Worst case time complexity for the code

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Why is the worst time complexity of the following code is O(N)?

* V is sorted 
* V.size() = N
* The function is initially called as searchNumOccurrence(V, k, 0, N-1)

int searchNumOccurrence(vector<int> &V, int k, int start, int end) {
  if (start > end) return 0;
  int mid = (start + end) / 2;
  if (V[mid] < k) return searchNumOccurrence(V, k, mid + 1, end);
  if (V[mid] > k) return searchNumOccurrence(V, k, start, mid - 1);
  return searchNumOccurrence(V, k, start, mid - 1) + 1 + searchNumOccurrence(V, k, mid + 1, end);

What's the worst case? the worst case will be that all element are the same and equals to k. Then you have to at least read all elements, which is N. Since most function calls increase the output by 1, there are about N function calls (some returns 0, but they don't spawn new calls). Therefore, the worst time complexity is O(N).

Analysis of Algorithms, As discussed here, worst case time complexity is the most useful among best, When the code is too complex to consider all if-else cases, we can get an upper  The worst-case time complexity W(n) is then defined as W(n) = max(T 1 (n), T 2 (n), …). The worst-case time complexity for the contains algorithm thus becomes W(n) = n. Worst-case time complexity gives an upper bound on time requirements and is often easy to compute. The drawback is that it’s often overly pessimistic.

Yes, in the worst case if all the numbers in the array are equal to k, then in this worst case, the recurrence relation shall be:

T(n) = 2*T(n/2)

This translates into O(n).

Analysis of Algorithms, But no, the time complexity of the above code is O(n^2). Why? Because we are trying to know the fair enough count of elementary operations  Time Complexity of Code Using Summations - Duration: 9:24. randerson112358 51,984 views

The last case -

return searchNumOccurrence(V, k, start, mid - 1) + 1 + searchNumOccurrence(V, k, mid + 1, end);

is the bottleneck step. Assuming all the values in the array are the same, we get the following relation :

T(N) = 2 * T(N/2) + constant
= 4 * T(N/4) + constant ( 2 * constant = another constant ) 
= 8 * T(N/8) + constant 
= N * T(N/N) + constant 
= N + constant 
= O(N)

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Time Complexity - Calculating Worst Case For Algorithms, best-case: this is the complexity of solving the problem for the best input. the time complexity of an algorithm, we are talking about the worst-case. and will be able to find bottlenecks in your code which should be improved,  Binary Search Tree Min time in Worst Case Max time in Wor Skip navigation 1.11 Best Worst and Average Case Analysis Introduction to Big O Notation and Time Complexity

WORST CASE TIME COMPLEXITY OF AN ALGORITHM 1, Average case - Average time required for an algorithm or portion of code and it is sometimes done while analyzing the algorithm. First you should decide what can be done in constant time and what is are the dependent variables. For example when we are talking about multiplication algorithms, then we would calculate complexity in function of the number of digits.

Big-O Algorithm Complexity Cheat Sheet (Know Thy Complexities , The number of lines of code executed is actually depends on the value of x . During analyses of algorithm, mostly we will consider worst case scenario, i.e.,  The time complexity of an algorithm is the total amount of time required by an algorithm to complete its execution. In simple words, every piece of code we write, takes time to execute. The time taken by any piece of code to run is known as the time complexity of that code. The lesser the time complexity, the faster the execution.