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Adaptive Sort
A sorting algorithm falls into the adaptive sort family if it takes advantage of existing order in its input. It benefits from the presortedness in the input sequence – or a limited amount of disorder for various definitions of measures of disorder – and sorts faster. Adaptive sorting is usually performed by modifying existing sorting algorithms. Motivation Comparison-based sorting algorithms have traditionally dealt with achieving an optimal bound of '' O''(''n'' log ''n'') when dealing with time complexity. Adaptive sort takes advantage of the existing order of the input to try to achieve better times, so that the time taken by the algorithm to sort is a smoothly growing function of the size of the sequence ''and'' the disorder in the sequence. In other words, the more presorted the input is, the faster it should be sorted. This is an attractive feature for a sorting algorithm because nearly sorted sequences are common in practice. Thus, the performance of existing sor ... [...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] |
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 Algorithms
In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is important for optimizing the efficiency of other algorithms (such as search and merge algorithms) that require input data to be in sorted lists. Sorting is also often useful for 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 rather than one that allows only sequential access. History and concepts From the beginning of computi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Tree
In computer science, a Cartesian tree is a binary tree derived from a sequence of numbers; it can be uniquely defined from the properties that it is heap-ordered and that a symmetric (in-order) traversal of the tree returns the original sequence. Introduced by in the context of geometric range searching data structures, Cartesian trees have also been used in the definition of the treap and randomized binary search tree data structures for binary search problems. The Cartesian tree for a sequence may be constructed in linear time using a stack-based algorithm for finding all nearest smaller values in a sequence. Definition The Cartesian tree for a sequence of distinct numbers can be uniquely defined by the following properties: #The Cartesian tree for a sequence has one node for each number in the sequence. Each node is associated with a single sequence value. #A symmetric (in-order) traversal of the tree results in the original sequence. That is, the left subtree consists ... [...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] |
Splaysort
In computer science, splaysort is an adaptive comparison sorting algorithm based on the splay tree data structure. Algorithm The steps of the algorithm are: # Initialize an empty splay tree # For each data item in the input order, insert it into the splay tree # Traverse the splay tree in inorder to find the sorted order of the data Thus, the algorithm may be seen as a form of insertion sort or tree sort, using a splay tree to speed up each insertion. Analysis Based on the amortized analysis of splay trees, the worst case running time of splaysort, on an input with ''n'' data items, is ''O''(''n'' log ''n''), matching the time bounds for efficient non-adaptive algorithms such as quicksort, heap sort, and merge sort. For an input sequence in which most items are placed close to their predecessor in the sorted order, or are out of order with only a small number of other items, splaysort can be faster than ''O''(''n'' log ''n''), showing that it is an adaptive s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothsort
In computer science, smoothsort is a comparison-based sorting algorithm. A variant of heapsort, it was invented and published by Edsger Dijkstra in 1981. Like heapsort, smoothsort is an in-place algorithm with an upper bound of , but it is not a stable sort. The advantage of smoothsort is that it comes closer to time if the input is already sorted to some degree, whereas heapsort averages regardless of the initial sorted state. Overview Like heapsort, smoothsort organizes the input into a priority queue and then repeatedly extracts the maximum. Also like heapsort, the priority queue is an implicit heap data structure (a heap-ordered implicit binary tree), which occupies a prefix of the array. Each extraction shrinks the prefix and adds the extracted element to a growing sorted suffix. When the prefix has shrunk to nothing, the array is completely sorted. Heapsort maps the binary tree to the array using a top-down breadth-first traversal of the tree; the array begins with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shellsort
Shellsort, also known as Shell sort or Shell's method, is an in-place comparison sort. It can be seen as either a generalization of sorting by exchange ( bubble sort) or sorting by insertion (insertion sort). The method starts by sorting pairs of elements far apart from each other, then progressively reducing the gap between elements to be compared. By starting with far apart elements, it can move some out-of-place elements into position faster than a simple nearest neighbor exchange. Donald Shell published the first version of this sort in 1959. The running time of Shellsort is heavily dependent on the gap sequence it uses. For many practical variants, determining their time complexity remains an open problem. Description Shellsort is an optimization of insertion sort that allows the exchange of items that are far apart. The idea is to arrange the list of elements so that, starting anywhere, taking every ''h''th element produces a sorted list. Such a list is said to be ''h''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patience Sort
In computer science, patience sorting is a sorting algorithm inspired by, and named after, the card game patience. A variant of the algorithm efficiently computes the length of a longest increasing subsequence in a given 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 right of all of the existing piles, thus forming a new pile. # When there are no more cards remaining to deal, the game ends. This card game is turned into a two-phase sorting algorithm, as follows. Given an array of elements from some totally ordered domain, consider this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Heap Sort
In computer science, adaptive heap sort is a comparison-based sorting algorithm of the adaptive sort family. It is a variant of heap sort that performs better when the data contains existing order. Published by Christos Levcopoulos and Ola Petersson in 1992, the algorithm utilizes a new measure of presortedness, ''Osc,'' as the number of oscillations. Instead of putting all the data into the heap as the traditional heap sort did, adaptive heap sort only take part of the data into the heap so that the run time will reduce significantly when the presortedness of the data is high. Heapsort Heap sort is a sorting algorithm that utilizes binary heap data structure. The method treats an array as a complete binary tree and builds up a Max-Heap/Min-Heap to achieve sorting. It usually involves the following four steps. # Build a Max-Heap(Min-Heap): put all the data into the heap so that all nodes are either greater than or equal (less than or equal to for ''Min-Heap'') to each of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy. The fields of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversion (discrete Mathematics)
In computer science and discrete mathematics, an inversion in a sequence is a pair of elements that are out of their natural order. Definitions Inversion Let \pi be a permutation. There is an inversion of \pi between i and j if i \pi(j). The inversion is indicated by an ordered pair containing either the places (i, j) or the elements \bigl(\pi(i), \pi(j)\bigr). The inversion set is the set of all inversions. A permutation's inversion set using place-based notation is the same as the inverse permutation's inversion set using element-based notation with the two components of each ordered pair exchanged. Likewise, a permutation's inversion set using element-based notation is the same as the inverse permutation's inversion set using place-based notation with the two components of each ordered pair exchanged. Inversions are usually defined for permutations, but may also be defined for sequences:Let S be a sequence (or multiset permutation). If i S(j), either the pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
