[Solved]: Complexity of dynamic programming algorithm for Knapsack

Problem Detail: 

Dynamic programming algorithm for Knapsack is stated to have complexity $\mathcal O (nW)$.

However, I've also seen the complexity stated as $\mathcal O (n^2V)$, where $V=\max v_i$.

(Here $n$ is the number of items and $W$ the weight limit).

I see from the algorithm that the first complexity measure is correct: http://en.wikipedia.org/wiki/Knapsack_problem

Can someone tell me, why the other complexity measure works ?

Asked By : Shuzheng

Answered By : Yuval Filmus

The first complexity measure is in terms of the target weight, the second in terms of the heaviest element. Since $W \leq nV$ (or rather, we can assume that $W \leq nV$), the first estimate $O(nW)$ implies the second $O(n^2V)$. So $O(nW)$ is a stronger estimate than $O(n^2V)$.

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Question Source : http://cs.stackexchange.com/questions/43175