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Pseudopolynomial Time Number Partitioning
In computer science, pseudopolynomial time number partitioning is a pseudopolynomial time algorithm for solving the partition problem. The problem can be solved using dynamic programming when the size of the set and the size of the sum of the integers in the set are not too big to render the storage requirements infeasible. Suppose the input to the algorithm is a multiset S of cardinality N: : ''S'' = Let ''K'' be the sum of all elements in ''S''. That is: ''K'' = ''x''1 + ... + ''xN''. We will build an algorithm that determines whether there is a subset of ''S'' that sums to \lfloor K/2 \rfloor . If there is a subset, then: : if ''K'' is even, the rest of ''S'' also sums to \lfloor K/2 \rfloor : if ''K'' is odd, then the rest of ''S'' sums to \lceil K/2 \rceil . This is as good a solution as possible. e.g.1 ''S'' = , ''K'' = ''sum(S)'' = 11, ''K/2'' = 5, Find a subset from ''S'' that is closest to ''K/2'' -> = 5, 11 - 5 * 2 = 1 e.g.2 ''S'' = , ''K'' = ''sum(S)'' = 11, '' ... [...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] |
Pseudo-polynomial Time
In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of the input (the number of bits required to represent it), which is the case for polynomial time algorithms.Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979. In general, the numeric value of the input is exponential in the input length, which is why a pseudo-polynomial time algorithm does not necessarily run in polynomial time with respect to the input length. An NP-complete problem with known pseudo-polynomial time algorithms is called weakly NP-complete. An NP-complete problem is called strongly NP-complete if it is proven that it cannot be solved by a pseudo-polynomial time algorithm unless . The strong/weak kinds of NP-hardness are defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem
In number theory and computer science, the partition problem, or number partitioning, is the task of deciding whether a given multiset ''S'' of positive integers can be partition of a set, partitioned into two subsets ''S''1 and ''S''2 such that the sum of the numbers in ''S''1 equals the sum of the numbers in ''S''2. Although the partition problem is NP-complete, there is a pseudo-polynomial time dynamic programming solution, and there are Heuristic, heuristics that solve the problem in many instances, either optimally or approximately. For this reason, it has been called "the easiest hard problem". There is an optimization problem, optimization version of the partition problem, which is to partition the multiset ''S'' into two subsets ''S''1, ''S''2 such that the difference between the sum of elements in ''S''1 and the sum of elements in ''S''2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. The partition problem is a special case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Problem DP Table Showing Dependencies
Partition may refer to: Arts and entertainment Film and television * ''Partition'' (1987 film), directed by Ken McMullen * ''Partition'' (2007 film), directed by Vic Sarin * '' Partition: 1947'', or ''Viceroy's House'', a 2017 film Music * Partition (music), a segment created from a set * "Partition" (song), by Beyoncé, 2014 * ''Partition'', soundtrack to the 2007 film by Brian Tyler Law and politics * Partition (law), the division of an estate * Partition (politics), a change of political borders, including: ** Partition of India ** Partition of Ireland ** Partitions of Poland Science and technology Computing * Disk partitioning, the division of a hard disk drive * Memory partition, a memory management technique * Partition (database), the division of a logical database * Logical partition, a subset of a computer's resources, virtualized as a separate computer * Binary space partitioning, in computer science * Partition problem, in number theory and computer science Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IJCAI
The International Joint Conference on Artificial Intelligence (IJCAI) is a conference in the field of artificial intelligence. The conference series has been organized by the nonprofit IJCAI Organization since 1969.Jointly sponsored by the IJCAI Organization and the hosting national AI societies. It was held biennially in odd-numbered years from 1969 to 2015 and annually starting from 2016. More recently, IJCAI was held jointly every four years with ECAI since 2018 and PRICAI since 2020 to promote collaboration of AI researchers and practitioners. IJCAI covers a broad range of research areas in the field of AI. It is a large and highly selective conference, with only about 20% or less of the submitted papers accepted after peer review in the 5 years leading up to 2022. Awards Three research awards are given at each IJCAI conference. * The IJCAI Computers and Thought Award is given to outstanding young scientists under the age of 35 in AI. * The Donald E. Walker Distinguished Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Partitioning
In computer science, multiway number partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are as similar as possible. It was first presented by Ronald Graham in 1969 in the context of the identical-machines scheduling problem. The problem is parametrized by a positive integer ''k'', and called ''k''-way number partitioning. The input to the problem is a multiset ''S'' of numbers (usually integers), whose sum is ''k*T''. The associated decision problem is to decide whether ''S'' can be partitioned into ''k'' subsets such that the sum of each subset is exactly ''T''. There is also an optimization problem: find a partition of ''S'' into ''k'' subsets, such that the ''k'' sums are "as near as possible". The exact optimization objective can be defined in several ways: * Minimize the difference between the largest sum and the smallest sum. This objective is common in papers about multiway number partitioning, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
