Question

A sequence of n distinct values A[O..n – 1] is said to be downup if there is an index p with 0 < p < n such that the values of A decrease up to Aſp) and then increase for the remainder of the sequence. The index p of value Aſp) is the valley of the sequence. For example sequence 50, 10, 5, 2, 1, 20, 30 is downup with valley 4, since A[5] = 60 and the sequence decreases to 1 and then increases. The sequence 5, 1, 4,3 is not downup. Design an o(n) worst-case running time algorithm that when given as input an downup sequence, it finds its valley p. Analyze the performance of your proposed algorithm and show that it is o(n) (this is a little-oh).

Answer 1

Algorithm :

For an array of size 'n', let's assume the valley index is p. Iterate over the array and keep updating the value of p, while array is still decreasing.

p = 0

min = arr[0]

for ( i = 1; i < n ; i++)

{

if ( arr[i] < min)

{

min = arr[i]

p = i

}

}

Time - Complexity Analysis : For worst case, array will be sorted in descending order. In this case, Algorithm will iterate over all the elements one by one, resulting in over all 'n' iterations in total. So, worst case time complexity is o(n).