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Transfer effects refer to the influence that learning or performance in one context or task has on performance in another, potentially unrelated, context or task. These effects can be positive, facilitating learning and performance, or negative, causing interference and confusion, and are crucial in understanding cognitive processes and educational practices.
Relevant Degrees
Concept
Computational complexity is a branch of computer science that studies the resources required for algorithms to solve problems, focusing on time and space as primary metrics. It categorizes problems based on their inherent difficulty and the efficiency of the best possible algorithms that solve them, providing a framework for understanding what can be computed feasibly.
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Polynomial time refers to the class of computational problems for which the time required to solve the problem using an algorithm is a polynomial function of the size of the input. This is significant in computer science because problems solvable in Polynomial time are considered efficiently solvable or 'tractable'.
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NP-Complete problems are a class of problems in computational complexity theory that are both in NP and as hard as any problem in NP, meaning that if any NP-Complete problem can be solved efficiently, then every problem in NP can be solved efficiently. They are central to understanding the limits of what can be efficiently computed, and no polynomial-time algorithm is known for any NP-Complete problem, making them a key focus in the study of P vs NP problem.
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Reduction refers to the process of simplifying a complex problem, system, or expression into a more manageable form, often by breaking it down into more fundamental components. This approach is widely used across various disciplines to enhance understanding, facilitate problem-solving, and improve efficiency in analysis and computation.
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The P vs NP Problem is a fundamental question in computer science that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. Solving this problem would have profound implications for fields such as cryptography, algorithm design, and computational complexity theory.
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Exponential time refers to the computational complexity of an algorithm whose growth doubles with each addition to the input data set, making it impractical for large inputs. This is often contrasted with polynomial time, where the growth of the algorithm is more manageable as the input size increases.
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Approximation algorithms are designed to find near-optimal solutions to optimization problems where finding the exact solution is computationally infeasible. They are particularly useful for NP-hard problems, providing solutions that are provably close to the best possible answer within a guaranteed performance ratio or approximation factor.
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Decision problems are questions with a yes or no answer, often used in computational complexity to determine whether a given problem can be solved within certain resource constraints. They are fundamental in distinguishing between different complexity classes, such as P, NP, and NP-complete, which help in understanding the efficiency of algorithms.
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Optimization problems involve finding the best solution from a set of feasible solutions, often under given constraints. They are fundamental in various fields such as operations research, economics, and computer science, where the goal is to maximize or minimize an objective function.
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The Cook-Levin Theorem, also known as the Cook's theorem, establishes that the Boolean satisfiability problem (SAT) is NP-complete, serving as the first known problem to be classified as such. This theorem laid the groundwork for the theory of NP-completeness, showing that if SAT can be solved in polynomial time, then every problem in NP can also be solved in polynomial time.
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The approximation ratio is a measure used in algorithm design to quantify how close the solution provided by an approximation algorithm is to the optimal solution. It is particularly important in the context of NP-hard problems, where finding exact solutions efficiently is often infeasible, and thus, approximate solutions with provable bounds are sought instead.
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Packing and covering are fundamental concepts in combinatorial optimization and geometry, focusing on how to efficiently fill or cover a space with geometric shapes without overlap (packing) or with complete coverage (covering). These concepts have applications in fields such as network design, coding theory, and resource allocation, where optimal space utilization or coverage is critical.
A Polynomial-Time Approximation Scheme (PTAS) is an algorithmic framework that provides solutions to optimization problems within any desired degree of accuracy, specified by a Parameter ε, in polynomial time. PTAS is significant for NP-hard problems where exact solutions are computationally infeasible, enabling approximate solutions that are practically useful in many applications.
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Exact algorithms are computational methods designed to solve problems with precision, delivering optimal solutions without approximation. They are essential for problems where accuracy is critical, often used in contexts like combinatorial optimization and integer programming.
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Computational infeasibility refers to problems that require an impractical amount of time or resources to solve with current technology, often due to their exponential growth in complexity as the problem size increases. This concept is crucial in cryptography, where security relies on the assumption that certain mathematical problems cannot be efficiently solved by adversaries.
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Branch and Bound is an algorithmic method for solving optimization problems, particularly useful in discrete and combinatorial optimization. It systematically explores the solution space by creating branches and uses bounds to prune sections that cannot contain optimal solutions, thus improving efficiency.
A Fully Polynomial Time Approximation Scheme (FPTAS) is an algorithm that provides solutions to optimization problems within any desired degree of accuracy and runs in Polynomial Time concerning both the input size and the reciprocal of the error parameter. It is particularly useful for NP-hard problems where exact solutions are computationally infeasible, offering a trade-off between solution accuracy and computational efficiency.
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