This document summarizes a lecture on computational complexity. It introduces the concepts of tractable and intractable problems, and the complexity classes P and NP. Key problems discussed include the subset sum and 3SAT problems. The document explains that the subset sum problem is NP-complete, meaning it is as hard as the hardest problems in NP. It also discusses several other NP-complete problems like traveling salesman and graph coloring. In summary, the document provides an overview of computational complexity theory and the classification of problems by difficulty.
Reachability Analysis "Control Of Dynamical Non-Linear Systems" M Reza Rahmati
The document summarizes an approach for algorithmically synthesizing control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using ellipsoidal calculus. The method uses a first-order Taylor approximation of the nonlinear dynamics combined with a conservative approximation of the Lagrange remainder to transform the system into an affine form. The reachable sets are then over-approximated using ellipsoidal operations. An iterative algorithm is proposed to compute stabilizing controllers by solving constrained optimization problems to drive the system state into a target ellipsoidal set within a finite number of steps while satisfying input constraints.
This document provides an overview of algebraic techniques in combinatorics, including linear algebra concepts, partially ordered sets (posets), and examples of problems solved using these techniques. Some key points discussed are:
- Useful linear algebra facts such as rank, determinants, and vector/matrix properties
- Definitions and representations of posets, including Dilworth's theorem relating chains and antichains
- Examples of combinatorial problems solved using linear algebra tools such as vectors/matrices or applying Dilworth's theorem to obtain a divisibility relation poset
W.-S. Luk of Fudan University gave a lecture on feasibility problems. He discussed the feasible flow problem and feasible potential problem. For feasible flow, the problem has a solution if and only if the demand is less than or equal to the capacity of every cut. For feasible potential, the problem has a solution if and only if the upper span is greater than or equal to zero for every cycle. Both problems can be solved efficiently using network flow algorithms or by finding a negative cycle. He also provided examples and remarks on reducing problems.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
The document discusses a new type of fractional differential equation that combines a multi-point boundary condition and an integral boundary condition. It presents existence results for solutions to this type of equation. The paper defines relevant fractional calculus concepts and establishes Green's functions for the boundary value problem. It then applies fixed point theorems to show the existence of multiple positive solutions under certain conditions on the continuous function f in the equation. Examples are also provided to illustrate the results.
This paper proposes a method called tensorizing neural networks to compress neural network models by decomposing weight matrices into tensor trains. It first introduces background on neural networks and backpropagation. The motivation is to address the large memory requirements of neural networks. It then formulates the problem and discusses a naive low-rank SVD approach. The main idea is to recursively apply low-rank SVD and adaptively reshape matrices into higher-dimensional tensors to decompose them into tensor train formats. Experiments on MNIST and ImageNet show the tensor train decomposition approach can effectively compress large fully-connected layers with better results than the naive method.
Linear Transformation Vector Matrices and SpacesSohaib H. Khan
The document discusses linear transformations between vector spaces. It defines a linear transformation as a mapping between vector spaces that satisfies two conditions: 1) it is additive and 2) it is homogeneous. It also defines the kernel as the set of vectors that map to the zero vector, and the image as the set of vectors in the target space that are the image of vectors in the domain space. The document is about linear transformations presented by Dr. Yasir Ali for an advanced engineering mathematics course.
- Bayesian techniques can be used for parameter estimation problems where parameters are considered random variables with associated densities rather than fixed unknown values.
- Markov chain Monte Carlo (MCMC) methods like the Metropolis algorithm are commonly used to sample from the posterior distribution when direct sampling is impossible due to high-dimensional integration. The algorithm constructs a Markov chain whose stationary distribution is the target posterior density.
- At each step, a candidate value is generated from a proposal distribution and accepted or rejected based on the posterior ratio to the previous value. Over many iterations, the chain samples converge to the posterior distribution.
Reachability Analysis "Control Of Dynamical Non-Linear Systems" M Reza Rahmati
The document summarizes an approach for algorithmically synthesizing control strategies for discrete-time nonlinear uncertain systems based on reachable set computations using ellipsoidal calculus. The method uses a first-order Taylor approximation of the nonlinear dynamics combined with a conservative approximation of the Lagrange remainder to transform the system into an affine form. The reachable sets are then over-approximated using ellipsoidal operations. An iterative algorithm is proposed to compute stabilizing controllers by solving constrained optimization problems to drive the system state into a target ellipsoidal set within a finite number of steps while satisfying input constraints.
