a) What is the Dynamic Programming recurrence formula for the 0/1 knapsack? Show thesubproblem overlapping of the following 0/1 knapsack problem?Max Weight of Bag = 7 kgItemWeightBenefit1312241532025b) MMM takes part in programming contests. His favourite topic is dynamic programming(DP).As he said, that he likes problems on DP, because "you spend a lot of time on thinking and alittle time on coding".In this problem you are to solve a version of the knapsack problem(link), one of the mostfamous examples of DP problem.You are given N items, each has two parameters: a weight and a cost. Let's define M as thesum of the weights of all the items.Your task is to determine the most expensive cost of a knapsack, which capacity equals to 1,2, .., M. A cost of a knapsack equals the sum of the costs of all the elements of the knapsack.Also, when you have a knapsack with a capacity equal to C, then you can fill it with items,****whose sum of weights is not greater than C.In the test case, M equals to 9.For C = 1, it's optimal to choose {1} items;For C = 2, it's optimal to choose {5} items;For C = 3, it's optimal to choose {1, 5} items;For C = 4, it's optimal to choose {4, 5} items;For C = 5, it's optimal to choose {1, 4, 5} items;For C = 6, it's optimal to choose {3, 4, 5} items;For C = 7, it's optimal to choose {1, 3, 4, 5} items;For C = 8, it's optimal to choose {2, 3, 4, 5} items;For C = 9, it's optimal to choose {1, 2, 3, 4, 5} items.Find the recurrence formula of the DP problem mentioned above3.

Note: Solving the first question as per the guidelines, i.e., a. Dynamic Programming of 0/1…

a) What is the Dynamic Programming recurrence formula for the 0/1 knapsack? Show the subproblem overlapping of the following 0/1 knapsack problem? Max Weight of Bag = 7 kg Item Weight Benefit 1 3 12 2 4 15 3 3 20 4 6 25

a) What is the Dynamic Programming recurrence formula for the 0/1 knapsack? Show the subproblem overlapping of the following 0/1 knapsack problem? Max Weight of Bag = 7 kg Item Weight Benefit 1 3 12 2 4 15 3 3 20 4 6 25

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Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

a) What is the Dynamic Programming recurrence formula for the 0/1 knapsack? Show the
subproblem overlapping of the following 0/1 knapsack problem?
Max Weight of Bag = 7 kg
Item
Weight
Benefit
1
3
12
2
4
15
3
20
25
b) MMM takes part in programming contests. His favourite topic is dynamic programming(DP).
As he said, that he likes problems on DP, because "you spend a lot of time on thinking and a
little time on coding".
In this problem you are to solve a version of the knapsack problem(link), one of the most
famous examples of DP problem.
You are given N items, each has two parameters: a weight and a cost. Let's define M as the
sum of the weights of all the items.
Your task is to determine the most expensive cost of a knapsack, which capacity equals to 1,
2, .., M. A cost of a knapsack equals the sum of the costs of all the elements of the knapsack.
Also, when you have a knapsack with a capacity equal to C, then you can fill it with items,
****
whose sum of weights is not greater than C.
In the test case, M equals to 9.
For C = 1, it's optimal to choose {1} items;
For C = 2, it's optimal to choose {5} items;
For C = 3, it's optimal to choose {1, 5} items;
For C = 4, it's optimal to choose {4, 5} items;
For C = 5, it's optimal to choose {1, 4, 5} items;
For C = 6, it's optimal to choose {3, 4, 5} items;
For C = 7, it's optimal to choose {1, 3, 4, 5} items;
For C = 8, it's optimal to choose {2, 3, 4, 5} items;
For C = 9, it's optimal to choose {1, 2, 3, 4, 5} items.
Find the recurrence formula of the DP problem mentioned above
3.

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