# Positive Subset Sumby Isai Damier, Android Engineer @ Google

```/*********************************************************************
* Author: Isai Damier
* Title: Positive Subset Sum
* Project: geekviewpoint
* Package: algorithms
*
* Statement:
*   Given a sequence of n positive numbers totaling to T, check
*   whether there exists a subsequence totaling to x, where x is less
*   than or equal to T.
*
* Time-complexity: pseudo-polynomial: O(n*x)
* Space-complexity: O(x)
*
* Dynamic Programming Strategy:
*
*   Let's call the given Sequence S for convenience. Solving this
*   problem, there are two approaches we could take. On the one hand,
*   we could look through all the possible sub-sequences of S to see if
*   any of them sum up to x. This approach, however, would take an
*   exponential amount of work since there are 2^n possible
*   sub-sequences in S. On the other hand, we could list all the sums
*   between 0 and x and then try to find a sub-sequence for each one
*   of them until we find one for x. This second approach turns out to
*   be quite a lot faster: O(n*T). Here are the steps:
*
*   0] Create a boolean array called sum of size x+1:
*     As you might guess, when we are done filling the array, all the
*     sub-sums between 0 and x that can be calculated from S will be
*     set to true and those that cannot be reached will be set to false.
*     For example if S={2,4,7,9} then sum=false while sum=true
*     since 4+9=13.
*
*   1] Initialize sum{} to false:
*      Before any computation is performed, assume/pretend that each
*      sub-sum is unreachable. We know that's not true, but for now
*      let's be outrageous.
*
*   2] Set sum at index 0 to true:
*     This truth is self-evident. By taking no elements from S, we end
*     up with an empty sub-sequence. Therefore we can mark sum=true,
*     since the sum of nothing is zero.
*
*   3] To fill the rest of the table, we are going to use the following
*     trick. Let S={2,4,7,9}. Then starting with 0, each time we find
*     a positive sum, we will add an element from S to that sum to get
*     a greater sum. For example, since sum=true and 2 is in S, then
*     sum[0+2] must also be true. Therefore, we set sum[0+2]=sum=true.
*     Then from sum=true and element 4, we can say sum[2+4]=sum=true,
*     and so on.
*
*  Step 3 is known as the relaxation step. First we started with an
*  absurd assumption that no sub-sequence of S can sum up to any
*  number. Then as we find evidence to the contrary, we relax our
*  assumption.
*
* Alternative implementation:
*   This alternative is easier to read, but it does not halt for small x.
*   In the actual code, each for-loop checks for "!sum[x]" since that's
*   really all we care about and should stop once we find it. Also
*   this time complexity is of O(n*T) and space complexity is O(T)
*
*   boolean sum[] = new boolean[T + 1];
*   sum = true;
*   for (int a : A)
*     for (int i = T; i >= a; i--)
*       if (!sum[i] && sum[i - a])
*         sum[i] = true;
*
**********************************************************************/
package algorithm.programming.dynamic;

public class PositiveSubsequenceSum {

public boolean positiveSubsetSum(int[] A, int x) {
//preliminary
int T = 0;
for (int a : A) {
T += a;
}
if (x < 0 || x > T) {
return false;
}
//algorithm
boolean sum[] = new boolean[x + 1];
sum = true;
for (int p = 0; !sum[x] && p < A.length; p++) {
int a = A[p];
for (int q = x; !sum[x] && q >= a; q--) {
if (!sum[q] && sum[q - a]) {
sum[q] = true;
}
}
}
return sum[x];
}
}
```
```package algorithm.programming.dynamic;

import org.junit.Test;
import static org.junit.Assert.*;

public class PositiveSubsequenceSumTest {

public PositiveSubsequenceSumTest() {
}

/**
* Test of positiveSubsetSum method, of class PositiveSubsequenceSum.
*/
@Test
public void testPositiveSubsetSum() {
System.out.println("positiveSubsetSum");
int[] S = {3, 5, 6};
PositiveSubsequenceSum instance = new PositiveSubsequenceSum();

int xTrue[] = {0, 3, 5, 6, 8, 9, 11, 14};
for (int x : xTrue) {
assertTrue(instance.positiveSubsetSum(S, x));
}

int xFalse[] = {1, 2, 4, 7, 10, 12, 13};
for (int x : xFalse) {
assertFalse(instance.positiveSubsetSum(S, x));
}
}
}```