Smallest number that can not be formed from sum of numbers from array

There's a beautiful algorithm for solving this problem in time O(n + Sort), where Sort is the amount of time required to sort the input array.

The idea behind the algorithm is to sort the array and then ask the following question: what is the smallest positive integer you cannot make using the first k elements of the array? You then scan forward through the array from left to right, updating your answer to this question, until you find the smallest number you can't make.

Here's how it works. Initially, the smallest number you can't make is 1. Then, going from left to right, do the following:

• If the current number is bigger than the smallest number you can't make so far, then you know the smallest number you can't make - it's the one you've got recorded, and you're done.
• Otherwise, the current number is less than or equal to the smallest number you can't make. The claim is that you can indeed make this number. Right now, you know the smallest number you can't make with the first k elements of the array (call it candidate) and are now looking at value A[k]. The number candidate - A[k] therefore must be some number that you can indeed make with the first k elements of the array, since otherwise candidate - A[k] would be a smaller number than the smallest number you allegedly can't make with the first k numbers in the array. Moreover, you can make any number in the range candidate to candidate + A[k], inclusive, because you can start with any number in the range from 1 to A[k], inclusive, and then add candidate - 1 to it. Therefore, set candidate to candidate + A[k] and increment k.

In pseudocode:

Sort(A)
candidate = 1
for i from 1 to length(A):
   if A[i] > candidate: return candidate
   else: candidate = candidate + A[i]
return candidate

Here's a test run on [4, 13, 2, 1, 3]. Sort the array to get [1, 2, 3, 4, 13]. Then, set candidate to 1. We then do the following:

• A[1] = 1, candidate = 1:
  • A[1] ≤ candidate, so set candidate = candidate + A[1] = 2
• A[2] = 2, candidate = 2:
  • A[2] ≤ candidate, so set candidate = candidate + A[2] = 4
• A[3] = 3, candidate = 4:
  • A[3] ≤ candidate, so set candidate = candidate + A[3] = 7
• A[4] = 4, candidate = 7:
  • A[4] ≤ candidate, so set candidate = candidate + A[4] = 11
• A[5] = 13, candidate = 11:
  • A[4] > candidate, so return candidate (11).

So the answer is 11.

The runtime here is O(n + Sort) because outside of sorting, the runtime is O(n). You can clearly sort in O(n log n) time using heapsort, and if you know some upper bound on the numbers you can sort in time O(n log U) (where U is the maximum possible number) by using radix sort. If U is a fixed constant, (say, 10⁹), then radix sort runs in time O(n) and this entire algorithm then runs in time O(n) as well.

Hope this helps!

Use bitvectors to accomplish this in linear time.

Start with an empty bitvector b. Then for each element k in your array, do this:

b = b | b << k | 2^(k-1)

To be clear, the i'th element is set to 1 to represent the number i, and | k is setting the k-th element to 1.

After you finish processing the array, the index of the first zero in b is your answer (counting from the right, starting at 1).

1. b=0
2. process 4: b = b | b<<4 | 1000 = 1000
3. process 13: b = b | b<<13 | 1000000000000 = 10001000000001000
4. process 2: b = b | b<<2 | 10 = 1010101000000101010
5. process 3: b = b | b<<3 | 100 = 1011111101000101111110
6. process 1: b = b | b<<1 | 1 = 11111111111001111111111

First zero: position 11.

Consider all integers in interval [2ⁱ .. 2ⁱ⁺¹ - 1]. And suppose all integers below 2ⁱ can be formed from sum of numbers from given array. Also suppose that we already know C, which is sum of all numbers below 2ⁱ. If C >= 2ⁱ⁺¹ - 1, every number in this interval may be represented as sum of given numbers. Otherwise we could check if interval [2ⁱ .. C + 1] contains any number from given array. And if there is no such number, C + 1 is what we searched for.

Here is a sketch of an algorithm:

1. For each input number, determine to which interval it belongs, and update corresponding sum: S[int_log(x)] += x.
2. Compute prefix sum for array S: foreach i: C[i] = C[i-1] + S[i].
3. Filter array C to keep only entries with values lower than next power of 2.
4. Scan input array once more and notice which of the intervals [2ⁱ .. C + 1] contain at least one input number: i = int_log(x) - 1; B[i] |= (x <= C[i] + 1).
5. Find first interval that is not filtered out on step #3 and corresponding element of B[] not set on step #4.

If it is not obvious why we can apply step 3, here is the proof. Choose any number between 2ⁱ and C, then sequentially subtract from it all the numbers below 2ⁱ in decreasing order. Eventually we get either some number less than the last subtracted number or zero. If the result is zero, just add together all the subtracted numbers and we have the representation of chosen number. If the result is non-zero and less than the last subtracted number, this result is also less than 2ⁱ, so it is "representable" and none of the subtracted numbers are used for its representation. When we add these subtracted numbers back, we have the representation of chosen number. This also suggests that instead of filtering intervals one by one we could skip several intervals at once by jumping directly to int_log of C.

Time complexity is determined by function int_log(), which is integer logarithm or index of the highest set bit in the number. If our instruction set contains integer logarithm or any its equivalent (count leading zeros, or tricks with floating point numbers), then complexity is O(n). Otherwise we could use some bit hacking to implement int_log() in O(log log U) and obtain O(n * log log U) time complexity. (Here U is largest number in the array).

If step 1 (in addition to updating the sum) will also update minimum value in given range, step 4 is not needed anymore. We could just compare C[i] to Min[i+1]. This means we need only single pass over input array. Or we could apply this algorithm not to array but to a stream of numbers.

Several examples:

Input:       [ 4 13  2  3  1]    [ 1  2  3  9]    [ 1  1  2  9]
int_log:       2  3  1  1  0       0  1  1  3       0  0  1  3

int_log:     0  1  2  3          0  1  2  3       0  1  2  3
S:           1  5  4 13          1  5  0  9       2  2  0  9
C:           1  6 10 23          1  6  6 15       2  4  4 13
filtered(C): n  n  n  n          n  n  n  n       n  n  n  n
number in
[2^i..C+1]:  2  4  -             2  -  -          2  -  -
C+1:              11                7                5

For multi-precision input numbers this approach needs O(n * log M) time and O(log M) space. Where M is largest number in the array. The same time is needed just to read all the numbers (and in the worst case we need every bit of them).

Still this result may be improved to O(n * log R) where R is the value found by this algorithm (actually, the output-sensitive variant of it). The only modification needed for this optimization is instead of processing whole numbers at once, process them digit-by-digit: first pass processes the low order bits of each number (like bits 0..63), second pass - next bits (like 64..127), etc. We could ignore all higher-order bits after result is found. Also this decreases space requirements to O(K) numbers, where K is number of bits in machine word.