NP-equivalent - NP-equivalent In complexity theory, NP-equivalent is the set of problems that are both NP-easy and NP-hard. NP-equivalent is similar to NP-complete, except the problems in NP-equivalent do not have to be decision problems. For example, the problem FIND-SUBSET-SUM is in NP-equivalent. Given a set of integers, FIND-SUBSET-SUM is the problem of finding some subset of the integers that adds up to zero (or returning the empty set if there is no such subset). This optimization problem is similar to the decision problem SUBSET-SUM. Given a set of integers, SUBSET-SUM is the problem of finding whether there exists a subset summing to zero. SUBSET-SUM is NP-complete. To show that FIND-SUBSET-SUM is NP-equivalent, we must show that it is both NP-hard and NP-easy. Clearly it is NP-hard. If.
NP-complete - NP-complete In complexity theory, the NP-complete problems are the hardest problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use that algorithm to solve all NP problems quickly. (A more formal definition is given below. See also Complexity classes P and NP). One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them adds up to zero. A supposed answer is very easy to verify for correctness, but no-one knows a faster way to solve the problem than to try every single possible.
NP-hard - NP-hard In computational complexity theory, NP-hard (Non-deterministic Polynomial-time hard) refers to the class of decision problems that contains all problems H such that for all decision problems L in NP there is a polynomial-time many-one reduction to H. Informally this class can be described as containing the decision problems that are at least as hard as any problem in NP. This intuition is supported by the fact that if we can find an algorithm A that solves one of these problems H in polynomial time then we can construct a polynomial time algorithm for every problem in NP by first executing the reduction from this problem to H and then executing the algorithm A. Assuming language L to be NP-complete, 1. L is in NP 2..
NP-easy - NP-easy In complexity theory, NP-easy is a set of problems similar to NP, except that NP-easy problems do not have to be decision problems. A problem X is in NP-easy if and only if there exists some problem Y in NP such that X is Turing reducible to Y. This means that given a black box that solves instances of Y in unit time, there exists an algorithm that solves X in polynomial time by repeatedly using that black box. An example of an NP-easy problem is the problem of sorting a list of strings. The decision problem "is string A greater than string B" is in NP. There are algorithms such as Quicksort that can sort the list using only a polynomial number of calls.
Knapsack problem - the knapsack problem is the question "can a value of at least V be achieved without exceeding the cost C?" The 0/1 knapsack problem restricts the number of each items to zero or one. Of particular interest is the special case of the problem with these properties: It is a decision problem It is a 0/1 problem For each item, the cost equals the value C = V Notice that in this special case, the problem is equivalent to this: given a set of integers, does any subset of it add up to exactly C? Or, if negative costs are allowed and C is chosen to be zero, the problem is: given a set of integers, does any subset add up to exactly 0? This special case is called the subset.
Graph coloring - of colors to the vertices such that no two adjacent vertices are assigned the same color. Here, "adjacent" means sharing the same edge. Graph coloring with colors is equivalent to the problem of partioning the vertex set into independent sets. The problem of coloring a graph has found a number of applications such as scheduling, register allocation in a microprocessor, frequency assignment in mobile radios, and pattern matching. In general, techniques for graph coloring concentrate on finding the least number of colors needed to color the graph ie. its chromatic number . For example the chromatic number of a complete graph of vertices(a graph with an edge between every two vertices), is . The problem of finding a minimum coloring of a graph is NP-hard. The corresponding decision problem(Is there a.
Algorithm - criteria for each case must be clear (and computable). Because an algorithm is a precise list of precise steps, the order of computation will almost always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting 'from the top' and going 'down to the bottom', an idea that is described more formally by flow of control. So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, 'mechanical' means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of 'memory' as a scratchpad. There is an.