This document provides an overview of algebraic techniques in combinatorics, including linear algebra concepts, partially ordered sets (posets), and examples of problems solved using these techniques. Some key points discussed are:
- Useful linear algebra facts such as rank, determinants, and vector/matrix properties
- Definitions and representations of posets, including Dilworth's theorem relating chains and antichains
- Examples of combinatorial problems solved using linear algebra tools such as vectors/matrices or applying Dilworth's theorem to obtain a divisibility relation poset
W.-S. Luk of Fudan University gave a lecture on feasibility problems. He discussed the feasible flow problem and feasible potential problem. For feasible flow, the problem has a solution if and only if the demand is less than or equal to the capacity of every cut. For feasible potential, the problem has a solution if and only if the upper span is greater than or equal to zero for every cycle. Both problems can be solved efficiently using network flow algorithms or by finding a negative cycle. He also provided examples and remarks on reducing problems.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Existence results for fractional q-differential equations with integral and m...journal ijrtem
The document discusses a new type of fractional differential equation that combines a multi-point boundary condition and an integral boundary condition. It presents existence results for solutions to this type of equation. The paper defines relevant fractional calculus concepts and establishes Green's functions for the boundary value problem. It then applies fixed point theorems to show the existence of multiple positive solutions under certain conditions on the continuous function f in the equation. Examples are also provided to illustrate the results.
This paper proposes a method called tensorizing neural networks to compress neural network models by decomposing weight matrices into tensor trains. It first introduces background on neural networks and backpropagation. The motivation is to address the large memory requirements of neural networks. It then formulates the problem and discusses a naive low-rank SVD approach. The main idea is to recursively apply low-rank SVD and adaptively reshape matrices into higher-dimensional tensors to decompose them into tensor train formats. Experiments on MNIST and ImageNet show the tensor train decomposition approach can effectively compress large fully-connected layers with better results than the naive method.
Linear Transformation Vector Matrices and SpacesSohaib H. Khan
The document discusses linear transformations between vector spaces. It defines a linear transformation as a mapping between vector spaces that satisfies two conditions: 1) it is additive and 2) it is homogeneous. It also defines the kernel as the set of vectors that map to the zero vector, and the image as the set of vectors in the target space that are the image of vectors in the domain space. The document is about linear transformations presented by Dr. Yasir Ali for an advanced engineering mathematics course.
- Bayesian techniques can be used for parameter estimation problems where parameters are considered random variables with associated densities rather than fixed unknown values.
- Markov chain Monte Carlo (MCMC) methods like the Metropolis algorithm are commonly used to sample from the posterior distribution when direct sampling is impossible due to high-dimensional integration. The algorithm constructs a Markov chain whose stationary distribution is the target posterior density.
- At each step, a candidate value is generated from a proposal distribution and accepted or rejected based on the posterior ratio to the previous value. Over many iterations, the chain samples converge to the posterior distribution.
IIT JAM MATH 2021 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2021 Question Paper
IIT JAM Preparation Strategy
For any query about exams feel free to contact us
Call - 9836793076
This lecture discusses dimensionality reduction techniques for big data, specifically the Johnson-Lindenstrauss lemma. It introduces linear sketching as a dimensionality reduction method from n dimensions to t dimensions (where t is logarithmic in n). It then proves the JL lemma, which shows that for t proportional to 1/ε^2, the l2 distances between points are preserved to within a 1±ε factor. As an application, it discusses locality sensitive hashing (LSH) for approximate nearest neighbor search, where points close in distance hash to the same bucket with high probability.
Nelly Litvak presents a document on degree-degree dependencies in random graphs with heavy-tailed degrees. She discusses Newman's assortativity coefficient ρ(G) which is a measure of correlations between the degrees of connected nodes. Positive values indicate assortative mixing where high degree nodes connect to other high degree nodes, while negative values indicate disassortative mixing. She reviews that technological and biological networks are typically disassortative while social networks are assortative. Litvak then presents theorems showing that in power law graphs with γ ∈ (1,3), the assortativity coefficient converges to a non-negative value, so these graphs are never strongly disassortative. She also discusses
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
The document discusses statistical representation of random inputs in continuum models. It provides examples of representing random fields using the Karhunen-Loeve expansion, which expresses a random field as the sum of orthogonal deterministic basis functions and random variables. Common choices for the covariance function in the expansion include the radial basis function and limiting cases of fully correlated and uncorrelated fields. The covariance function can be approximated from samples of the random field to enable representation in applications.
The Kuhn-Munkres algorithm is used to find the maximum weighted matching in a weighted bipartite graph. It works by finding an initial feasible labeling and matching. It then searches for augmenting paths in the equality graph to improve the matching, or improves the labeling if no augmenting path exists to obtain a larger equality graph. The algorithm repeats this process until finding a perfect matching. An example applies the algorithm to find the optimal matching for a weighted bipartite graph.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
Density theorems for anisotropic point configurationsVjekoslavKovac1
This document discusses density theorems for anisotropic point configurations. Specifically:
- It summarizes previous results on density theorems for linear configurations in Euclidean spaces.
- It then presents new results on density theorems for anisotropic power-type scalings, where points are scaled by different powers in different coordinates.
- Theorems are proven for anisotropic simplices and boxes in such spaces, showing that any set of positive density must contain scaled copies of these configurations for scales above a certain threshold.
