51Nod Limited Backpack Counting problem report

First of all, this problem can theoretically be done by O (Nlogn), because there is a formula that can be accelerated by polynomial multiplication on oeis.

But because the modulus is not very magnetic, so it makes nlogn difficult to write

Here's an O (N*SQRT (n)) approach

First, it's easy to see that when the size of an item is >SQRT (n), the limit of the number of items is a dummy.

This means that the size of the item >sqrt (n) is actually a full backpack

And for a complete backpack, there is another approach (refer to the division of NOIP2001 number)

Since we know that if we only use >SQRT (n) items, we use up to sqrt (n) items

You may wish to set f[i][j] to show that I have used the items of I >sqrt (n) to spell out the scheme of J

We classify the discussion:

1. At least one of these I items is sqrt (n) +1

2, this I items are >sqrt (n) +1

can easily get a recursive f[i][j]=f[i][j-i]+f[i-1][j-sqrt (n)-1]

Since a maximum of sqrt (n) items are used, the number of States is N*SQRT (n), and the Transfer is O (1)

So the time complexity of this part is O (n*sqrt (n))

We then consider using only the <=SQRT (n) condition

This is obviously a simple multi-knapsack problem.

We all know that the time complexity of multiple knapsack problems is O (number of items * Backpack size)

So the complexity of Time is O (n*sqrt (n))

After merging two cases, Time complexity O (n*sqrt (n))

Well, the upper part of the problem is almost the NOIP2001 number division.

The lower part is a bare multi-pack.

So the overall is noip difficult, and I do 51Nod when I did not want to come out

So my level is below Noip

Certificate of Completion Qaq

By the way, the final merged formula is a convolution form

51Nod Limited Backpack Counting problem report