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Dynamic Programming

Dynamic programming is a strategy for developing an algorithm where each subproblem is solved and the results recorded for use in solving larger problems. In this exercise you will write a pair of dynamic programming methods.

Wave 1 Newman-Conway Sequence

[Newman-Conway sequence] is the one which generates the following integer sequence. 1 1 2 2 3 4 4 4 5 6 7 7….. and follows below recursive formula.

P(1) = 1
P(2) = 1
for all n > 2
P(n) = P(P(n - 1)) + P(n - P(n - 1))

Given a number n then print n terms of Newman-Conway Sequence

Examples:

Input : 13
Output : 1 1 2 2 3 4 4 4 5 6 7 7 8

Input : 20
Output : 1 1 2 2 3 4 4 4 5 6 7 7 8 8 8 8 9 10 11 12

You should be able to do this in O(n) time complexity.

Wave 2 Largest Sum Contiguous Subarray

Write a method to find the contiguous subarray in a 1-dimensional array with the largest sum.

Largest subarray

This can be solved using Kadane's Algorithm

Initialize:
    max_so_far = 0
    max_ending_here = 0

Loop for each element of the array
  (a) max_ending_here = max_ending_here + a[i]
  (b) if(max_ending_here < 0)
            max_ending_here = 0
  (c) if(max_so_far < max_ending_here)
            max_so_far = max_ending_here
return max_so_far

Explanation

The idea of the Kadane’s algorithm is to look for all positive contiguous segments of the array (max_ending_here is used for this). And keep track of the maximum sum contiguous segment among all positive segments (max_so_far is used for this). Each time we get a positive sum compare it with max_so_far and update max_so_far if it is greater than max_so_far

There is also a subtle divide & conquer algorithm for this.

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