- The proofs use a multiscale approach involving pattern counting forms, smoothed counting forms, and analyzing the structured, uniform, and error parts that arise from decomposing the counting forms. Mult
Backtracking is a technique for solving problems by incrementally building candidates to the solutions, and abandoning each partial candidate ("backtracking") as soon as it is determined that the candidate cannot possibly be completed to a valid solution. It is useful for problems with constraints or complex conditions that are difficult to test incrementally. The key steps are: 1) systematically generate potential solutions; 2) test if a solution is complete and satisfies all constraints; 3) if not, backtrack and vary the previous choice. Backtracking has been used to solve problems like the N-queens puzzle, maze generation, Sudoku puzzles, and finding Hamiltonian cycles in graphs.
A Szemerédi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on establishing a Szemerédi-type theorem for subsets of the unit cube. It discusses using a "largeness/smoothness multiscale approach" to control three key quantities: the structured part using known bounds on Szemerédi's theorem, the error part involving multilinear singular integrals, and the uniform part using Gowers uniformity norms. The proof strategy is to show the existence of progression gaps in subsets of positive measure by bounding the sum of these quantities below a threshold. An open problem is establishing a stronger property for subsets of positive upper Banach density.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
This document discusses Newton's method for solving systems of nonlinear equations. It begins by introducing Newton's method in one dimension and extending it to multiple dimensions using a Jacobian matrix. It then proves that under certain conditions, Newton's method will converge quadratically to the solution. An example is provided to illustrate computing the Jacobian and using Newton's method. The document also discusses shooting methods for solving boundary value problems by converting them into initial value problems through an initial guess of a boundary condition.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document discusses how lack of access to information on weather, agriculture techniques and markets has hindered livelihood development in Kampong Thom province in Cambodia. It proposes providing this information to farmers and villagers through SMS on mobile phones to address issues like unproductive farming. An implementation in two groups in Kampong Thom provided agriculture knowledge, weather updates, market prices and meeting announcements via SMS. However, further funding, capacity building and official support are needed for nationwide scale-up of the mobile learning solution.
Plants have four basic parts - leaves, stems, roots, and flowers. Leaves use photosynthesis to produce food for the plant from sunlight, water and carbon dioxide. Stems transport water and nutrients from the roots and food from the leaves. Roots anchor the plant and absorb water and minerals from the soil. Flowers allow reproduction through pollination and development of fruits and seeds.
This document provides information and guidance for MHC class web coordinators. It discusses different communication methods and shows that classes prefer websites, letters, and the MHC Quarterly. It also outlines how to create a class website using WordPress and the roles and challenges of being a web coordinator. Tips are provided such as recruiting assistance, reviewing content for errors, and designing the site for low maintenance. Overall, the document aims to equip and encourage web coordinators in developing and maintaining class websites.
Vídeo director nuevas herram. 2011_revisadoaropaifo
Este curso sobre nuevas herramientas de gestión pública en instituciones docentes tiene como objetivos presentar conceptos clave y metodologías de gestión a través de un mapa conceptual. Participan varias instituciones docentes en el desarrollo del curso.
IIT JAM MATH 2021 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2021 Question Paper
IIT JAM Preparation Strategy
For any query about exams feel free to contact us
Call - 9836793076
This lecture discusses dimensionality reduction techniques for big data, specifically the Johnson-Lindenstrauss lemma. It introduces linear sketching as a dimensionality reduction method from n dimensions to t dimensions (where t is logarithmic in n). It then proves the JL lemma, which shows that for t proportional to 1/ε^2, the l2 distances between points are preserved to within a 1±ε factor. As an application, it discusses locality sensitive hashing (LSH) for approximate nearest neighbor search, where points close in distance hash to the same bucket with high probability.
Nelly Litvak presents a document on degree-degree dependencies in random graphs with heavy-tailed degrees. She discusses Newman's assortativity coefficient ρ(G) which is a measure of correlations between the degrees of connected nodes. Positive values indicate assortative mixing where high degree nodes connect to other high degree nodes, while negative values indicate disassortative mixing. She reviews that technological and biological networks are typically disassortative while social networks are assortative. Litvak then presents theorems showing that in power law graphs with γ ∈ (1,3), the assortativity coefficient converges to a non-negative value, so these graphs are never strongly disassortative. She also discusses
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
This document discusses orthogonal subspaces and inner products in advanced engineering mathematics. It defines the inner product of two vectors u and v in Rn as the transpose of u dotted with v, which results in a scalar. Two vectors are orthogonal if their inner product is 0. An orthogonal basis for a subspace W is a basis for W that is also an orthogonal set. The document also discusses orthogonal complements, projections, and inner products on function spaces.
The document discusses statistical representation of random inputs in continuum models. It provides examples of representing random fields using the Karhunen-Loeve expansion, which expresses a random field as the sum of orthogonal deterministic basis functions and random variables. Common choices for the covariance function in the expansion include the radial basis function and limiting cases of fully correlated and uncorrelated fields. The covariance function can be approximated from samples of the random field to enable representation in applications.
