What’s the minimum number of weights required to count between 1 to 121 pounds ?

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Find the minimum number of weights required to count anything between 1 to 121 pounds ?

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This type of problems have a standard formula to compute the minimum number of weights and it’s given by Ceiling of the log to the base 2 of the maximum weight to be counted. 

Thus, the formula would be where W is the max weight which is 121 pounds in this case which would give a value of 7. 

Hence the minimum number of weights required for this would be 7 and those weights would be powers of 2 from 0 until N where N will be the value such that 2^N doesn’t exceed 121,  which gives the weights as [1, 2, 4, 8, 16, 32, 64]

Below is an algorithm to count all possible weights from 1 to 121 pounds using binary search.

init();
 function init() {	
 var output = [];
 var weights = [1, 2, 4, 8, 16, 32, 64];
 var maxWeightToCount = 121;
   for (var i = 1; i <= maxWeightToCount; i++) {
  searchWeights(weights, i, output);
  console.log('Sequence of minimum number of weights with sum equal = ', i, 
  'pounds from the given list of available weights ', weights, '=', output);
  output = [];
 }
}
 function searchWeights(arr, value, result) {
 var val =  binSearch(arr, value);
   if (value - val === 0) {
  result.push(val);
  return;
 } 
   if (val === -1) {
  val = getNearestValue(arr, value);
  result.push(val);
 }
   searchWeights(arr, value - val, result);
}
 function binSearch(arr, W, bounds=[]) {
 var l = 0, r = arr.length - 1;
  while (l <= r) { 
        var m = Math.floor(l + (r - l) / 2); 
           // Check if W is present at mid 
        if (arr[m] == W) 
            return arr[m]; 
           // If W is greater, ignore left half 
        if (arr[m] < W) 
            l = m + 1; 
           // If W is smaller, ignore right half 
        else
            r = m - 1; 
    } 
   bounds[0] = l, bounds[1] = r;
 return -1;
}
  function getNearestValue(a, value) {
 var bounds = [];
   if(value < a[0]) {
  return a[0];
 }
 if(value > a[a.length-1]) {
  return a[a.length-1];
 }
   var retVal = binSearch(a, value, bounds);
   var lo = bounds[0], hi = bounds[1];
   if (retVal === -1) {
  retVal = (a[lo] - value) < (value - a[hi] && a[lo] < value) ? a[lo] : a[hi];
 }
   return retVal;
}

Explanation:-

  1. Use binary search to find if the weight to be counted is there in the available list of weights. If a match is found, push it to the output array and return.
  2. If the previous step fails, find the closest possible value from the given set of weights which is lower than the weight to be counted and push that to an array.
  3. Compute the difference between, intended weight to be counted and the closest possible lower value obtained in step 2 and repeat steps 1 & 2.
  4. Repeat step 3 until the difference obtained in step 3 is zero.

Here is the demo of the above algorithm:- http://code.golibrary.co/EjXBOJn1O72e

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