问题
You are given all subset sums of an array. You are then supposed to recover the original array from the subset sums provided.
Every element in the original array is guaranteed to be non-negative and less than 10^5. There are no more than 20 elements in the original array. The original array is also sorted. The input is guaranteed to be valid.
Example 1
If the subset sums provided are this:
0 1 5 6 6 7 11 12
We can quickly deduce that the size of the original array is 3 since there are 8 (2^3) subsets. The output (i.e original array) for the above input is this:
1 5 6
Example 2
Input:
0 1 1 2 8 9 9 10
Output:
1 1 8
What I Tried
Since all elements are guaranteed to be non-negative, the largest integer in the input must be the total of the array. However, I am not sure as to how do I proceed from there. By logic, I thought that the next (2^2 - 1) largest subset sums must include all except one element from the array.
However, the above logic does not work when the original array is this:
1 1 8
That's why I am stuck and am not sure on how to proceed on.
回答1:
Say S is the subset sum array and A is the original array. I'm assuming S is sorted.
|A| = log2(|S|)
S[0] = 0
S[1] = A[0]
S[2] = A[1]
S[3] = EITHER A[2] OR A[0] + A[1].
In general, S[i] for i >= 3 is either an element of A or a combination of the elements of A that you've already encountered. When processing S, skip once per combination of known elements of A that generate a given number, add any remaining numbers to A. Stop when A gets to the right size.
E.g., if A=[1,2,7,8,9] then S will include [1,2,1+2=3,...,1+8=9, 2+7=9,9,...]. When processing S we skip over two 9s because of 1+8 and 2+7, then see a third 9 which we know must belong to A.
E.g., if S=[0,1,1,2,8,9,9,10] then we know A has 3 elements, that the first 2 elements of A are [1,1], when we get to 2 we skip it because 1+1=2, we append 8 and we're done because we have 3 elements.
回答2:
Here's an easy algorithm that doesn't require finding which subset sums to a given number.
S ← input sequence
X ← empty sequence
While S has a non-zero element:
- d ← second smallest element of S (the smallest one is always zero)
- Insert d in X
- N ← empty sequence
- While S is not empty:
- z ← smallest element of S
- Remove both z and z+d from S (if S does not contain z+d, it's an error; remove only one instance of both z and z+d if there are several).
- Insert z in N.
- S ← N
Output X.
来源：https://stackoverflow.com/questions/50663548/recover-original-array-from-all-subsets