The Kuhn-Munkres algorithm is used to find the maximum weighted matching in a weighted bipartite graph. It works by finding an initial feasible labeling and matching. It then searches for augmenting paths in the equality graph to improve the matching, or improves the labeling if no augmenting path exists to obtain a larger equality graph. The algorithm repeats this process until finding a perfect matching. An example applies the algorithm to find the optimal matching for a weighted bipartite graph.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
Density theorems for anisotropic point configurationsVjekoslavKovac1
This document discusses density theorems for anisotropic point configurations. Specifically:
- It summarizes previous results on density theorems for linear configurations in Euclidean spaces.
- It then presents new results on density theorems for anisotropic power-type scalings, where points are scaled by different powers in different coordinates.
- Theorems are proven for anisotropic simplices and boxes in such spaces, showing that any set of positive density must contain scaled copies of these configurations for scales above a certain threshold.
- The proofs use a multiscale approach involving pattern counting forms, smoothed counting forms, and analyzing the structured, uniform, and error parts that arise from decomposing the counting forms. Mult
Backtracking is a technique for solving problems by incrementally building candidates to the solutions, and abandoning each partial candidate ("backtracking") as soon as it is determined that the candidate cannot possibly be completed to a valid solution. It is useful for problems with constraints or complex conditions that are difficult to test incrementally. The key steps are: 1) systematically generate potential solutions; 2) test if a solution is complete and satisfies all constraints; 3) if not, backtrack and vary the previous choice. Backtracking has been used to solve problems like the N-queens puzzle, maze generation, Sudoku puzzles, and finding Hamiltonian cycles in graphs.
A Szemerédi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on establishing a Szemerédi-type theorem for subsets of the unit cube. It discusses using a "largeness/smoothness multiscale approach" to control three key quantities: the structured part using known bounds on Szemerédi's theorem, the error part involving multilinear singular integrals, and the uniform part using Gowers uniformity norms. The proof strategy is to show the existence of progression gaps in subsets of positive measure by bounding the sum of these quantities below a threshold. An open problem is establishing a stronger property for subsets of positive upper Banach density.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
This document discusses Newton's method for solving systems of nonlinear equations. It begins by introducing Newton's method in one dimension and extending it to multiple dimensions using a Jacobian matrix. It then proves that under certain conditions, Newton's method will converge quadratically to the solution. An example is provided to illustrate computing the Jacobian and using Newton's method. The document also discusses shooting methods for solving boundary value problems by converting them into initial value problems through an initial guess of a boundary condition.
On an Optimal control Problem for Parabolic Equationsijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document discusses how lack of access to information on weather, agriculture techniques and markets has hindered livelihood development in Kampong Thom province in Cambodia. It proposes providing this information to farmers and villagers through SMS on mobile phones to address issues like unproductive farming. An implementation in two groups in Kampong Thom provided agriculture knowledge, weather updates, market prices and meeting announcements via SMS. However, further funding, capacity building and official support are needed for nationwide scale-up of the mobile learning solution.
Plants have four basic parts - leaves, stems, roots, and flowers. Leaves use photosynthesis to produce food for the plant from sunlight, water and carbon dioxide. Stems transport water and nutrients from the roots and food from the leaves. Roots anchor the plant and absorb water and minerals from the soil. Flowers allow reproduction through pollination and development of fruits and seeds.
This document provides information and guidance for MHC class web coordinators. It discusses different communication methods and shows that classes prefer websites, letters, and the MHC Quarterly. It also outlines how to create a class website using WordPress and the roles and challenges of being a web coordinator. Tips are provided such as recruiting assistance, reviewing content for errors, and designing the site for low maintenance. Overall, the document aims to equip and encourage web coordinators in developing and maintaining class websites.
Vídeo director nuevas herram. 2011_revisadoaropaifo
Este curso sobre nuevas herramientas de gestión pública en instituciones docentes tiene como objetivos presentar conceptos clave y metodologías de gestión a través de un mapa conceptual. Participan varias instituciones docentes en el desarrollo del curso.
The document discusses creating a five-year plan for class officers of Mount Holyoke alumni. It provides examples of goals that could be included in each year such as encouraging involvement after reunion, planning mini-reunions, and increasing participation in class activities and alumni fundraising. The class officers are responsible for creating and executing the plan with support from the Alumnae Association. A sample year one plan outlines goals and action steps for officers including updating the class website and encouraging participation in alumni events and donations.
Orca 34: a gentleman's yacht with excellent seakeeping characteristics.
Tiira 500GT: Go anywhere and enjoy the surroundings to the fullest. This yacht is rubost to the elements (bow) yet open to her surroundings on "the entertainment side".
