[Algorithm] Dynamic programming: Find Sets Of Numbers That Add Up To 16

For a given array, we try to find set of pair which sums up as the given target number.

For example, we are given the array and target as:

const data = [2, 4, 6, 10];
const target = 16;

We should be able to find 2 pair, which sum up to 16:

{2,4,10}
{6,10}

We need to create a function to return the number of pair we found, in this case, the answer is: 2

const data = [2, 4, 6, 10];
/**
 * Return number of pair found
 */
function DP(data, target = 16) {
  let memo = {};
  return rc(data, target, data.length - 1, memo);
  function rc(data, target, i, memo) {
    // Construct the key - value for memo
    let key = `${target}-${i}`;
    // Store the result
    let res = 0;
    // return the value if already calcualte
    if (memo[key]) {
      return memo[key];
    }
    // if the target is zero, then it is empty set, return 1
    if (target === 0) {
      return 1;
    } else if (target < 0) {
      // if target < 0, we don't consider the case, return 0
      return 0;
    } else if (i < 0) {
      // if i <0, means we run out of element, return 0
      return 0;
    }
    // if current value is larger than targer, we continue with next value
    if (data[i] > target) {
      res = rc(data, target, i - 1, memo);
    } else {
      /**
       * Two cases:
       * 1. The current value is not include:
       *  rc(data, target, i - 1, memo)
       * 2. The current value is include: the rest of els should sum up to new target value
       *  rc(data, target - data[i], i - 1, memo)
       */
      // i is not included + i is included
      res =
        rc(data, target, i - 1, memo) + rc(data, target - data[i], i - 1, memo);
    }
    memo[key] = res;
    return res;
  }
}

console.log(DP(data, 16)); // 2

Time complexity: O(target * n * 2 + 1) = O(T*N)