Computation - by these early researchers. One model, the Turing machine, stores characters on an infinitely long tape, with one square at any given time being scanned by a read/write head. Another model, recursive functions, uses functions and function composition to operate on numbers. The lambda calculus uses a similar approach. Still others, including Markov algorithms and Post systems, use grammar-like rules to operate on strings. All of these formalisms were shown to be equivalent in computational power -- that is, any computation that can be performed with one can be performed with any of the others. They are also equivalent in power to the familiar electronic computer, if one pretends that electronic computers have infinite memory. Indeed, it is widely believed that all "proper" formalizations of the concept of algorithm will be.
Computational complexity theory - machine or language is being used. To avoid that problem, we generally use Big O notation. If a problem has time complexity O(n2) on one typical computer, then it will also have complexity O(n2) on most other computers, so this notation allows us to generalize away from the details of a particular computer. Example: Mowing grass has linear complexity because it takes double the time to mow double the area. However, looking up something in a dictionary has only logarithmic complexity because for a double sized dictionary you have to open it only one time more (e.g. exactly in the middle - then the problem is reduced to the half). Decision problems Much of complexity theory deals with decision problems. A decision problem is a problem where the answer is always.
Completeness - if every such functor has a colimit. In logic, a formal calculus (often just specified by a set of additional axioms used to formalize some theory within the underlying logic) is said to be complete if, for any statement P, a proof exists for P or for not P. A system is consistent if a proof never exists for both P and not P. Gödel's incompleteness theorem proved that no system as powerful as the Peano axioms can be both consistent and complete. See also below for another notion of completeness in logic. In proof theory and related fields of mathematical logic, a formal calculus is said to be complete with respect to a certain logic (i.e. wrt its semantics), if every statement P, that follows sematically from a set of.
Traveling salesman problem - problem can be stated as: Given a number of cities and the costs of travelling from one to the other, what is the cheapest roundtrip route that visits each city and then returns to the starting city? An equivalent formulation in terms of graph theory is: Find the shortest Hamiltonian cycle in a weighted graph. A related problem is the Bottleneck traveling salesman problem (bottleneck TSP): Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge. Table of contents showTocToggle("show","hide") 1 Computational complexity 2 Algorithms 2.1 Exact algorithms 2.2 Heuristics 2.3 Special cases 2.3.1 TSP with triangle inequality 3 External Links Computational complexity The most direct solution would be to try all the combinations and see which one is cheapest, but given that the number.
Stephen Cook - a noted computer scientist. Cook formalised the notion of NP-completeness in a famous 1971 paper "The Complexity of Theorem Proving Procedures", which also contained a proof that the boolean satisfiability problem was NP-complete. The paper left open theoretical computer science's greatest unsolved question - whether complexity classes P and NP are equivalent, the answer to which has eluded researchers since. Cook received the Turing Award in 1982 for his discovery. His citation reads: For his advancement of our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, Laid the foundations for the theory of NP-Completeness. The ensuing exploration of the boundaries and nature of NP-complete class of.
Standard enthalpy change of reaction - enthalpy change of reaction is the enthalpy change that occurs in a system when 1 equivalent of matter is transformed by a chemical reaction under standard conditions. For a generic chemical reaction nA A + nB B + ... → nP P + nQ Q ... the standard enthalpy change of reaction ΔHr is related with the standard enthalpy change of formation ΔHf of the reactants and products by the equation.
Subset sum problem - any subset sum to exactly zero? For example, given the set {-7, -3, -2, 5, 8}, the answer is YES because the subset {-3, -2, 5} sums to zero. The problem is NP-Complete, and is perhaps the simplest such problem to describe. An equivalent problem is this: given a set of integers and an integer s, does any subset sum to s? Subset sum can also be thought of as a special case of the knapsack problem. Although the problem was described above in terms of integers and addition, it can actually be defined using any group. For example, the problem could be: given an integer n and a set of integers in the range [0,n-1], does any subset sum to zero modulo n? This form of the problem has been.
P-adic number - expressed in base 5 as the limit of the sequence 0.1313131313...5. In this formulation, the integers are precisely those numbers which can be represented in the form where ai = 0 for all i<0. As an alternative, if we extend the p-adic expansions by allowing infinite sums of the form where k is some (not necessarily positive) integer, we obtain the field Qp of p-adic numbers. Those p-adic numbers for which ai = 0 for all i<0 are also called the p-adic integers. Intuitively, as opposed to p-adic expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p (as is done for the real numbers as described above), these are numbers whose p-adic expansion to the left are allowed to go on forever..