The document provides guidance on developing an effective engagement and communications plan for a college alumni class. It recommends communicating on a regular basis to foster ongoing connections between alumni through various channels such as newsletters, websites, social media, and personal outreach. The plan should identify key messages to share including class and college updates and milestones, and establish a rhythm of communication tied to important events. It also offers best practices and examples of communications strategies and activities that can help build unity within the alumni class.
Problem | Problem v/s Algorithm v/s Program | Types of Problems | Computational complexity | P class v/s NP class Problems | Polynomial time v/s Exponential time | Deterministic v/s non-deterministic Algorithms | Functions of non-deterministic Algorithms | Non-deterministic searching Algorithm | Non-deterministic sorting Algorithm | NP - Hard and NP - Complete Problems | Reduction | properties of reduction | Satisfiability problem and Algorithm
This document discusses P, NP, NP-hard and NP-complete problems. It begins by defining tractable problems that can be solved in polynomial time as class P problems. Intractable problems that cannot be solved in reasonable time with increasing input size are also introduced. NP is the class of problems that can be solved by a non-deterministic machine in polynomial time. NP-hard problems are those that are at least as hard as the hardest problems in NP, and NP-complete problems are NP-hard problems that are also in NP. Common NP-complete problems like 3-SAT and the clique problem are reduced to each other to demonstrate their equivalence. Prior questions related to complexity classes are also addressed.
This document discusses P, NP, NP-hard and NP-complete problems. It begins by defining tractable problems that can be solved in polynomial time as class P problems. Intractable problems that cannot be solved in reasonable time with increasing input size are also introduced. NP is the class of problems that can be solved by a non-deterministic machine in polynomial time. NP-hard problems are those that are at least as hard as the hardest problems in NP, and NP-complete problems are NP-hard problems that are also in NP. Common NP-complete problems like 3-SAT and the clique problem are reduced to each other to demonstrate their equivalence. Prior questions related to complexity classes are also addressed.
Algorithm Design and Complexity - Course 6Traian Rebedea
This document provides an overview of algorithm design and complexity. It discusses different classes of problems including P vs NP problems. P problems can be solved in polynomial time, while NP problems can be verified in polynomial time but may not be solvable in polynomial time. NP-hard problems are at least as hard as NP problems, and NP-complete problems are NP-hard problems that are also in NP. The document describes techniques for solving difficult problems like backtracking and discusses examples like the n-queens problem.
1) The document discusses the complexity classes P, NP, NP-hard and NP-complete. P refers to problems that can be solved in polynomial time, while NP includes problems that can be verified in polynomial time.
2) NP-hard problems are at least as hard as the hardest problems in NP. NP-complete problems are the hardest problems in NP. If any NP-complete problem could be solved in polynomial time, then P would be equal to NP.
3) Common NP-complete problems discussed include the traveling salesman problem and integer knapsack problem. Reductions are used to show that one problem is at least as hard as another.
The document discusses computational complexity problems that are solvable in polynomial time but for which no significantly faster algorithms are known. It presents several such problems from areas like graph algorithms, computational biology, and computational geometry. It then discusses recent work that aims to establish conditional lower bounds for the runtime of such problems by relating their hardness to standard conjectures like 3SUM, APSP, SETH, orthogonal vectors, and small universe hitting set. Fine-grained reductions are used to show relationships between problems. Overall, the document outlines an approach for proving conditional lower bounds for problems solvable in polynomial time based on reasonable complexity theoretic conjectures.
1) NP-Completeness refers to problems that are in NP (can be verified in polynomial time) and are as hard as any problem in NP.
2) The first problem proven to be NP-Complete was the Circuit Satisfiability problem, which asks whether there exists an input assignment that makes a Boolean circuit output 1.
3) To prove a problem P is NP-Complete, it must be shown that P is in NP and that any problem in NP can be reduced to P in polynomial time. This establishes P as at least as hard as any problem in NP.
This document discusses branch and bound algorithms and NP-hard and NP-complete problems. It provides examples and proofs related to these topics.
1) Branch and bound is an algorithm that systematically enumerates candidate solutions by discarding subsets that are provably suboptimal. The knapsack problem is used as an example problem.
2) NP-hard and NP-complete problems are those whose best known algorithms run in non-polynomial time. If a problem can be solved in polynomial time, then all NP-complete problems can be. Proving problems NP-complete involves reducing other known NP-complete problems to the target problem.
3) Trees are connected graphs without cycles. They are used to represent hierarchies and
A Stochastic Limit Approach To The SAT ProblemValerie Felton
This document proposes using quantum adaptive stochastic systems to solve NP-complete problems like SAT in polynomial time. It summarizes the SAT problem, discusses existing quantum algorithms for it, and introduces the concept of using channels instead of just unitary operators to model more realistic quantum computations. It argues that combining a quantum SAT algorithm with a stochastic limit approach using channels could provide a method to distinguish computation results that existing algorithms cannot, potentially solving NP-complete problems efficiently.
