title: Codility Peaks Complexity
body: <p>I've just done the following Codility <a href="http://codility.com/demo/take-sample-test/peaks" rel="noreferrer">Peaks</a> problem. The problem is as follows:</p>
  
  <hr>
  
  <p>A non-empty zero-indexed array A consisting of N integers is given.
  A peak is an array element which is larger than its neighbors. More precisely, it is an index P such that 0 &lt; P &lt; N − 1,  A[P − 1] &lt; A[P] and A[P] > A[P + 1].
  For example, the following array A:</p>
  
  <pre><code>A[0] = 1
  A[1] = 2
  A[2] = 3
  A[3] = 4
  A[4] = 3
  A[5] = 4
  A[6] = 1
  A[7] = 2
  A[8] = 3
  A[9] = 4
  A[10] = 6
  A[11] = 2
  </code></pre>
  
  <p>has exactly three peaks: 3, 5, 10.
  We want to divide this array into blocks containing the same number of elements. More precisely, we want to choose a number K that will yield the following blocks:
  A[0], A[1], ..., A[K − 1],
  A[K], A[K + 1], ..., A[2K − 1],
  ...
  A[N − K], A[N − K + 1], ..., A[N − 1].
  What's more, every block should contain at least one peak. Notice that extreme elements of the blocks (for example A[K − 1] or A[K]) can also be peaks, but only if they have both neighbors (including one in an adjacent blocks).
  The goal is to find the maximum number of blocks into which the array A can be divided.
  Array A can be divided into blocks as follows:</p>
  
  <p>one block (1, 2, 3, 4, 3, 4, 1, 2, 3, 4, 6, 2). This block contains three peaks.</p>
  
  <p>two blocks (1, 2, 3, 4, 3, 4) and (1, 2, 3, 4, 6, 2). Every block has a peak.</p>
  
  <p>three blocks (1, 2, 3, 4), (3, 4, 1, 2), (3, 4, 6, 2). Every block has a peak. </p>
  
  <p>Notice in particular that the first block (1, 2, 3, 4) has a peak at A[3], because A[2] &lt; A[3] > A[4], even though A[4] is in the adjacent block.
  However, array A cannot be divided into four blocks, (1, 2, 3), (4, 3, 4), (1, 2, 3) and (4, 6, 2), because the (1, 2, 3) blocks do not contain a peak. Notice in particular that the (4, 3, 4) block contains two peaks: A[3] and A[5].
  The maximum number of blocks that array A can be divided into is three.</p>
  
  <p>Write a function:
  class Solution { public int solution(int[] A); }
  that, given a non-empty zero-indexed array A consisting of N integers, returns the maximum number of blocks into which A can be divided.
  If A cannot be divided into some number of blocks, the function should return 0.
  For example, given:</p>
  
  <pre><code>A[0] = 1
  A[1] = 2 
  A[2] = 3 
  A[3] = 4 
  A[4] = 3 
  A[5] = 4 
  A[6] = 1 
  A[7] = 2 
  A[8] = 3 
  A[9] = 4 
  A[10] = 6 
  A[11] = 2
  </code></pre>
  
  <p>the function should return 3, as explained above.
  Assume that:</p>
  
  <p>N is an integer within the range [1..100,000];
  each element of array A is an integer within the range [0..1,000,000,000].</p>
  
  <p>Complexity:</p>
  
  <p>expected worst-case time complexity is O(N*log(log(N)))</p>
  
  <p>expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).</p>
  
  <p>Elements of input arrays can be modified.</p>
  
  <hr>
  
  <h2>My Question</h2>
  
  <p>So I solve this with what to me appears to be the brute force solution – go through every group size from <code>1..N</code>, and check whether every group has at least one peak. The first 15 minutes I was trying to solve this I was trying to figure out some more optimal way, since the required complexity is O(N*log(log(N))).</p>
  
  <p>This is my "brute-force" code that passes all the tests, including the large ones, for a score of 100/100:</p>
  
  <pre><code>public int solution(int[] A) {
      int N = A.length;
  
      ArrayList&lt;Integer&gt; peaks = new ArrayList&lt;Integer&gt;();
      for(int i = 1; i &lt; N-1; i++){
          if(A[i] &gt; A[i-1] &amp;&amp; A[i] &gt; A[i+1]) peaks.add(i);
      }
  
      for(int size = 1; size &lt;= N; size++){
          if(N % size != 0) continue;
          int find = 0;
          int groups = N/size;
          boolean ok = true;
          for(int peakIdx : peaks){
              if(peakIdx/size &gt; find){
                  ok = false;
                  break;
              }
              if(peakIdx/size == find) find++;
          }
          if(find != groups) ok = false;
          if(ok) return groups;
      }
  
      return 0;
  }
  </code></pre>
  
  <p>My question is how do I deduce that this is in fact O(N*log(log(N))), as it's not at all obvious to me, and I was surprised I pass the test cases. I'm looking for even the simplest complexity proof sketch that would convince me of this runtime. I would assume that a log(log(N)) factor means some kind of reduction of a problem by a square root on each iteration, but I have no idea how this applies to my problem. Thanks a lot for any help</p>