Find the Smallest Integer Not in a List

Question

An interesting interview question that a colleague of mine uses:

Suppose that you are given a very long, unsorted list of unsigned 64-bit integers. How would you find the smallest non-negative integer that does not occur in the list?

FOLLOW-UP: Now that the obvious solution by sorting has been proposed, can you do it faster than O(n log n)?

FOLLOW-UP: Your algorithm has to run on a computer with, say, 1GB of memory

CLARIFICATION: The list is in RAM, though it might consume a large amount of it. You are given the size of the list, say N, in advance.

Answer 1

Sort the list, look at the first and second elements, and start going up until there is a gap.

Answer 2

I'd just sort them then run through the sequence until I find a gap (including the gap at the start between zero and the first number).

In terms of an algorithm, something like this would do it:

def smallest_not_in_list(list):
    sort(list)
    if list[0] != 0:
        return 0
    for i = 1 to list.last:
        if list[i] != list[i-1] + 1:
            return list[i-1] + 1
    if list[list.last] == 2^64 - 1:
        assert ("No gaps")
    return list[list.last] + 1

Of course, if you have a lot more memory than CPU grunt, you could create a bitmask of all possible 64-bit values and just set the bits for every number in the list. Then look for the first 0-bit in that bitmask. That turns it into an O(n) operation in terms of time but pretty damned expensive in terms of memory requirements :-)

I doubt you could improve on O(n) since I can't see a way of doing it that doesn't involve looking at each number at least once.

The algorithm for that one would be along the lines of:

def smallest_not_in_list(list):
    bitmask = mask_make(2^64) // might take a while :-)
    mask_clear_all (bitmask)
    for i = 1 to list.last:
        mask_set (bitmask, list[i])
    for i = 0 to 2^64 - 1:
        if mask_is_clear (bitmask, i):
            return i
    assert ("No gaps")

Answer 3

Since the numbers are all 64 bits long, we can use radix sort on them, which is O(n). Sort 'em, then scan 'em until you find what you're looking for.

if the smallest number is zero, scan forward until you find a gap. If the smallest number is not zero, the answer is zero.

Answer 4

As pointed out in other answers you can do a sort, and then simply scan up until you find a gap.

You can improve the algorithmic complexity to O(N) and keep O(N) space by using a modified QuickSort where you eliminate partitions which are not potential candidates for containing the gap.

• On the first partition phase, remove duplicates.
• Once the partitioning is complete look at the number of items in the lower partition
• Is this value equal to the value used for creating the partition?
  • If so then it implies that the gap is in the higher partition.
    • Continue with the quicksort, ignoring the lower partition
  • Otherwise the gap is in the lower partition
    • Continue with the quicksort, ignoring the higher partition

This saves a large number of computations.

Answer 5

I am not sure if I got the question. But if for list 1,2,3,5,6 and the missing number is 4, then the missing number can be found in O(n) by: (n+2)(n+1)/2-(n+1)n/2

EDIT: sorry, I guess I was thinking too fast last night. Anyway, The second part should actually be replaced by sum(list), which is where O(n) comes. The formula reveals the idea behind it: for n sequential integers, the sum should be (n+1)*n/2. If there is a missing number, the sum would be equal to the sum of (n+1) sequential integers minus the missing number.

Thanks for pointing out the fact that I was putting some middle pieces in my mind.

Answer 6

Here's a simple O(N) solution that uses O(N) space. I'm assuming that we are restricting the input list to non-negative numbers and that we want to find the first non-negative number that is not in the list.

1. Find the length of the list; lets say it is N.
2. Allocate an array of N booleans, initialized to all false.
3. For each number X in the list, if X is less than N, set the X'th element of the array to true.
4. Scan the array starting from index 0, looking for the first element that is false. If you find the first false at index I, then I is the answer. Otherwise (i.e. when all elements are true) the answer is N.

In practice, the "array of N booleans" would probably be encoded as a "bitmap" or "bitset" represented as a byte or int array. This typically uses less space (depending on the programming language) and allows the scan for the first false to be done more quickly.

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This is how / why the algorithm works.

Suppose that the N numbers in the list are not distinct, or that one or more of them is greater than N. This means that there must be at least one number in the range 0 .. N - 1 that is not in the list. So the problem of find the smallest missing number must therefore reduce to the problem of finding the smallest missing number less than N. This means that we don't need to keep track of numbers that are greater or equal to N ... because they won't be the answer.

The alternative to the previous paragraph is that the list is a permutation of the numbers from 0 .. N - 1. In this case, step 3 sets all elements of the array to true, and step 4 tells us that the first "missing" number is N.