This document provides an overview of NP-completeness and polynomial time reductions. It defines the classes P and NP, and explains that the core question is whether P=NP. NP-complete problems are the hardest problems in NP, and to prove a problem is NP-complete it must be shown to be in NP and there must be a polynomial time reduction from a known NP-complete problem like 3-SAT. Examples of NP-complete problems discussed include Clique, Independent Set, and Minesweeper. The document outlines the method for proving a problem is NP-complete using a reduction from 3-SAT.
The document provides an overview of computational complexity and discusses the Clay Mathematics Institute's Millennium Prize Problems. It notes that the Clay Mathematics Institute selected seven important unsolved problems in mathematics and is offering a $1 million prize for solving each one. The problems were announced at a 2000 meeting in Paris to celebrate their importance. The rules for awarding the prizes were developed by the Institute's Scientific Advisory Board and approved by its Directors. Inquiries about the Millennium Prize Problems can be sent to the listed email address.
Skiena algorithm 2007 lecture19 introduction to np completezukun
1. The document introduces the concept of NP-completeness and discusses how it can be used to show that many problems that cannot be solved efficiently are essentially the same problem.
2. It describes how reductions can be used to show that if one problem can be transformed or reduced to another problem, then finding an efficient algorithm for one would imply an efficient algorithm for the other.
3. The traveling salesman problem and the satisfiability problem are used as examples to illustrate decision problems, instances, encodings, and reductions.
This document discusses algorithms for solving satisfiability (SAT) problems. It first defines several types of SAT problems, including GSAT, SAT, 1-SAT, 2-SAT, and 3-SAT. It then explains that any GSAT problem can be transformed into an equivalent 3-SAT problem in polynomial time. The document goes on to describe algorithms for solving 1-SAT, 2-SAT and 3-SAT problems, noting that 1-SAT and 2-SAT can be solved in linear time while 3-SAT is NP-complete.
This document discusses the famous P vs NP problem in computer science. It begins by defining efficient ("P") vs inefficient ("NP") algorithms. P problems can be solved quickly in polynomial time, while NP problems are easy to check but hard to solve with no known fast algorithms. Examples are given like sorting vs integer factorization. The document goes on to explain NP-completeness, that if any NP problem like 3-SAT could be solved quickly, then all NP problems would be considered P. It concludes that proving whether P equals NP or not is extremely challenging and would have profound implications.
This document discusses NP-hard and NP-complete problems. It begins by defining the classes P, NP, NP-hard, and NP-complete. It then provides examples of NP-hard problems like the traveling salesperson problem, satisfiability problem, and chromatic number problem. It explains that to show a problem is NP-hard, one shows it is at least as hard as another known NP-hard problem. The document concludes by discussing how restricting NP-hard problems can result in problems that are solvable in polynomial time.
1) Complexity classes categorize problems based on the time and space complexity of their solutions. P represents problems solvable in polynomial time, while NP includes problems verifiable in polynomial time.
2) NP-hard problems are at least as hard as any problem in NP, and NP-complete problems are both in NP and NP-hard - they are the most difficult problems in NP.
3) Reducibility is used to prove problems are NP-complete - if problem A can be reduced to problem B in polynomial time, and B is NP-complete, then A is also NP-complete. 3-SAT is reduced to the clique problem by creating graph vertices for each literal.
This document provides an overview of complexity theory, including:
- Asymptotic notation like Big-O, Big-Omega, and Big-Theta for analyzing algorithm runtime.
- Deterministic algorithms that always produce the same output for a given input.
- Non-deterministic algorithms that may produce different outputs for the same input.
- The classes P and NP, where P contains problems solvable in polynomial time and NP contains problems verifiable in polynomial time.
- NP-complete problems, the hardest problems in NP, like 3-Satisfiability and the Hamiltonian Cycle problem.
This document provides an overview of complexity theory concepts including:
- Asymptotic notation like Big-O, Big-Omega, and Big-Theta for analyzing algorithm runtime.
- The difference between deterministic and non-deterministic algorithms, with deterministic algorithms always providing the same output for a given input, and non-deterministic algorithms possibly providing different outputs.
- The classes P and NP, with P containing problems solvable in polynomial time by a deterministic algorithm, and NP containing problems verifiable in polynomial time by a non-deterministic algorithm.
- NP-complete problems being the hardest problems in NP, with examples like the knapsack problem, Hamiltonian path problem, and Boolean satisfiability problem.
The document discusses NP-completeness and algorithms. It introduces dynamic programming and greedy algorithms. It then discusses the activity selection problem and shows it can be solved greedily by choosing the activity with the earliest finish time at each step. Finally, it defines the classes P and NP, introduces the concept of reductions to show problems are NP-complete, and states that if any NP-complete problem can be solved in polynomial time, then P=NP.
