Question

You are given a list of n positive integers a1, a2, . . . an and a positive integer t. Use dynamic programming to design an algorithm that examines whether a subset of these n numbers add up to exactly t, under the constraint that you use each ai at most once. If there is a solution, your program should output the subset of selected numbers. Recurrence Relation (and base cases)

Answer 1

Answer:

See explaination for the program code

Explanation:

The code below

Pseudo-code:

//each item ai is used at most once

isSubsetSum(A[],n,t)//takes array of items of size n, and sum t

{

boolean subset[n+1][t+1];//creating a boolean mtraix

for i=1 to n+1

subset[i][1] = true; //initially setting all first column values as true

for i = 2 to t+1

subset[1][i] = false; //initialy setting all first row values as false

for i=2 to n

{

for j=2 to t

{

if(j<A[i-1])

subset[i][j] = subset[i-1][j];

if (j >= A[i-1])

subset[i][j] = subset[i-1][j] ||

subset[i - 1][j-set[i-1]];

}

}

//returns true if there is a subset with given sum t

//other wise returns false

return subset[n][t];

}

Recurrence relation:

T(n) =T(n-1)+ t//here t is runtime of inner loop, and innner loop will run n times

T(1)=1

solving recurrence:

T(n)=T(n-1)+t

T(n)=T(n-2)+t+t

T(n)=T(n-2)+2t

T(n)=T(n-3)+3t

,,

,

T(n)=T(n-n-1)+(n-1)t

T(n)=T(1)+(n-1)t

T(n)=1+(n-1)t = O(nt)

//so complexity is :O(nt)//where n is number of element, t is given sum