title: Fastest Algorithm to Find the Minimum Sum of Absolute Differences through List Rotation
body: <p>By rotating 2 lists either from left to right, Find the smallest possible sum of the absolute value of the differences between each corresponding item in the two lists given they're the <em><strong>same</strong></em> length.</p>
  <p>Rotation Sample:</p>
  <pre><code>List [0, 1, 2, 3, 4, 5] rotated to the left = [1, 2, 3, 4, 5, 0]
  
  List [0, 1, 2, 3, 4, 5] rotated to the right= [5, 0, 1, 2, 3, 4]
  </code></pre>
  <p>Sum of Absolute Difference:</p>
  <pre><code>List 1 = [1, 2, 3, 4]
  List 2 = [5, 6, 7, 8]
  
  Sum of Abs. Diff. = |1-5| + |2-6| + |3-7| + |4-8| = 16
  </code></pre>
  <p>Once again, for any arbitrary length of list and integer values, task is to look for the least possible sum by simply rotating to the left/right of either or both lists.</p>
  <p>I had no problem with the rotation and acquiring of the minimum sum of absolute difference. I just want to know the smarter approach since my algorithm checks for every possible combination which is quite slow.</p>
  <p>Here is my bruteforce approach:</p>
  <pre><code>list1 = [45, 21, 64, 33, 49]
  list2 = [90, 12, 77, 52, 28]
  choices = []                # Put all possible sums into a list to find the minimum value.
  for j in range(len(list1)):  # List1 does a full rotation
      total = 0
      for k in range(len(list1)):
          total += abs(list1[k] - list2[k])
      list1.append(list1.pop(0))
      choices.append(total)
  print(min(choices))
  </code></pre>
  <p>What's a smarter approach? I would appreciate a shorter code and time complexity as well.</p>
  <p>I managed to make it faster by applying generators. Credits to @kuriboh for the idea! But since I'm still new in generator implementation, just want to know if this is the best way of implementing it to reduce my time complexity especially for my loop. <strong>Can we still go faster than this configuration?</strong></p>
  <pre><code>list1 = [45, 21, 64, 33, 49]
  list2 = [90, 12, 77, 52, 28]
  choices = []
  l = len(list1)
  for j in range(l):
      total = sum([abs(int(list1[k])-int(list2[k])) for k in range(l)])
      list1.append(list1.pop(0))
      choices.append(total)
  print(min(choices))
  </code></pre>