One-time pad - of the key (the traditional "Caesar cipher" used a single offset for the whole message). He then sends the encoded message to the receiver, who decrypts it with his copy of the first page of the pad. Both now discard the used key page, having used it only 'one-time'. One-time pads are information-theoretically secure, in that if all the conditions are met properly (i.e., the keys are chosen randomly, are at least as long as the message, and are never reused), then the ciphertext gives the attacking cryptanalyst no information about the message other than its length. This is a very strong notion of security, and implies that one-time pads are secure even against cryptanalysts with infinite computational power. Also, they are one of very few cryptosystems which can be implemented.
Oracle machine - a black box that computes a function in a single step. This could be a function solving an NP-complete problem such as the subset sum problem. It could even be an "uncomputable" (non-recursive) function like the halting problem. Table of contents showTocToggle("show","hide") 1 Oracle machines 2 Oracles and Complexity Theory 3 Oracles and Halting Problems 4 References in Popular Culture Oracle machines An oracle machine is a Turing machine connected to an oracle. The Turing machine can write on its own tape an input for the oracle, then tell the oracle to execute. In a single step, the oracle computes its function, erases its input, and writes its output to the tape. Sometimes the Turing machine is described as having two tapes, one of which is reserved for oracle inputs and.
Margin of error - interval from 57% to 63% is a 90% confidence interval for the proportion of voters in the whole population who will vote "yes". The radius of that interval is 3%; that is the margin of error. How to compute a margin of error Let n be the number of voters in the sample. Suppose them to have been drawn randomly and independently from the whole population of voters. This is perhaps optimistic, but if care is taken it can be at least approximated in reality. Let p be the proportion of voters in the whole population who will vote "yes". Then the number X of voters in the sample who will vote "yes" is a random variable with a binomial distribution with parameters n and p. If n is large enough,.
List of mathematical topics (M-O) - Morse theory -- Moser polygon notation -- Mrs. Miniver's problem -- M-theory -- Multidimensional scaling -- Multigrid -- Multilinear algebra -- Multiperfect number -- Multiplication -- Multiplication algorithm -- Multiplication table -- Multiplicative function -- Multiplicative order -- Multiplier -- Multiply perfect number -- Multiset -- Multivalued function -- Multivariate normal distribution -- Mumford, David -- Mutually exclusive -- Myhill-Nerode Theorem N N-body problem -- N-Mahlo cardinal -- N-sequence -- Naive Bayesian classification -- Naive set theory -- Napier -- Napier, John -- Napier's bones -- Nash embedding theorem -- Nash equilibrium -- Nash, John Forbes -- Natarajan Saradha -- Natural deduction -- Natural logarithm -- Natural number -- Natural numbers -- Natural transformation -- Navier-Stokes equations -- NC -- Nearest neighbour relation -- Necessary and sufficient -- Necker cube --.
List of computability and complexity topics - Halting problem Correctness Post correspondence problem Decidable language Undecidable language Word problem for groups Wang tile Definability questions Computable number Definable number Halting probability Algorithmic information theory Data compression Complexity theory Polynomial time Exponential time Time hierarchy theorem Polynomial-time many-one reduction Polynomial-time Turing reduction Complexity classes Complexity classes P and NP P-complete NP, NP-complete, NP-hard, NP-equivalent Co-NP, Co-NP-complete Sharp-P, Sharp-P-complete PSPACE, PSPACE-complete EXPSPACE, EXPTIME, EXPTIME-complete BPP BQP Class NC RP UP (complexity) ZPP Named problems Clique problem Hamiltonian cycle problem Integer factorization Knapsack problem Satisfiability problem 3-satisfiability Subset sum problem Traveling salesman problem Extensions Probabilistic algorithm Non-deterministic Turing machine Probabilistic Turing Machine Hypercomputation.
