Solving sudoku as an Integer Programming problem
... Solving sudoku as an Integer Programming problem ⊡ A standard way to solve sudoku is by applying recursion, an algorithm where the solution depends on solutions to smaller instances of the reference problem. One checks all the combinations (for every cell it considers all integers between 1 and 9) a ...
... Solving sudoku as an Integer Programming problem ⊡ A standard way to solve sudoku is by applying recursion, an algorithm where the solution depends on solutions to smaller instances of the reference problem. One checks all the combinations (for every cell it considers all integers between 1 and 9) a ...
Determining Optimal Parameters in Magnetic
... at quickly finding good solutions in the vicinity of x0 . A (condensed) description of the proposed approach is: 1. Grid partition the whole feasible set K into a collection of subsets {K1 , . . . , K p }. 2. For each K j , compute the value f j of the solution obtained by maxeval iterations of Sdbo ...
... at quickly finding good solutions in the vicinity of x0 . A (condensed) description of the proposed approach is: 1. Grid partition the whole feasible set K into a collection of subsets {K1 , . . . , K p }. 2. For each K j , compute the value f j of the solution obtained by maxeval iterations of Sdbo ...
LP and Approximation
... 1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2. If the optimal value of the primal is unbounded then the dual is infeasible. 3. If the opti ...
... 1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2. If the optimal value of the primal is unbounded then the dual is infeasible. 3. If the opti ...
Chapter 2: Fundamentals of the Analysis of Algorithm Efficiency
... Decide on a parameter indicating an input’s size. ...
... Decide on a parameter indicating an input’s size. ...
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... Obtain a bfs (if possible) from the standard form. Determine whether the current bfs is optimal. If the current bfs is not optimal, then determine which nonbasic basic variable should become a basic variable and which basic variable should become a nonbasic variable to find a new bfs with a better o ...
... Obtain a bfs (if possible) from the standard form. Determine whether the current bfs is optimal. If the current bfs is not optimal, then determine which nonbasic basic variable should become a basic variable and which basic variable should become a nonbasic variable to find a new bfs with a better o ...
Lecture
... Recursion is a repetitive process in which an algorithm calls itself. Usually recursion is organized in such a way that a subroutine calls itself or a function calls itself ...
... Recursion is a repetitive process in which an algorithm calls itself. Usually recursion is organized in such a way that a subroutine calls itself or a function calls itself ...
ALGORITHMS - HOMEWORK 5 - SOLUTIONS 1. Consider the
... 1. Consider the following closest-point algorithm for building an approximate traveling-salesman tour whose cost function satisfies the triangle inequality. Begin with a trivial cycle consisting of a single arbitrarily chosen vertex. At each step, identify the vertex u that is not on the cycle but w ...
... 1. Consider the following closest-point algorithm for building an approximate traveling-salesman tour whose cost function satisfies the triangle inequality. Begin with a trivial cycle consisting of a single arbitrarily chosen vertex. At each step, identify the vertex u that is not on the cycle but w ...
Chapter 2: Fundamentals of the Analysis of Algorithm
... Asymptotic order of growth A way of comparing functions that ignores constant factors and small input sizes • O(g(n)): class of functions f(n) that grow no faster than g(n) • Θ(g(n)): class of functions f(n) that grow at same rate as g(n) • Ω(g(n)): class of functions f(n) that grow at least as fas ...
... Asymptotic order of growth A way of comparing functions that ignores constant factors and small input sizes • O(g(n)): class of functions f(n) that grow no faster than g(n) • Θ(g(n)): class of functions f(n) that grow at same rate as g(n) • Ω(g(n)): class of functions f(n) that grow at least as fas ...
Proximity Inversion Functions on the Non
... We would like to generalize the results of this approach, so it can be used to solve different constraint problems. In particular, can all metric space BSPs be solved this way? If not, what are the necessary and sufficient conditions on the distance constraints for such a solution to exist? ...
... We would like to generalize the results of this approach, so it can be used to solve different constraint problems. In particular, can all metric space BSPs be solved this way? If not, what are the necessary and sufficient conditions on the distance constraints for such a solution to exist? ...
Linear Programming
... 1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2. If the optimal value of the primal is unbounded then the dual is infeasible. 3. If the opti ...
... 1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2. If the optimal value of the primal is unbounded then the dual is infeasible. 3. If the opti ...
ùñ dP 2 ¡ 2A S x 1 ùñ dS 1 ¡ 1 c r2a2 ¡a4 ùñ dA
... 16 4π Using the fact that a b 10, we can rewrite a with a 10 b, so then we have A ...
