Question

I have some random integers like

99 20 30 1 100 400 5 10

I have to find a sum from any combination of these integers that is closest(equal or more but not less) to a given number like

183

what is the fastest and accurate way of doing this?

Answer 1

If your numbers are small, you can use a simple Dynamic Programming(DP) technique. Don’t let this name scare you. The technique is fairly understandable. Basically you break the larger problem into subproblems.

Here we define the problem to be can[number]. If the number can be constructed from the integers in your file, then can[number] is true, otherwise it is false. It is obvious that 0 is constructable by not using any numbers at all, so can[0] is true. Now you try to use every number from the input file. We try to see if the sum j is achievable. If an already achieved sum + current number we try == j, then j is clearly achievable. If you want to keep track of what numbers made a particular sum, use an additional prev array, which stores the last used number to make the sum. See the code below for an implementation of this idea:

int UPPER_BOUND = number1 + number2 + ... + numbern //The largest number you can construct
bool can[UPPER_BOUND + 1]; //can[number] is true if number can be constructed
can[0] = true; //0 is achievable always by not using any number

int prev[UPPER_BOUND + 1]; //prev[number] is the last number used to achieve sum "number"
for (int i = 0; i < N; i++) //Try to use every number(numbers[i]) from the input file
{
   for (int j = UPPER_BOUND; j >= 1; j--) //Try to see if j is an achievable sum
   {
      if (can[j]) continue; //It is already an achieved sum, so go to the next j

      if (j - numbers[i] >= 0 && can[j - numbers[i]]) //If an (already achievable sum) + (numbers[i]) == j, then j is obviously achievable
      {
          can[j] = true;
          prev[j] = numbers[i]; //To achieve j we used numbers[i]
      } 
   }
}

int CLOSEST_SUM = -1;
for (int i = SUM; i <= UPPER_BOUND; i++)
   if (can[i])
   {
       //the closest number to SUM(larger than SUM) is i
       CLOSEST_SUM = i;
       break; 
   }

int currentSum = CLOSEST_SUM;    
do
{
    int usedNumber = prev[currentSum];
    Console.WriteLine(usedNumber);

    currentSum -= usedNumber;
} while (currentSum > 0);