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The computational complexity of the algorithm is O(N) with a relatively small constant of proportionality. It makes two linear passes through the list, or just one pass if the list length is known to start with. There is no need to represent the hold the entire list in memory, so the algorithm's asymptotic memory usage is just what is needed to represent the array of booleans; i.e. O(N) bits.

(By contrast, algorithms that rely on in-memory sorting or partitioning assume that you can represent the entire list in memory. In the form the question was asked, this would require O(N) 64-bit words.)

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@Jorn comments that steps 1 through 3 are a variation on counting sort. In a sense he is right, but the differences are significant:

• A counting sort requires an array of (at least) Xmin - Xmin counters where Xmax is the largest number in the list and Xmin is the smallest number in the list. Each counter has to be able to represent N states; i.e. assuming a binary representation it has to have an integer type (at least) ceiling(log2(N)) bits.
• To determine the array size, a counting sort needs to make an initial pass through the list to determine Xmax and Xmin.
• The minimum worst-case space requirement is therefore ceiling(log2(N)) * (Xmax - Xmin) bits.

By contrast, the algorithm presented above simply requires N bits in the worst and best cases.

However, this analysis leads to the intuition that if the algorithm made an initial pass through the list looking for a zero (and counting the list elements if required), it would give a quicker answer using no space at all if it found the zero. It is definitely worth doing this if there is a high probability of finding at least one zero in the list. And this extra pass doesn't change the overall complexity.

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EDIT: I've changed the description of the algorithm to use "array of booleans" since people apparently found my original description using bits and bitmaps to be confusing.

Answer 7

Since the OP has now specified that the original list is held in RAM and that the computer has only, say, 1GB of memory, I'm going to go out on a limb and predict that the answer is zero.

1GB of RAM means the list can have at most 134,217,728 numbers in it. But there are 2⁶⁴ = 18,446,744,073,709,551,616 possible numbers. So the probability that zero is in the list is 1 in 137,438,953,472.

In contrast, my odds of being struck by lightning this year are 1 in 700,000. And my odds of getting hit by a meteorite are about 1 in 10 trillion. So I'm about ten times more likely to be written up in a scientific journal due to my untimely death by a celestial object than the answer not being zero.

Answer 8

To illustrate one of the pitfalls of O(N) thinking, here is an O(N) algorithm that uses O(1) space.

for i in [0..2^64):
  if i not in list: return i

print "no 64-bit integers are missing"

Answer 9

You can do it in O(n) time and O(1) additional space, although the hidden factor is quite large. This isn't a practical way to solve the problem, but it might be interesting nonetheless.

For every unsigned 64-bit integer (in ascending order) iterate over the list until you find the target integer or you reach the end of the list. If you reach the end of the list, the target integer is the smallest integer not in the list. If you reach the end of the 64-bit integers, every 64-bit integer is in the list.

Here it is as a Python function:

def smallest_missing_uint64(source_list):
    the_answer = None

    target = 0L
    while target < 2L**64:

        target_found = False
        for item in source_list:
            if item == target:
                target_found = True

        if not target_found and the_answer is None:
            the_answer = target

        target += 1L

    return the_answer

This function is deliberately inefficient to keep it O(n). Note especially that the function keeps checking target integers even after the answer has been found. If the function returned as soon as the answer was found, the number of times the outer loop ran would be bound by the size of the answer, which is bound by n. That change would make the run time O(n^2), even though it would be a lot faster.

Answer 10

For a space efficient method and all values are distinct you can do it in space O( k ) and time O( k*log(N)*N ). It's space efficient and there's no data moving and all operations are elementary (adding subtracting).

1. set U = N; L=0
2. First partition the number space in k regions. Like this:
   • 0->(1/k)*(U-L) + L, 0->(2/k)*(U-L) + L, 0->(3/k)*(U-L) + L ... 0->(U-L) + L
3. Find how many numbers (count{i}) are in each region. (N*k steps)
4. Find the first region (h) that isn't full. That means count{h} < upper_limit{h}. (k steps)
5. if h - count{h-1} = 1 you've got your answer
6. set U = count{h}; L = count{h-1}
7. goto 2

this can be improved using hashing (thanks for Nic this idea).

1. same
2. First partition the number space in k regions. Like this:
   • `L + (i/k)->L + (i+1/k)*(U-L)'
3. inc count{j} using j = (number - L)/k (if L < number < U)
4. find first region (h) that doesn't have k elements in it
5. if count{h} = 1 h is your answer
6. set U = maximum value in region h L = minimum value in region h

This will run in O(log(N)*N).

Answer 11

Thanks to egon, swilden, and Stephen C for my inspiration. First, we know the bounds of the goal value because it cannot be greater than the size of the list. Also, a 1GB list could contain at most 134217728 (128 * 2^20) 64-bit integers.

