What is the big-O of this recursive algorithm

Question

We have a problem of size n

An algorithm that recursively solves a subproblem of size n − 1 and performs a linear amount of work on the solution.

I tried using plug n chug and found that the big-O is n, linear, but that does not seem right to me. What else could I try?

is 1 + 2 + 3 + ... + n linear just because each term is linear? — John Coleman
– John Coleman, Commented Mar 11, 2016 at 3:41

displayName · Accepted Answer · 2016-03-11 13:53:32Z

2

The formula for the recursion mentioned by you is:

T(n) = T(n-1) + O(n)

It implies that:

T(n) = kn + k(n-1) + k(n-2) + .. + k, which is equal to k * n * (n + 1) / 2.

So, the complexity of the algorithm is O(n²).

edited Mar 11, 2016 at 13:53

answered Mar 11, 2016 at 4:14

displayName

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Mac O'Brien · Accepted Answer · 2016-03-11 04:02:52Z

2

The people at Mathematics Stack Exchange will probably be able to do this better than I can, but I'll give it a shot.

The description of the algorithm is ambiguous, so there are two possibilities:

The algorithm performs a constant amount of work for each subproblem. In this case, the algorithm will in fact run in O(n) time (technically 2n, but constant factors are ignored).
The algorithm performs a linear amount of work for each subproblem. In this case, you're looking at a linear loop that does linear work each execution. n executions, n work per execution = O(n^2). Obviously each successive execution does less work, and this would manifest in a recurrence relation solution as something around T(n) = (1/2)n^2, assuming I remember that pattern of problem right.

answered Mar 11, 2016 at 4:02

Mac O'Brien

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John Coleman Over a year ago

The wording of the problem seems to suggest the second interpretation (the algorithm calls itself on n-1 and then does a linear amount of work "on the solution" which suggests that there is a linear amount of work after the recursive call.

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