... 16 4π Using the fact that a b 10, we can rewrite a with a 10 b, so then we have A ...
Chapter 8 Primal-Dual Method and Local Ratio
... • A probe can determine if a connected link is working correctly. • The goal is to minimize the number of used probes to check all the links. ...
... • A probe can determine if a connected link is working correctly. • The goal is to minimize the number of used probes to check all the links. ...
A Complete Characterization of a Family of Solutions to a
... and the overall mean m equals ∑ki=1 pi mi . Finally, Si is the covariance matrix of class i. A solution to these optimization problems can be obtained by means of a generalized eigenvalue decomposition, which Fukunaga (1990) relates to a simultaneous diagonalization of the two matrices involved (see ...
... and the overall mean m equals ∑ki=1 pi mi . Finally, Si is the covariance matrix of class i. A solution to these optimization problems can be obtained by means of a generalized eigenvalue decomposition, which Fukunaga (1990) relates to a simultaneous diagonalization of the two matrices involved (see ...
chemistry log: solutions
... PROBLEM: The formula and the molecular weight of an unknown hydrocarbon compound are to be determined by elemental analysis and the freezing point depression method. a) The hydrocarbon is found to contain 93.46 % carbon and 6.54 % hydrogen. Calculate the empirical formula of the unknown hydrocarbon. ...
... PROBLEM: The formula and the molecular weight of an unknown hydrocarbon compound are to be determined by elemental analysis and the freezing point depression method. a) The hydrocarbon is found to contain 93.46 % carbon and 6.54 % hydrogen. Calculate the empirical formula of the unknown hydrocarbon. ...
out!
... Problem 2 (15 points) We have discussed several different alignment algorithms. The algorithms differ in scope (global, local), end- treatment (ends-free, ends-full), and gap-penalty (constant, linear, affine, convex, custom). We discussed overlap detection, bounded dynamic programming, linear space ...
... Problem 2 (15 points) We have discussed several different alignment algorithms. The algorithms differ in scope (global, local), end- treatment (ends-free, ends-full), and gap-penalty (constant, linear, affine, convex, custom). We discussed overlap detection, bounded dynamic programming, linear space ...
CS 262 Tuesday, January 13, 2015 Computational Genomics
... We have discussed several different alignment algorithms. The algorithms differ in scope (global, local), end- treatment (ends-free, ends-full), and gap-penalty (constant, linear, affine, convex, custom). We discussed overlap detection, bounded dynamic programming, linear space dynamic programming, ...
... We have discussed several different alignment algorithms. The algorithms differ in scope (global, local), end- treatment (ends-free, ends-full), and gap-penalty (constant, linear, affine, convex, custom). We discussed overlap detection, bounded dynamic programming, linear space dynamic programming, ...
Sample Problems 1 Problem 1: Find the value of each of the
... print *, "enter the side lengths of a triangle:(a, b, c)" read *, a, b, c s = (a + b + c) / 2.0 area = sqrt(s * (s-a) * (s-b) * (s-c)) print *, "The area of the triangle having side lengths" print *, "a = ", a, "b = ", b, "c = ", c print *, "is" print *, "Area = ", area end program midterm_prob3 ...
... print *, "enter the side lengths of a triangle:(a, b, c)" read *, a, b, c s = (a + b + c) / 2.0 area = sqrt(s * (s-a) * (s-b) * (s-c)) print *, "The area of the triangle having side lengths" print *, "a = ", a, "b = ", b, "c = ", c print *, "is" print *, "Area = ", area end program midterm_prob3 ...
Calculus Quiz week 37.1 - Personal Web pages at the Department
... University of Aarhus Department of Mathematical Sciences ...
... University of Aarhus Department of Mathematical Sciences ...
Source - Department of Computing Science
... A lesson for AI: The Power of a “Visible” Goal • In MDPs, the goal (reward) is part of the data, part of the agent’s normal operation • The agent can tell for itself how well it is doing • This is very powerful… we should do more of it in AI • Can we give AI tasks visible goals? ...
... A lesson for AI: The Power of a “Visible” Goal • In MDPs, the goal (reward) is part of the data, part of the agent’s normal operation • The agent can tell for itself how well it is doing • This is very powerful… we should do more of it in AI • Can we give AI tasks visible goals? ...
Chapter 1 Introduction to Recursive Methods
... ct + it ≤ F (kt , nt ) with equality. Moreover, since the agent does not value leisure (and labor is productive), an optimal path is such that nt = 1 for all t. Using the law of motion for kt , one can rewrite the above problem by dropping the leisure and investment variables, and obtain: ...
