Question

Data Structures:

Suppose we sort an array of numbers, but it turns out every element of the array is the same, e.g., {17, 17, 17, ..., 17}. (So, in hindsight, the sorting is useless.)

(a) What is the asymptotic running time of insertion sort in this case?

(b) What is the asymptotic running time of selection sort in this case?

(c) What is the asymptotic running time of merge sort in this case?

(d) What is the asymptotic running time of quick sort in this case?

Answer 1

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a)

if all the elements in the array are same then inner loop of insertion sort only takes O(1) time so the overall complexity now depends on the outer loop which runs for O(n) time

so in this case the overall complexity = O(n)

b)

since selection sort , sorts the array by repeatedly find the minimum element even if all the elements are same it still takes

O(n²) time since finding minimum element takes O(n) and we have to do this for n elements

c)

merge sort irrespective of the elements present in it follows the recurrence relation T(n)= 2T(n/2) + O(n) which solves to O(nlgon) so the complexity for this case is O(nlogn)

d)

if all the elements in the array are same then the quick sort will behave in the worst case making the recurrence to

T(n)= T(n-1)+ O(n) (the pivot will stay there and recurrence will follow for n-1 elements) which by solving we get O(n²)