earliest programming languages, Lisp pioneered many ideas in computer science, including tree data structures, automatic storage management, dynamic typing, conditionals, higher-order functions, recursion, the self-hosting compiler, and the read†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quicksort
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer algorithm. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. For this reason, it is sometimes called partition-exchange sort. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. Most implementations of quicksort are not stable, meaning that the relative order of equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. Volumes 1–5 are intended to represent the central core of computer programming for sequential machines. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment of Volume 4 appeared in paperback as Fascicle 2 in 2005. The hardback Volume 4A, combining Volume 4, Fascicles 0–4, was published in 2011. Volume 4, Fascicle 6 ("Satisfiability") was rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
