Time complexity explained

What is the running time of the following algorithm?

Algorithm max(arr):
   max ← arr[0]
   for i = 1 to len(arr)-1
       if arr[i] > max
            max ← arr[i]
   return max

The answer depends on factors such as input, programming language and runtime, coding skill, compiler, operating system, and hardware.

We often want to reason about execution time in a way that depends only on the algorithm and its input. This can be achieved by choosing an elementary operation, which the algorithm performs repeatedly, and define the time complexity T(n) as the number of such operations the algorithm performs given an array of length n.

For the algorithm above we can choose the comparison arr[i] > max as an elementary operation.

  1. This captures the running time of the algorithm well, since comparisons dominate all other operations in this particular algorithm.
  2. Also, the time to perform a comparison is constant: it doesn’t depend on the size of arr.

The time complexity, measured in the number of comparisons, then becomes T(n) = n - 1.

In general, an elementary operation must have two properties:

  1. There can’t be any other operations that are performed more frequently as the size of the input grows.
  2. The time to execute an elementary operation must be constant: it mustn’t increase as the size of the input grows.

Worst-case time

Consider this algorithm:

Algorithm contains(arr, x):
   for i = 0 to len(arr)-1
       if x == arr[i]
            return true
   return false

The comparison x == arr[i] can be used as an elementary operation in this case. However, for this algorithm the number of comparisons depends not only on the number of elements, n, in the array but also on the value of x and the values in arr:

  • If x isn’t found in arr the algorithm makes n comparisons,
  • but if x equals arr[0] there is only one comparison.

Because of this, we often choose to study worst-case time complexity:

  • Let T1(n), T2(n), … be the execution times for all possible inputs of size n.
  • The worst-cast time complexity W(n) is then defined as W(n) = max(T1(n), T2(n), …).

The worst-cast 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.

Average-case time

Average-case time complexity is a less common measure:

  • Let T1(n), T2(n), … be the execution times for all possible inputs of size n,
    and let P1(n), P2(n), … be the probabilities of these inputs.
  • The average-case time complexity is then defined as P1(n)T1(n) + P2(n)T2(n) + …

Average-case time is often harder to compute, and it also requires knowledge of how the input is distributed.

Linear vs. quadratic time

Finally, we’ll look at an algorithm with poor time complexity.

Algorithm reverse(arr):
   for i = 1 to len(arr)-1
       x ← arr[i]
       for j = i downto 1
           arr[j] ← arr[j-1]
       arr[0] ← x

We choose the assignment arr[j] ← arr[j-1] as elementary operation. Updating an element in an array is a constant-time operation, and the assignment dominates the cost of the algorithm.

The number of elementary operations only depends on n in this case, and the time complexity becomes

W(n) = 1 + 2 + … + n - 1 = n(n - 1)/2 = n2/2 - n/2.

The quadratic term dominates for large n, and we therefore say that this algorithm has quadratic time complexity. This means that the algorithm scales poorly and can be used only for small input: to reverse the elements of an array with 10,000 elements, the algorithm will perform about 50,000,000 assignments.

In this case it’s easy to find an algorithm with linear time complexity.

Algorithm reverse(arr):
   for i = 0 to n/2
       swap arr[i] and arr[n-i-1]

This is a huge improvement over the previous algorithm: an array with 10,000 elements can now be reversed with only 5,000 swaps, i.e. 10,000 assignments. That’s roughly a 5,000-fold speed improvement, and the improvement keeps growing as the the input gets larger.

It’s common to use Big O notation when talking about time complexity. We could then say that the time complexity of the first algorithm is Θ(n2), and that the improved algorithm has Θ(n) time complexity.

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