... ct + it ≤ F (kt , nt ) with equality. Moreover, since the agent does not value leisure (and labor is productive), an optimal path is such that nt = 1 for all t. Using the law of motion for kt , one can rewrite the above problem by dropping the leisure and investment variables, and obtain: ...
Optimization and Control: Examples Sheet 1
... unattended or attended, at costs of 0 or c respectively. In either case there is a chance that the machine will break down and need repair the following day, with probabilities p and p0 respectively, where p0 < (7/8)p. Costs are discounted by a factor β ∈ (0, 1), and it is desired to minimize the to ...
... unattended or attended, at costs of 0 or c respectively. In either case there is a chance that the machine will break down and need repair the following day, with probabilities p and p0 respectively, where p0 < (7/8)p. Costs are discounted by a factor β ∈ (0, 1), and it is desired to minimize the to ...
Introducing a New Product
... Feasible solution – A solution for which all constraints are satisfied Infeasible solution – A solution for which at least one constraint is violated Feasible region – The collection of all feasible solutions Optimal solution – A feasible solution that has the most favorable value of the objective f ...
... Feasible solution – A solution for which all constraints are satisfied Infeasible solution – A solution for which at least one constraint is violated Feasible region – The collection of all feasible solutions Optimal solution – A feasible solution that has the most favorable value of the objective f ...
Summary of big ideas
... • For many discrete optimization problems, there are benchmarks of instances on which algorithms are tested. • For TSP, such a benchmark is TSPLIB. • On TSPLIB instances, the Christofides’ algorithm outputs solutions which are on average 1.09 times the optimum. For comparison, the nearest neighbor a ...
... • For many discrete optimization problems, there are benchmarks of instances on which algorithms are tested. • For TSP, such a benchmark is TSPLIB. • On TSPLIB instances, the Christofides’ algorithm outputs solutions which are on average 1.09 times the optimum. For comparison, the nearest neighbor a ...
Project on Pick`s Formula
... Problem 2. Let P be a lattice rectangle with sides parallel to the coordinate axes. We can then assume that the vertices of P are at the points (0, 0), (a, 0), (0, b), and (a, b) for some integers a and b. Check that Pick’s formula holds for P . Problem 3. Now let T be a lattice triangle with two si ...
... Problem 2. Let P be a lattice rectangle with sides parallel to the coordinate axes. We can then assume that the vertices of P are at the points (0, 0), (a, 0), (0, b), and (a, b) for some integers a and b. Check that Pick’s formula holds for P . Problem 3. Now let T be a lattice triangle with two si ...
Dynamic programming
In mathematics, computer science, economics, and bioinformatics, dynamic programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems and optimal substructure (described below). When applicable, the method takes far less time than other methods that don't take advantage of the subproblem overlap (like depth-first search).In order to solve a given problem using a dynamic programming approach, we need to solve different parts of the problem (subproblems), then combine the solutions of the subproblems to reach an overall solution. Often when using a more naive method, many of the subproblems are generated and solved many times. The dynamic programming approach seeks to solve each subproblem only once, thus reducing the number of computations: once the solution to a given subproblem has been computed, it is stored or ""memoized"": the next time the same solution is needed, it is simply looked up. This approach is especially useful when the number of repeating subproblems grows exponentially as a function of the size of the input.Dynamic programming algorithms are used for optimization (for example, finding the shortest path between two points, or the fastest way to multiply many matrices). A dynamic programming algorithm will examine the previously solved subproblems and will combine their solutions to give the best solution for the given problem. The alternatives are many, such as using a greedy algorithm, which picks the locally optimal choice at each branch in the road. The locally optimal choice may be a poor choice for the overall solution. While a greedy algorithm does not guarantee an optimal solution, it is often faster to calculate. Fortunately, some greedy algorithms (such as minimum spanning trees) are proven to lead to the optimal solution.For example, let's say that you have to get from point A to point B as fast as possible, in a given city, during rush hour. A dynamic programming algorithm will look at finding the shortest paths to points close to A, and use those solutions to eventually find the shortest path to B. On the other hand, a greedy algorithm will start you driving immediately and will pick the road that looks the fastest at every intersection. As you can imagine, this strategy might not lead to the fastest arrival time, since you might take some ""easy"" streets and then find yourself hopelessly stuck in a traffic jam.Sometimes, applying memoization to a naive basic recursive solution already results in a dynamic programming solution with asymptotically optimal time complexity; however, the optimal solution to some problems requires more sophisticated dynamic programming algorithms. Some of these may be recursive as well but parametrized differently from the naive solution. Others can be more complicated and cannot be implemented as a recursive function with memoization. Examples of these are the two solutions to the Egg Dropping puzzle below.