Problem Description

In this problem, we're given an array called nums which contains n + 1 integers, where every integer is within the range of 1 to n, both inclusive. There's an important constraint in the problem: there is exactly one number in the array nums that appears more than once, and our task is to find that one repeated number. The challenge is to solve this problem under the following two conditions: we are not permitted to alter the original array, and we have to solve it using only a constant amount of space, which eliminates the possibility of using additional data structures that can grow with the size of the input.

Intuition

We can approach this problem with a binary search technique despite the fact that the array is not sorted. This might seem counterintuitive at first because binary search usually requires a sorted array. However, the key insight here is to use binary search not on the elements of the array itself, but rather on the range of numbers between 1 to n.

The intuition is based on the Pigeonhole Principle which states that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In this context, if there are more numbers in the array than the range it covers (n numbers in the range 1 to n), one of the numbers must be a duplicate.

We start by considering the entire range of numbers from 1 to n. Then, we use binary search to split this range into two halves: the first half (from 1 to mid) and the second half (from mid + 1 to n). The helper function, f, calculates how many numbers in the array are less than or equal to a given middle value, x. If the count is greater than x, we know the duplicate number must be in the first half; otherwise, it's in the second half.

By repeatedly halving the search space, we can eventually narrow down the range to a single number, which is the duplicate we're looking for. The bisect_left function in Python assists us in performing this binary search, and the key f, is passed to determine whether we should go left or right in our search.

Learn more about Two Pointers and Binary Search patterns.

Solution Approach

The solution approach for finding the duplicate number in the array leverages binary search, which is an efficient algorithm for finding an item from a sorted list by repeatedly dividing the search interval in half. Although the array itself is not sorted, we use binary search on the range of possible numbers (1 to n) to find the duplicate.

Let's walk through the implemented solution step by step:

1. Define the Helper Function f(x): This function takes an integer x and returns True if the number of elements in nums less than or equal to x is greater than x itself. Otherwise, it returns False. This function is essential because it determines whether the duplicate number lies in the lower half (1 to x) or the upper half (x + 1 to n) of the current search interval.

2. Binary Search with bisect_left: We use Python's bisect_left function from the bisect module to apply binary search. The bisect_left function takes three arguments:

   • The range to perform the search on, which is range(len(nums)). Note that this range goes from 0 to n inclusive since the array nums has n + 1 elements.
   • A boolean value that we are trying to find in the hypothetical sorted array of booleans, True in this case since we are looking for the point where f(x) transitions from False to True.
   • The key function, which in our solution is the helper function f. This function is applied to the middle value in the current search interval to guide the binary search process.

3. Finding the Duplicate Number: The binary search proceeds by checking the middle of the current interval. If f(mid) is True, it means there are more numbers in nums that are less than or equal to mid than there should be, indicating that the duplicate number must be less than or equal to mid. If f(mid) is False, it means that the duplicate is greater than mid and we shift our search to the upper half. This process continues until the algorithm converges on the duplicate number, which will be the point at which f(x) changes from False to True.

By repeatedly narrowing the search interval, we eventually find the duplicate number with O(log n) search iterations, with each iteration involving an O(n) operation to calculate the sum within the helper function. Overall, the solution thus takes O(n log n) time with constant space complexity, as we do not use any additional data structures that are dependent on the size of the input.

Example Walkthrough

Let's consider an example to understand the solution approach. Imagine we have an array nums with the size of n + 1, and it looks like this: [1, 3, 4, 2, 2]. The n in this case is 4 since the range of numbers is from 1 to 4.

Here's how we would walk through the problem step by step:

1. Define the Helper Function f(x):

   • We need to implement a function f(x) that returns True if the count of numbers in nums that are less than or equal to x is greater than x. For example, f(3) would count how many numbers in nums are <= 3. In our example, f(3) would return True because there are 4 numbers that fit the condition (1, 2, 2, and 3), which is greater than 3.

2. Binary Search with bisect_left:

   • Now we initiate a binary search on the range of numbers from 1 to n (1 to 4 in our example).
   • The bisect_left function will effectively split this range and use our function f to decide whether to look in the lower half or the upper half.
   • In the first iteration, the middle value mid between 1 and 4 is 2. We calculate f(2), which is False since there are only 2 numbers in nums that are less than or equal to 2, which is not greater than 2. This tells us that the duplicate must be larger than 2.
   • In the next iteration, the middle value between 3 and 4 is 3. We calculate f(3) and, as stated earlier, it returns True. This tells us to search in the lower half, but since only 3 and 4 is left, we have narrowed down 3 as the potential duplicate.

3. Finding the Duplicate Number:

   • Once we have the bounds narrowed down to a single number where f(x) transitions from False to True, or vice versa, we know we have found the duplicate number. In our example, when f(3) returns True and f(2) returned False, we know that 3 is the value where the transition happens, thus 3 is the duplicate number.

In conclusion, even though the number array is not sorted, we used the properties of binary search on the range 1 to n to efficiently find the duplicate number. Since f(x) only involves counting elements, and we only needed a range to apply bisect_left, we maintained constant space usage as per the problem's constraints.

Solution Implementation

Time and Space Complexity

The time complexity of the given code snippet is O(n * log n). This is because the bisect_left function performs binary search, which has a time complexity of O(log n), and it calls the f function on each step of the binary search. The f function has a time complexity of O(n) since it iterates over all n elements in the nums list to calculate the sum of all elements less than or equal to x. Since the binary search is performed in the range of len(nums), which is n, the f function is called O(log n) times, resulting in an overall time complexity of O(n * log n).

The space complexity of the code is O(1). The code uses a constant amount of additional space: the f function and the binary search do not use any extra space that grows with the input size. Therefore, regardless of the size of the input list nums, the additional space required by the algorithm remains constant.

Learn more about how to find time and space complexity quickly using problem constraints.