1. 1
CS216: Program and Data Representation
University of Virginia Computer Science
Spring 2006 David Evans
Lecture 8:
Crash Course
in
Computational
Complexity
http://www.cs.virginia.edu/cs216 2UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Procedures and Problems
• So far we have been talking about
procedures (how much work is our
brute force subset sum algorithm?)
• We can also talk about problems:
how much work is the subset sum
problem?
What is a problem?
What does it mean to describe the
work of a problem?
3UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Problems and Solutions
• A problem defines a desired output
for a given input.
• A solution to a problem is a
procedure for finding the correct
output for all possible inputs.
• The time complexity of a problem
is the running time of the best
possible solution to the problem
4UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Subset Sum Problem
• Input: set of n positive integers, {w0,
…, wn-1}, maximum weight W
• Output: a subset S of the input set
such that the sum of the elements of
S ≤ W and there is no subset of the
input set whose sum is greater than
the sum of S and ≤ W
What is the time complexity of the
subset sum problem?
5UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Brute Force Subset Sum
Solution
def subsetsum (items, maxweight):
best = {}
for s in allPossibleSubsets (items):
if sum (s) <= maxweight
and sum (s) > sum (best)
best = s
return best
Running time ∈ Θ(n2n)
What does this tell us about the time
complexity of the subset sum problem?
6UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Problems and Procedures
• If we know a procedure that is that is
Θ(f (n)) that solves a problem then we
know the problem is O (f(n)).
• The subset sum problem is in Θ(n2n)
since we know a procedure that
solves it in Θ(n2n)
• Is the subset sum problem in Θ(n2n)?
No, we would need to prove there is no
better procedure.
2. 2
7UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Lower Bound
• Can we find an bound for the
subset sum problem?
• It is in (n) since we know we need
to at least look at every input
element
• Getting a higher lower bound is
tough
8UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
How much work is the
Subset Sum Problem?
• Upper bound: O (2n)
Try all possible subsets
• Lower bound: Ω (n)
Must at least look at every element
• Tight bound: Θ (?)
No one knows!
9UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Tractable/Intractable
0
200000
400000
600000
800000
1000000
1200000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
logn n
nlogn n^2
n^3 2^n
Sequence Alignment
Subset Sum
“tractable”
“intractable”
I do nothing that a man of unlimited funds, superb physical
endurance, and maximum scientific knowledge could not do.
– Batman (may be able to solve intractable problems, but
computer scientists can only solve tractable ones for large n)
10UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Complexity Class P
“Tractable”
Class P: problems that can be
solved in polynomial time
O (nk) for some constant k.
Easy problems like sorting,
sequence alignment, simulating
the universe are all in P.
11UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Complexity Class NP
Class NP: problems that can be solved
in nondeterministic polynomial time
If we could try all possible solutions at
once, we could identify the solution in
polynomial time.
Alternately: If we had a magic guess-
correctly procedure that makes every
decision correctly, we could devise a
procedure that solves the problem in
polynomial time.
12UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Complexity Classes
Class P: problems that can be solved in
polynomial time (O(nk) for some constant
k): “myopic” problems like sequence
alignment, interval scheduling are all in P.
Class NP: problems that can be solved in
polynomial time by a nondeterministic
machine: includes all problems in P and
some problems possibly outside P like
subset sum
3. 3
13UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Complexity Classes: Possible View
P
NP
Interval
Scheduling:
O(n log n)
Sequence
Alignment: O(n2)
Subset Sum:
O(2n) and Ω(n)
O(n)
How many problems
are in the O(n) class?
How many problems
are in P but not
in the O(n) class?
How many problems
are in NP but not
in P?
infinite
Either 0 or infinite!
infinite
14UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
P = NP?
• Is P different from NP: is there a
problem in NP that is not also in P
– If there is one, there are infinitely many
• Is the “hardest” problem in NP also in P
– If it is, then every problem in NP is also in P
• No one knows the answer!
• The most famous unsolved problem in
computer science and math
– Listed first on Millennium Prize Problems
15UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Problem Classes if P ⊂ NP:
P
NP
Interval
Scheduling:
O(n log n)
Sequence
Alignment: O(n2)
O(n)
Subset Sum:
O(2n) and Ω(n)
How many problems
are in NP but not
in P?
Infinite!
16UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Problem Classes if P = NP:
P
Interval
Scheduling:
O(n log n)
Sequence
Alignment: O(n2)
Subset Sum:
O(nk)
O(n)
How many problems
are in NP but not
in P?
