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The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?
- a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]
- b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]
- c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]
- d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]
Correct answer is option 'B'. Can you explain this answer?
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The subset-sum problem is defined as follows. Given a set of n positiv...
X[I, j] (2 <= i <= n and ai <= j <= W), is true if any of the following is true 1) Sum of weights excluding ai is equal to j, i.e., if X[i-1, j] is true. 2) Sum of weights including ai is equal to j, i.e., if X[i-1, j-ai] is true so that we get (j – ai) + ai as j
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The subset-sum problem is defined as follows. Given a set of n positiv...
Subset-Sum Problem and Dynamic Programming
The subset-sum problem is a well-known problem in computer science and mathematics. It involves finding a subset of a given set of positive integers whose sum is equal to a given target value. This problem has applications in various fields, such as cryptography, finance, and computer science.
Dynamic programming is a popular technique used to solve the subset-sum problem. It involves breaking down the problem into smaller subproblems and solving them recursively. The solution to the original problem is then obtained by combining the solutions to the subproblems.
2-Dimensional Boolean Array X
To solve the subset-sum problem using dynamic programming, we use a 2-dimensional Boolean array X. This array has n rows and W+1 columns, where n is the number of elements in the given set S and W is the target value. The element X[i,j] of the array is true if and only if there exists a subset of the first i elements of S whose sum is j.
The base case of the dynamic program is when i=0, which means that we have no elements to consider. In this case, X[0,j] is false for all j. The final solution to the problem is given by X[n,W].
Valid Equation for 2 = i = n and ai = j = W
To compute the values of the elements of the array X, we use the following recurrence relation:
X[i,j] = X[i-1,j] OR X[i-1,j-ai]
This equation is valid for 2 ≤ i ≤ n and ai ≤ j ≤ W. The OR operator represents the logical OR function, which returns true if either of its operands is true.
Explanation of the Valid Equation
The recurrence relation can be explained as follows:
- X[i,j] = X[i-1,j] OR X[i-1,j-ai]: To compute X[i,j], we first consider the case where the i-th element is not included in the subset. In this case, X[i,j] is true if and only if there exists a subset of the first i-1 elements whose sum is j. This is represented by X[i-1,j]. However, if we include the i-th element in the subset, then the remaining sum that we need to achieve is j-ai. Therefore, X[i,j] is also true if there exists a subset of the first i-1 elements whose sum is j-ai, and the i-th element is included in the subset. This is represented by X[i-1,j-ai].
- Using the OR operator: We use the OR operator to combine these two cases because we only need to find one subset whose sum is equal to j. If either of the two cases is true, then we can include the i-th element in the subset.
- The range of i and j: The recurrence relation is valid for 2 ≤ i ≤ n and ai ≤ j ≤ W. This is because we need at least two elements in the set to form a subset, and the sum of the elements cannot exceed the target value W.
Therefore, option B is the correct answer, which states that X[i,j] = X[i-1,j] OR X[i-1,j-ai].
The subset-sum problem is a well-known problem in computer science and mathematics. It involves finding a subset of a given set of positive integers whose sum is equal to a given target value. This problem has applications in various fields, such as cryptography, finance, and computer science.
Dynamic programming is a popular technique used to solve the subset-sum problem. It involves breaking down the problem into smaller subproblems and solving them recursively. The solution to the original problem is then obtained by combining the solutions to the subproblems.
2-Dimensional Boolean Array X
To solve the subset-sum problem using dynamic programming, we use a 2-dimensional Boolean array X. This array has n rows and W+1 columns, where n is the number of elements in the given set S and W is the target value. The element X[i,j] of the array is true if and only if there exists a subset of the first i elements of S whose sum is j.
The base case of the dynamic program is when i=0, which means that we have no elements to consider. In this case, X[0,j] is false for all j. The final solution to the problem is given by X[n,W].
Valid Equation for 2 = i = n and ai = j = W
To compute the values of the elements of the array X, we use the following recurrence relation:
X[i,j] = X[i-1,j] OR X[i-1,j-ai]
This equation is valid for 2 ≤ i ≤ n and ai ≤ j ≤ W. The OR operator represents the logical OR function, which returns true if either of its operands is true.
Explanation of the Valid Equation
The recurrence relation can be explained as follows:
- X[i,j] = X[i-1,j] OR X[i-1,j-ai]: To compute X[i,j], we first consider the case where the i-th element is not included in the subset. In this case, X[i,j] is true if and only if there exists a subset of the first i-1 elements whose sum is j. This is represented by X[i-1,j]. However, if we include the i-th element in the subset, then the remaining sum that we need to achieve is j-ai. Therefore, X[i,j] is also true if there exists a subset of the first i-1 elements whose sum is j-ai, and the i-th element is included in the subset. This is represented by X[i-1,j-ai].
- Using the OR operator: We use the OR operator to combine these two cases because we only need to find one subset whose sum is equal to j. If either of the two cases is true, then we can include the i-th element in the subset.
- The range of i and j: The recurrence relation is valid for 2 ≤ i ≤ n and ai ≤ j ≤ W. This is because we need at least two elements in the set to form a subset, and the sum of the elements cannot exceed the target value W.
Therefore, option B is the correct answer, which states that X[i,j] = X[i-1,j] OR X[i-1,j-ai].
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The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer?
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The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer?.
The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer?.
Solutions for The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Computer Science Engineering (CSE).
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The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer?, a detailed solution for The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The subset-sum problem is defined as follows. Given a set of n positive integers, S = {a1 ,a2 ,a3 ,…,an} and positive integer W, is there a subset of S whose elements sum to W? A dynamic program for solving this problem uses a 2-dimensional Boolean array X, with n rows and W+1 columns. X[i, j],1 <= i <= n, 0 <= j <= W, is TRUE if and only if there is a subset of {a1 ,a2 ,...,ai} whose elements sum to j. Which of the following is valid for 2 <= i <= n and ai <= j <= W?a)X[i, j] = X[i - 1, j] ∨ X[i, j -ai]b)X[i, j] = X[i - 1, j] ∨ X[i - 1, j - ai]c)X[i, j] = X[i - 1, j] ∧ X[i, j - ai]d)X[i, j] = X[i - 1, j] ∧ X[i -1, j - ai]Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Computer Science Engineering (CSE) tests.
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