Proposition is attached ### Proposition 6.5. Let \( (M, S) \) be a subset-sum problem in which the integers in \( M \) form a superincreasing sequence. Assuming that a solution \( x \) exists, it is unique and may be computed by the following fast algorithm:---### 6.2. Subset-sum problems and knapsack cryptosystems Page 355```plaintextLoop i from n down to 1 If S ≥ Mi, set xi = 1 and subtract Mi from S Else set xi = 0End Loop```In this section, the text describes an algorithm for solving a subset-sum problem where the sequence \( M \) is superincreasing, meaning each element is greater than the sum of all previous elements. The provided algorithm determines the binary vector \( x \) that solves the problem, ensuring that solution is unique. The uniqueness arises from the superincreasing property of the sequence \( M \), as there is no ambiguity in selecting the terms to match \( S \). The algorithm operates as follows:- A loop iterates from the last element of \( M \) to the first.- In each iteration, it checks whether the remaining sum \( S \) is greater than or equal to the current element \( Mi \). - If true, it sets the corresponding binary variable \( xi \) to 1 and reduces \( S \) by \( Mi \). - If false, it sets \( xi \) to 0.The loop continues until all elements of the sequence have been considered.This text is useful for understanding efficient ways to solve certain types of knapsack problems and has applications in cryptography, particularly in knapsack-based cryptosystems. **6.2 Subset-Sum Problem Exercises****Objective:** Use the algorithm described in Proposition 6.5 to solve each of the following subset-sum problems. If the “solution” that you get is not correct, explain what went wrong.**Problem Statements:**(a) Given a set \( M = \{3, 7, 19, 43, 89, 195\} \) and target sum \( S = 260 \).(b) Given a set \( M = \{5, 11, 25, 61, 125, 261\} \) and target sum \( S = 408 \).(c) Given a set \( M = \{2, 5, 12, 28, 60, 131, 257\} \) and target sum \( S = 334 \).(d) Given a set \( M = \{4, 12, 15, 36, 75, 162\} \) and target sum \( S = 214 \).**Instructions:**1. Use the specified algorithm from Proposition 6.5 to determine if there exists any subset of \( M \) that sums to \( S \).2. If you find such a subset, ensure your solution is correct.3. If no correct subset is found, analyze and explain any discrepancies or errors in the process.**Educational Note:**The subset-sum problem is a classic problem in computer science and mathematics, where the goal is to determine whether a subset of a given set of integers sums up to a particular value. Solving these problems often involves understanding and implementing algorithms designed to efficiently compute potential subsets and their sums.

The subset-sum problem is a computational problem in computer science and mathematics. It involves…

6.2. Use the algorithm described in Proposition 6.5 to solve each of the following subset-sum problems. If the "solution" that you get is not correct, explain what went wrong. (a) M = (3,7, 19, 43, 89, 195), S = 260. (b) M = (5,11,25, 61, 125,261), S = 408. (c) M = (2, 5, 12, 28, 60, 131, 257), S = 334. (d) M = (4, 12, 15, 36, 75, 162), S = 214.

6.2. Use the algorithm described in Proposition 6.5 to solve each of the following subset-sum problems. If the "solution" that you get is not correct, explain what went wrong. (a) M = (3,7, 19, 43, 89, 195), S = 260. (b) M = (5,11,25, 61, 125,261), S = 408. (c) M = (2, 5, 12, 28, 60, 131, 257), S = 334. (d) M = (4, 12, 15, 36, 75, 162), S = 214.

Database System Concepts

7th Edition

ISBN:9780078022159

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

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

Chapter1: Introduction

Section: Chapter Questions

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Question

Proposition is attached

### Proposition 6.5.

Let \( (M, S) \) be a subset-sum problem in which the integers in \( M \) form a superincreasing sequence. Assuming that a solution \( x \) exists, it is unique and may be computed by the following fast algorithm:

---

### 6.2. Subset-sum problems and knapsack cryptosystems

Page 355

```plaintext
Loop i from n down to 1
If S ≥ Mi, set xi = 1 and subtract Mi from S
Else set xi = 0
End Loop
```

In this section, the text describes an algorithm for solving a subset-sum problem where the sequence \( M \) is superincreasing, meaning each element is greater than the sum of all previous elements. The provided algorithm determines the binary vector \( x \) that solves the problem, ensuring that solution is unique. The uniqueness arises from the superincreasing property of the sequence \( M \), as there is no ambiguity in selecting the terms to match \( S \).

The algorithm operates as follows:
- A loop iterates from the last element of \( M \) to the first.
- In each iteration, it checks whether the remaining sum \( S \) is greater than or equal to the current element \( Mi \).
- If true, it sets the corresponding binary variable \( xi \) to 1 and reduces \( S \) by \( Mi \).
- If false, it sets \( xi \) to 0.
The loop continues until all elements of the sequence have been considered.

This text is useful for understanding efficient ways to solve certain types of knapsack problems and has applications in cryptography, particularly in knapsack-based cryptosystems.

**6.2 Subset-Sum Problem Exercises**

**Objective:** Use the algorithm described in Proposition 6.5 to solve each of the following subset-sum problems. If the “solution” that you get is not correct, explain what went wrong.

**Problem Statements:**

(a) Given a set \( M = \{3, 7, 19, 43, 89, 195\} \) and target sum \( S = 260 \).

(b) Given a set \( M = \{5, 11, 25, 61, 125, 261\} \) and target sum \( S = 408 \).

(c) Given a set \( M = \{2, 5, 12, 28, 60, 131, 257\} \) and target sum \( S = 334 \).

(d) Given a set \( M = \{4, 12, 15, 36, 75, 162\} \) and target sum \( S = 214 \).

**Instructions:**
1. Use the specified algorithm from Proposition 6.5 to determine if there exists any subset of \( M \) that sums to \( S \).
2. If you find such a subset, ensure your solution is correct.
3. If no correct subset is found, analyze and explain any discrepancies or errors in the process.

**Educational Note:**
The subset-sum problem is a classic problem in computer science and mathematics, where the goal is to determine whether a subset of a given set of integers sums up to a particular value. Solving these problems often involves understanding and implementing algorithms designed to efficiently compute potential subsets and their sums.

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