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Shortest Common Supersequence Problem
In computer science, the shortest common supersequence of two sequences X and Y is the shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subsequence problem. Given two sequences X = and Y = , a sequence U = is a common supersequence of X and Y if items can be removed from U to produce X and Y. A shortest common supersequence (SCS) is a common supersequence of minimal length. In the SCS problem, two sequences X and Y are given, and the task is to find a shortest possible common supersequence of these sequences. In general, an SCS is not unique. For two input sequences, an SCS can be formed from a longest common subsequence (LCS) easily. For example, the longest common subsequence of X ..m= abcbdab and Y ..n= bdcaba is Z ..L= bcba. By inserting the non-LCS symbols into Z while preserving their original order, we obtain a shortest common supersequence U ..S= abdcabdab. In particular, the equation L + S = m + n holds for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a partial order. Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as \langle B,C,D \rangle, from \langle A,B,C,D,E,F \rangle, is a substring. The substring is a refinement of the subsequence. The list of all subsequences for the word "apple" would be "''a''", "''ap''", "''al''", "''ae''", "''app''", "''apl''", "''ape''", "''ale''", "''appl''", "''appe''", "''aple''", "''apple''", "''p''", "''pp''", "''pl''", "''pe''", "''ppl''", "''ppe''", " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Longest Common Subsequence Problem
A longest common subsequence (LCS) is the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from the longest common substring: unlike substrings, subsequences are not required to occupy consecutive positions within the original sequences. The problem of computing longest common subsequences is a classic computer science problem, the basis of data comparison programs such as the diff utility, and has applications in computational linguistics and bioinformatics. It is also widely used by revision control systems such as Git for reconciling multiple changes made to a revision-controlled collection of files. For example, consider the sequences (ABCD) and (ACBAD). They have five length-2 common subsequences: (AB), (AC), (AD), (BD), and (CD); two length-3 common subsequences: (ABD) and (ACD); and no longer common subsequences. So (ABD) and (ACD) are their longest common subsequences. Complexity For the general case of an ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Problem
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Cover Problem
The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory. Given a set of elements (henceforth referred to as the universe, specifying all possible elements under consideration) and a collection, referred to as , of a given subsets whose union equals the universe, the set cover problem is to identify a smallest sub-collection of whose union equals the universe. For example, consider the universe, and the collection of sets In this example, is equal to 4, as there are four subsets that comprise this collection. The union of is equal to . However, we can cover all elements with only two sets: , see picture, but not with only one set. Therefore, the solution to the set cover problem for this and has size 2. More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a set cover is a subfamily \mathcal\subseteq\mathcal of sets whose union is \mathcal. * In the set cover deci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximation Algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problems On Strings
A problem is a difficulty which may be resolved by problem solving. Problem(s) or The Problem may also refer to: People * Problem (rapper), (born 1985) American rapper Books * Problems (Aristotle), ''Problems'' (Aristotle), an Aristotelian (or pseudo-Aristotelian) collection of problems in question and answer form * The Problem (play), ''The Problem'' (play), by A. R. Gurney Film and TV * Problems (TV series), ''Problems'' (TV series), a 2012 Australian comedy television series. * ''The Problem with Jon Stewart'', a former American current affairs television series. Music Albums * The Problem (album), ''The Problem'' (album), by Mathematics * Problems (album), ''Problems'' (album), a 2019 album by The Get Up Kids Songs *Problem (Ariana Grande song), "Problem" (Ariana Grande song), 2014 *Problems (Matt Corby song), "Problems" (Matt Corby song), 2022 *Problem (Natalia Kills song), "Problem" (Natalia Kills song), 2013 *Problems (The Everly Brothers song), "Problems" (The Everly Bro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Languages
In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a set called "#Definition, alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also called "words"). Words that belong to a particular formal language are sometimes called Formal language#Definition, ''well-formed words''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar. In computer science, formal languages are used, among others, as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages, in which the words of the language represent concepts that are associated with meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
