Minimum least common multiplier for random combinations

Question

TLDR: I'm looking for an algo that returns the smallest possible least common multiplier for a variable array of numbers while knowing:

• one of the numbers
• the size of my array
• the min and max values possible for the numbers

I’m working with a music app and have an algo problem: When mixing different rhythms (each with a different number of steps), I need to compute the resulting number of steps for the result to loop. This is done easily with a least common multiplier calculation. Lets assume I have a lengths array that contains all the different lengths in steps

var lengths = [4,5,6,8]

//greatest common denominator
function gcd(a,b){
  var t,b,a
  while(b != 0){
    t = b;
    b = a%b
    a=t
  }
  return a;
}
//least common multiplier
function lcm(a,b){
  return a*b/gcd(a,b)
}
function getLoopLength(arr{
  var result = 1;
  for(var i = 0;i<arr.length;i++)
    result = lcm(result,arr[i])
  return m
}


getLoopLength(lengths)
==> 120
// superimposing 4 rhythm with length 4,5,6 and 8 will result in a a rhythms that loops in 120 steps

Now I need a function that computes the minimum number of steps for the following hypotheses:

• The possible step lengths are bounded (between 2 and 11 in my case - might change)
• All the step lengths values must different
• 1 length value is known (will be a variable)
• The size of my lengths array can vary (between 1 and 4 in my case - will not change)

So what I'm after is a function that looks like this:

var minPossibleLength(knownLength, lengthsSize){
  ...
  return min
}

For example minPossibleLength(4,4) should return 24 (when my lengths are [2,4,8,3] or [2,4,8,6])

Now I tried brute forcing it, loop through all possible lengths combinations and find the minimum lcm, and it does work with my conditions, but I'd love to know if I can find a more elegant and efficient solution.

Thx

Answer 1

The following algorithm for minPossibleLength(4,4) finds better solution than 24: least common multiple for [4, 2, 3, 6] is 12.

var lengths = [4,5,6,8]

    //greatest common denominator
    function gcd(a,b){
      var t,b,a
      while(b != 0){
        t = b;
        b = a%b
        a=t
      }
      return a;
    }
    //least common multiplier
    function lcm(a,b){
      return a*b/gcd(a,b)
    }
    function getLoopLength(arr, length){
      var result = 1;
      for(var i = 0;i<arr.length && i<length;i++)
        result = lcm(result,arr[i])
      return result
    }

    var minBound = 2;
    var maxBound = 11;

    function minPossibleLength(knownLength, lengthsSize) {      
      var min = 27720; // Maximum for bound range [2..11]
      var newmin; // Newly computed minimum.
      if (lengthsSize == 1)
        return knownLength;
      lengths[0] = knownLength;
      for(var i = minBound; i<=maxBound; i++) {
        if (i != knownLength) {
          lengths[1] = i;
          for(var j = (lengthsSize>2?i+1:maxBound); j<=maxBound; j++) {
            if (lengthsSize<3 || (i != j && j!= knownLength)) {
              lengths[2] = j;
              for(var k = (lengthsSize>3?j+1:maxBound); k<=maxBound; k++) {
                if (lengthsSize<4 || (i != k && j != k && k!= knownLength)) {
                  lengths[3] = k;
                  newmin = getLoopLength(lengths, lengthsSize)
                  if (newmin < min) {
                    min = newmin;
                    console.log('Minimum lcm so far for (['+knownLength+', '+i+(lengthsSize>2?', '+j+(lengthsSize>3?', '+k:''):'')+']) = '+min); 
                  }
                }
              }
            }
          }
        }
      }
      return min;
    }

    minPossibleLength(4,4);