Question

Given a set of numbers and a number k, find the maximum sum such that if you pick a number at index i you should not pick any number from index i - K to index i + K.

This problem was asked in google to my friend. I am not able to figure out a solution better then a naive O(n^2).

Answer 1

You can do this in O(n) by keeping track of the maximum of all values seen in the first i-K-1 entries in the array.

Python code:

A=[3,9,10,3,6,7,1,5]
K=2
m=A[0]
bestsum=0
for i in xrange(K+1,len(A)):
    m=max(A[i-K-1],m) # stores maximum of values in A[0],A[1],...,A[i-K-1]
    bestsum=max(bestsum,A[i]+m)
print bestsum

For each index i we combine A[i] with m which is the highest value seen in the initial values of the array A[0],..,A[i-K-1].