问题
I'm using DP algorithm, i.e. storing sub-problem values in 2D array where one axis
means n items and other - w values from 0 to W where W is the maximum
capacity of knapsack. Therefore T[n-1][W] value is the optimum I need to
calculate. I've read in other sources that time complexity of this algorithm is
O(nW). My quesiton would be: is it possible to reduce this time complexity even more?
I found other answer which talks about pretty much same thing but I can't understant it without example: how to understand about reducing time complexity on 0~1 knapsack
I tells that we de not need to to calculate T[i][w] with small w values as they are not used in the optimum, but I can't get this properly, could anyone give detailed and visual example? This would benefit me a lot.
回答1:
The 2D array you're trying to fill is of size n by W (actually, W+1 since the values go from 0..W, but off-by-one doesn't affect the asymptotic complexity here). Therefore, to fill that array, you would need to do at least n*W work (even if you just initialize the array to all zeroes!).
Therefore, Θ(nW) (tightly bound, which is both O(nW) and Ω(nW)) is the best you can do in terms of asymptotic algorithmic time complexity.
This is what makes the dynamic programming solution so cool, is that you spend constant time on each element of the solution array (in this case, 2D) doing some constant work, from the bottom up (contrast this to the complexity of the top-down recursive solution!).
来源：https://stackoverflow.com/questions/42879631/reducing-time-complexity-of-knapsack-01-while-using-dp-algorithm