0
17UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Distinguishing P and NP
• Suppose we identify the hardest
problem in NP - let’s call it Super
Arduous Task (SAT)
• Then deciding is P = NP should be
easy:
– Find a P-time solution to SAT ⇒ P = NP
– Prove there is no P-time solution to SAT
⇒ P ⊂ NP
18UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
The Satisfiability Problem (SAT)
• Input: a sentence in propositional
grammar
• Output: Either a mapping from
names to values that satisfies the
input sentence or no way
(meaning there is no possible
assignment that satisfies the input
sentence)
4. 4
19UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
SAT
Example
SAT (a ∨ (b ∧ c) ∨ ¬b ∧ c)
→ { a: true, b: false, c: true }
→ { a: true, b: true, c: false }
SAT (a ∧ ¬a)
→ no way
Sentence ::= Clause
Clause ::= Clause1 ∨ Clause2 (or)
Clause ::= Clause1 ∧ Clause2 (and)
Clause ::= ¬Clause (not)
Clause ::= ( Clause )
Clause ::= Name
20UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
The 3SAT Problem
• Input: a sentence in propositional
grammar, where each clause is a
disjunction of 3 names which may be
negated.
• Output: Either a mapping from names
to values that satisfies the input
sentence or no way (meaning there
is no possible assignment that
satisfies the input sentence)
21UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
3SAT / SAT
Is 3SAT easier or harder than
SAT?
It is definitely not harder than
SAT, since all 3SAT problems
are also SAT problems. Some
SAT problems are not 3SAT
problems.
22UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
3SAT
Example
3SAT ( (a ∨ b ∨ ¬ c)
∧ (¬a ∨ ¬ b ∨ d)
∧ (¬a ∨ b ∨ ¬ d)
∧ (b ∨ ¬ c ∨ d ) )
→ { a: true, b: false, c: false, d: false}
Sentence ::= Clause
Clause ::= Clause1 ∨ Clause2 (or)
Clause ::= Clause1 ∧ Clause2 (and)
Clause ::= ¬Clause (not)
Clause ::= ( Clause )
Clause ::= Name
23UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
NP Completeness
• Cook and Levin proved that 3SAT was
NP-Complete (1971): as hard as the
hardest problem in NP
• If any 3SAT problem can be
transformed into an instance of
problem Q in polynomial time, than
that problem must be no harder than
3SAT: Problem Q is NP-hard
• Need to show in NP also to prove Q is
NP-complete.
24UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Subset Sum is NP-Complete
• Subset Sum is in NP
–Easy to check a solution is correct?
• 3SAT can be transformed into
Subset Sum
–Transformation is complicated, but
still polynomial time.
–A fast Subset Sum solution could
be used to solve 3SAT problems
5. 5
25UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Problem Classes if P ≠ NP:
P
Interval
Scheduling:
Θ(n log n)
Sequence
Alignment: O(n2)
Subset Sum
O(n)
How many problems
are in the Θ(n) class?
How many problems
are in P but not
in the Θ(n) class?
How many problems
are in NP but not
in P?
infinite
infinite
infinite
NP
3SAT
NP-Complete
Note the
NP-
Complete
class is a
ring – others
are circles
26UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
NP-Complete Problems
• Easy way to solve by trying all
possible guesses
• If given the “yes” answer, quick (in P)
way to check if it is right
– Assignments of values to names (evaluate
logical proposition in linear time)
– Subset – check if it has correct sum
• If given the “no” answer, no quick way
to check if it is right
– No solution (can’t tell there isn’t one)
27UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Traveling Salesperson Problem
–Input: a graph of cities and roads
with distance connecting them and a
minimum total distant
–Output: either a path that visits each
with a cost less than the minimum,
or “no”.
• If given a path, easy to check if it
visits every city with less than
minimum distance traveled
28UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Graph (Map) Coloring Problem
–Input: a graph of nodes with edges
connecting them and a minimum
number of colors
–Output: either a coloring of the nodes
such that no connected nodes have
the same color, or “no”.
If given a coloring, easy to check if it no
connected nodes have the same color,
and the number of colors used.
29UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Minesweeper Consistency
Problem
–Input: a position of n
squares in the game
Minesweeper
–Output: either a
assignment of bombs to
squares, or “no”.
• If given a bomb assignment,
easy to check if it is consistent.
30UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Pegboard Problem
6. 6
31UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Pegboard Problem
- Input: a configuration of n pegs on
a cracker barrel style pegboard
- Output: if there is a sequence of
jumps that leaves a single peg,
output that sequence of jumps.
Otherwise, output false.
If given the sequence of jumps, easy
(O(n)) to check it is correct. If not,
hard to know if there is a solution.
32UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Drug Discovery Problem
–Input: a set of
proteins, a desired
3D shape
–Output: a sequence
of proteins that
produces the shape
(or impossible)
If given a sequence, easy (not really –
this may actually be NP-Complete too!) to
check if sequence has the right shape.
Caffeine
33UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Is it ever useful to be
confident that a problem is
hard?
34UVa CS216 Spring 2006 - Lecture 8: Computational Complexity
Charge
• PS3 can be turned in up till 4:50pm
Friday: turn in to Brenda Perkins in
CS office (she has folders for each
section)
• Exam 2 will be handed out
Wednesday
– Send me email questions you want
reviewed