Let r[i][j]r[i][j] be the maximum value we can carry using only items 1 to i given a knapsack of сарacity j. Suppose there are 4 items in the shop. The values of the items are [5, 7, 10, 6] and the weights of the items are [3, 2, 4, 1]. 1. If our knapsack has capacity 1, compute r[i][1] for each 1<=i<=4. Which items should we take?

Let r[i][j]r[i][j] be the maximum value we can carry using only items 1 to i given a knapsack of сарacity j. Suppose there are 4 items in the shop. The values of the items are [5, 7, 10, 6] and the weights of the items are [3, 2, 4, 1]. 1. If our knapsack has capacity 1, compute r[i][1] for each 1<=i<=4. Which items should we take?

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter2: Basic Linear Algebra

Section: Chapter Questions

Problem 15RP

See similar textbooks

Similar questions

Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?
Let E : y2 = x3 + 3x + 4 be an elliptic curve over F37. 1, Find all the elements of the elliptice curve group 2, Find the order of the 3, Find a primitive element of this group and call it G. 4, Compute [30]G = G ⊕ G ⊕ · · ⊕ G (addition of 30 many G’s)
Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.
Question: Suppose there are n people in a group, each aware of a scandal no one else in the group knows about. These people communicate by telephone; when two people in the group talk, they share information about all scandals each knows about. For example, on the first call, two people share information, so by the end of the call, each of these people knows about two scandals. The gossip problem asks for G(n), the minimum number of telephone calls that are needed for all n people to learn about all the scandals. a). Find G(1), G(2), G(3), and G(4). b). Use mathematical induction to prove that G(n) 4. [Hint: In the inductive step, have a new person call a particular person at the start and at the end.] c). Prove that G(n)= 2n – 4 for n> 4.
; 3. Let T U V and S: V W. We have that SoT: U → W. Prove that (SoT)* = T* o S*. 4. Let v₁,.., Un EV be a basis of V. (a) Define the dual basis vi,.., v € V*. (b) Prove that formula: a = Σi-na (v₁) vi for every a € V*. i=1:n (c) Prove the reciprocal formula: v= Ei=1:n v (v) v₁ for every v € V.
Let f(x) = x¹ Hx-x¹b, where H and b are constant, independent of x, and H is symmetric positive definite. Given vectors x0) and p0), find the value of the scalar a that minimizes f(x0) + ap0)). This is the formula for the stepsize ak in the linear conjugate gradient algorithm.
Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but c is relatively prime to both a and b . Let n = s × a + t × b be some linear combination of a and b, where s and t are integers. Prove that n cannot be a divisor of c. Follow the definition of relative primes, and use contradiction.
There are n ≥ 2 married couples who need to cross a river. They have a boat that can hold no more than two people at a time. To complicate matters, all the husbands are jealous and will not agree on any crossing procedure that would put a wife on the same bank of the river with another woman's husband without the wife's husband being there too, even if there are other people on the same bank. Can they cross the river under such constraints? Solve the problem for n = 2.
suppose a computer solves a 100x100 matrix using Gauss elimination with partial pivoting in 1 second, how long will it take to solve a 300x300 matrix using Gauss elimination with partial pivoting on the same computer? and if you have a limit of 100 seconds to solve a matrix of size (N x N) using Gauss elimination with partial pivoting, what is the largest N can you do? show all the steps of the solution
USING PYTHON A tridiagonal matrix is one where the only nonzero elements are the ones on the main diagonal (i.e., ai,j where j = i) and the ones immediately above and belowit(i.e.,ai,j wherej=i+1orj=i−1). Write a function that solves a linear system whose coefficient matrix is tridiag- onal. In this case, Gauss elimination can be made much more efficient because most elements are already zero and don’t need to be modified or added. Please show steps and explain.
A ski rental agency has n pairs of skis, where the height of the the ith pair of skis is si . There are n skiers who wish to rent skis, where the height of the ith skier is hi. Ideally, each skier should obtain a pair of skis whose height matches her/his own height as closely as possible. We would like to assign skis to skiers so that the sum of the absolute differences of the heights of each skier and her/his skis is minimized. Design a greedy algorithm for the problem. Prove the correctness of your algorithm. (Hint: Start with two skis and two skiers. How would you match them? Continue to three skis and three skiers, and identify a strategy.)
Prove that f(x) = x is O(x3).