Algorithms for Coding Interviews in C++/

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Solution: Big (O) of Nested Loop with Multiplication

Solution: Big (O) of Nested Loop with Multiplication

This review provides a detailed analysis of how to solve the Big (O) of Nested Loop with Multiplication Challenge.

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Solution #

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int main() {
  int n = 10;    
  int sum = 0;   
  float pie = 3.14;  
  int var = 1;
  while (var < n){      // O(log n)
    cout << pie << endl; // O(log n)
    for (int j=0; j<var; j++)  // 8n - 12
      sum+=1;    //  (4n-6)
    var*=2;  // O(log n)
  }
  cout<<sum; 
}

Time Complexity

The outer loop here runs log(n)log(n) times. In the first iteration of the outer loop, the body of the inner loop runs once. In the second iteration, it runs twice, and so on. The number of executions of the body of the inner loop increases in powers of 2. So, if kk is the number of iterations of the outer loop, the body of the inner loop runs a total of 1+2+4+8+⋯+2k1+2+4+8+\cdots+2^k times. This geometric series sums to 2k+1−12^{k+1}-1. The inner loop condition requires that in the last time the inner loop runs, it runs at most nn times. This requires 2k<n2^k < n ...