Dynamic Programming Problem.. Array Partitioning..

问题

The question says,

That given an array of size n, we have to output/partition the array into subsets which sum to N.

For E,g, 
    I/p  arr{2,4,5,7}, n=4, N(sum) = 7(given)
    O/p = {2,5}, {7}

I saw similar kind of problem/explanation in the url Dynamic Programming3

And I have the following queries in the pdf:-

How could we find the subsets which sum to N, as the logic only tells whether the subset exist or not?

Also, if we change the question a bit, can we find two subsets which has equal average using the same ideology?

Can anybody thrown some light on this Dynamic Programming problem.. :)

Thanks in Advance..

回答1:

You can try to process recursively:

Given a SORTED array X={x1 ... xn} xi !=0 and an intger N.

First find all the possibilities "made" with just one element:

here if N=xp, eliminate all xi s.t i>=p

second find all the possibilities made with 2 elements:

{ (x1,x2) .... (xp-2,xp-1)}

Sort by sum and elminate all the sums >=N and you had the rules: xi cannot go with xj when xi+xj >= N

Third with 3 elments: You create all the part that respect the above rule. And idem step 2 etc...

Example:

X={1,2,4,7,9,10} N=9

step one:
{9}
X'={1,2,4,7,9}

step 2: cannot chose 9 and 10
X={(1,2) (1,4) (2,4) (1,7) (2,7) (4,7)}
{2,7}
X'={(1,2) (1,4) (2,4) (1,7)}

step 3: 4 and 2 cannot go with 7:
X={(1,2,4)}
no sol

{9} {2,7} are the only solutions

This diminishes the total number of comparaison (that would be 2^n = 2^6=64) you only did : 12 comparaisons

hope it helps

回答2:

Unfortunately, this is a very difficult problem. Even determining if there exists a single subset summing to your target value is NP-Complete.

If the problem is more restricted, you might be able to find a good algorithm. For example:

Do the subsets have to be contiguous?
Can you ignore subsets with more than K values?
Are the array values guaranteed to be positive?
Are the array values guaranteed to be distinct? What about differing from the other values by at least some constant factor?
Is there some bound on the difference between the smallest and largest value?

回答3:

The proposed algorithm stores only a single bit of information in the temporary array T[N], namely whether it's reachable at all. Obviously, you can store more information at each index [N], such as the values C[i] used to get there. (It's a variation of the "Dealing with Unlimited Copies" chapter in the PDF)

来源：https://stackoverflow.com/questions/6493120/dynamic-programming-problem-array-partitioning

标签

algorithm

dynamic-programming

bin-packing