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We have a list of integers, where:
It follows that our list has at least one integer which appears at least twice. But it may have several duplicates, and each duplicate may appear more than twice.
Write a function which finds an integer that appears more than once in our list. Don’t modify the input! (If there are multiple duplicates, you only need to find one of them.)
We’re going to run this function on our new, super-hip MacBook Pro With Retina Display™. Thing is, the damn thing came with the RAM soldered right to the motherboard, so we can’t upgrade our RAM. So we need to optimize for space!
We can do this in O(1) space.
We can do this in less than O(n2) time while keeping O(1) space.
We can do this in O(nlgn) time and O(1) space.
We can do this without modifying the input.
Most O(nlgn) algorithms double something or cut something in half. How can we rule out half of the numbers each time we iterate through the list?
This one’s a classic! We just do one walk through the list, using a set to keep track of which items we’ve seen!
numbers_seen = set() for number in numbers: if number in numbers_seen: return number else: numbers_seen.add(number) # Whoops--no duplicate raise Exception('no duplicate!')
Bam. O(n) time and … O(n) space …
Right, we’re supposed to optimize for space. O(n) is actually kinda high space-wise. Hm. We can probably get O(1)…
We can “brute force” this by taking each number in the range 1..n and, for each, walking through the list to see if it appears twice.
for needle in range(1, len(numbers)): has_been_seen = False for number in numbers: if number == needle: if has_been_seen: return number else: has_been_seen = True # Whoops--no duplicate raise Exception('no duplicate!')
This is O(1) space and O(n2) time.
That space complexity can’t be beat, but the time cost seems a bit high. Can we do better?
One way to beat O(n2) time is to get O(nlgn) time. Sorting takes O(nlgn) time. And if we sorted the list, any duplicates would be right next to each-other!
But if we start off by sorting our list we’ll need to take O(n) space to store the sorted list…
…unless we sort the input list in place! ↴
Okay, so this’ll work:
This’ll keep us at O(1) space and bring us down to O(nlgn) time.
But modifying the input is kind of a drag—it might cause problems elsewhere in our code. Can we maintain this time and space cost without modifying the input?
Let’s take a step back. How can we break this problem down into subproblems?
If we’re going to do O(nlgn) time, we’ll probably be iteratively doubling something or iteratively cutting something in half. That’s how we usually get a “lgn“. So what if we could cut the problem in half somehow?
Well, binary search ↴ works by cutting the problem in half after figuring out which half of our input list holds the answer.
But in a binary search, the reason we can confidently say which half has the answer is because the list is sorted. For this problem, when we cut our unsorted list in half we can’t really make any strong statements about which elements are in the left half and which are in the right half.
What if we could cut the problem in half a different way, other than cutting the list in half?
With this problem, we’re looking for a needle (a repeated number) in a haystack (list). What if instead of cutting the haystack in half, we cut the set of possibilities for the needle in half?
The full range of possibilities for our needle is 1..n. How could we test whether the actual needle is in the first half of that range (1..2n) or the second half (2n+1..n)?
A quick note about how we’re defining our ranges: when we take 2n we’re doing integer division, so we throw away the remainder. To see what’s going on, we should look at what happens when n is even and when n is odd:
So we can notice a few properties about our ranges:
So, how do we know if the needle is in 1..2n or 2n+1..n?
Think about the original problem statement. We know that we have at least one repeat because there are n+1 items and they are all in the range 1..n, which contains only n distinct integers.
This notion of “we have more items than we have possibilities, so we must have at least one repeat” is pretty powerful. It’s sometimes called the pigeonhole principle. ↴ Can we exploit the pigeonhole principle to see which half of our range contains a repeat?
Imagine that we separated the input list into two sublists—one containing the items in the range 1..2n and the other containing the items in the range 2n+1..n.
Each sublist has a number of elements as well as a number of possible distinct integers (that is, the length of the range of possible integers it holds).
Given what we know about the number of elements vs the number of possible distinct integers in the original input list, what can we say about the number of elements vs the number of distinct possible integers in these sublists?
In the original list, there are n+1 elements but only n possible distinct integers. When we divide the list into two sublists, the total number of elements doesn’t change. Neither does the range of possible values. One of the sublists must have at least one more element than it has possible distinct integers.
Not convinced? We can prove this by contradiction. Suppose neither list had more elements than it had possible distinct integers. In other words, both lists have at most the same number of items as they have distinct possibilities. Then, across both lists, the number of items would then be at most the total number of possibilities, which is n. This is a contradiction—we know that our total number of items from the original input list is n+1, which is greater than n.
Now that we know one of our sublists has 1 or more items more than it has distinct possibilities, we know that sublist must have at least one duplicate, by the same pigeonhole argument that we use to know that the original input list has at least one duplicate.
So once we know which sublist has the count higher than its number of distinct possibilities, we can use this same approach recursively, cutting that sublist into two halves, etc, until we have just 1 item left in our range.
Of course, we don’t need to actually separate our list into sublists. All we care about is how long each sublist would be. So we can simply do one walk through the input list, counting the number of items that would be in each sublist.
Can you formalize this in code?
Careful—if we do this recursively, we’ll incur a space cost in the call stack! Do it iteratively instead.
Our approach is similar to a binary search, except we divide the range of possible answers in half at each step, rather than dividing the list in half.
floor = 1 ceiling = len(numbers) - 1 while floor < ceiling: # Divide our range 1..n into an upper range and lower range # (such that they don't overlap) # Lower range is floor..midpoint # Upper range is midpoint+1..ceiling midpoint = floor + ((ceiling - floor) / 2) lower_range_floor, lower_range_ceiling = floor, midpoint upper_range_floor, upper_range_ceiling = midpoint+1, ceiling # Count number of items in lower range items_in_lower_range = 0 for item in numbers: # Is it in the lower range? if item >= lower_range_floor and item <= lower_range_ceiling: items_in_lower_range += 1 distinct_possible_integers_in_lower_range = ( lower_range_ceiling - lower_range_floor + 1 ) if items_in_lower_range > distinct_possible_integers_in_lower_range: # There must be a duplicate in the lower range # so use the same approach iteratively on that range floor, ceiling = lower_range_floor, lower_range_ceiling else: # There must be a duplicate in the upper range # so use the same approach iteratively on that range floor, ceiling = upper_range_floor, upper_range_ceiling # Floor and ceiling have converged # We found a number that repeats! return floor
O(1) space and O(nlgn) time.
Tricky as this solution is, we can actually do even better, getting our runtime down to O(n) while keeping our space cost at O(1). The solution is NUTS; it’s probably outside the scope of what most interviewers would expect. But for the curious…here it is!
This function always returns one duplicate, but there may be several duplicates. Write a function that returns all duplicates.
Our answer was a modified binary search. We got there by reasoning about the expected runtime:
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