Hashing part
I propose using hashing to dramatically reduce our search space. First, square root the size of the list. For a 1GB list, that's N=11,586. Set up an integer array of size N. Iterate through the list, and take the square root* of each number you find as your hash. In your hash table, increment the counter for that hash. Next, iterate through your hash table. The first bucket you find that is not equal to it's max size defines your new search space.

Bitmap part
Now set up a regular bit map equal to the size of your new search space, and again iterate through the source list, filling out the bitmap as you find each number in your search space. When you're done, the first unset bit in your bitmap will give you your answer.

This will be completed in O(n) time and O(sqrt(n)) space.

(*You could use use something like bit shifting to do this a lot more efficiently, and just vary the number and size of buckets accordingly.)

Answer 12

How would you find the smallest non-negative integer that does not occur in the list?

That should be easy, and in O(1), the answer is 0 or the question cannot be answered.

If 0 is included in the list, no non-negative integer smaller extists. If 0 is not included in the list, it must be the smallest non-negative integer :-)

Answer 13

Well if there is only one missing number in a list of numbers, the easiest way to find the missing number is to sum the series and subtract each value in the list. The final value is the missing number.

Answer 14

As Stephen C smartly pointed out, the answer must be a number smaller than the length of the array. I would then find the answer by binary search. This optimizes the worst case (so the interviewer can't catch you in a 'what if' pathological scenario). In an interview, do point out you are doing this to optimize for the worst case.

The way to use binary search is to subtract the number you are looking for from each element of the array, and check for negative results.

Answer 15

I like the "guess zero" apprach. If the numbers were random, zero is highly probable. If the "examiner" set a non-random list, then add one and guess again:

LowNum=0
i=0
do forever {
  if i == N then leave /* Processed entire array */
  if array[i] == LowNum {
     LowNum++
     i=0
     }
   else {
     i++
   }
}
display LowNum

The worst case is n*N with n=N, but in practice n is highly likely to be a small number (eg. 1)

Answer 16

If the datastructure can be mutated in place and supports random access then you can do it in O(N) time and O(1) additional space. Just go through the array sequentially and for every index write the value at the index to the index specified by value, recursively placing any value at that location to its place and throwing away values > N. Then go again through the array looking for the spot where value doesn't match the index - that's the smallest value not in the array. This results in at most 3N comparisons and only uses a few values worth of temporary space.

# Pass 1, move every value to the position of its value
for cursor in range(N):
    target = array[cursor]
    while target < N and target != array[target]:
        new_target = array[target]
        array[target] = target
        target = new_target

# Pass 2, find first location where the index doesn't match the value
for cursor in range(N):
    if array[cursor] != cursor:
        return cursor
return N

Answer 17

Well done Ants Aasma! I thought about the answer for about 15 minutes and independently came up with an answer in a similar vein of thinking to yours:

#define SWAP(x,y) { numerictype_t tmp = x; x = y; y = tmp; }
int minNonNegativeNotInArr (numerictype_t * a, size_t n) {
    int m = n;
    for (int i = 0; i < m;) {
        if (a[i] >= m || a[i] < i || a[i] == a[a[i]]) {
            m--;
            SWAP (a[i], a[m]);
            continue;
        }
        if (a[i] > i) {
            SWAP (a[i], a[a[i]]);
            continue;
        }
        i++;
    }
    return m;
}

m represents "the current maximum possible output given what I know about the first i inputs and assuming nothing else about the values until the entry at m-1".

This value of m will be returned only if (a[i], ..., a[m-1]) is a permutation of the values (i, ..., m-1). Thus if a[i] >= m or if a[i] < i or if a[i] == a[a[i]] we know that m is the wrong output and must be at least one element lower. So decrementing m and swapping a[i] with the a[m] we can recurse.

If this is not true but a[i] > i then knowing that a[i] != a[a[i]] we know that swapping a[i] with a[a[i]] will increase the number of elements in their own place.

Otherwise a[i] must be equal to i in which case we can increment i knowing that all the values of up to and including this index are equal to their index.

The proof that this cannot enter an infinite loop is left as an exercise to the reader. :)

Answer 18

What about using a hash table to hold the numbers? Once all numbers are done, run a counter from 0 till we find the lowest... A reasonably good hash will hash and store in constant time, and retrieves in constant time.

for every i in X          // One scan Θ(1)
    hashtable.put(i, i);    // O(1)

low = 0;

while (hashtable.get(i) <> null)    // utmost n+1 times
    low ++;

print low;


The worst case if there are n elements in the array, and are {0, 1, ... n-1} In which case, the answer will be obtained at n, still keeping O(n).