Kth Missing Positive Number

The problem requires us to find the kth missing positive integer from a given strictly increasing array.

The simplest way to solve this problem is to use a counter to track the current number and a second counter to track how many missing numbers we’ve found. We’ll increment the first counter from 1 upwards and check if the current number exists in the given array. If not, we increment the second counter. When the second counter equals k, we return the current number.

This can be implemented in Python as follows:

class Solution:
    def findKthPositive(self, arr, k):
        missing_counter = 0
        num_counter = 0
        while missing_counter < k:
            num_counter += 1
            if num_counter not in arr:
                missing_counter += 1
        return num_counter

This solution works by iterating over the positive integers, checking for each one whether it is in the array, and counting the ones that are not until it has found k missing numbers. It has a time complexity of O(n) where n is the maximum value in the array or the value of k, whichever is larger.

Please note that in a large-scale scenario, this approach might not be the most efficient, as the operation num_counter not in arr itself is an O(n) operation, resulting in an overall complexity of O(n^2). A more efficient solution could involve binary search or other techniques.

The termination condition is either reaching the end of the array or we have reached the value of k. Is this correct?

Yes, that’s correct. The process continues until one of two conditions is met:

  1. The number of missing integers found (missing_counter) is equal to k. This means that we have found the kth missing positive integer, so we can stop and return the current number (num_counter).

  2. If you iterate over the array instead of positive integers, the process would continue until you reach the end of the array. In this case, if you haven’t found the kth missing positive integer within the array, you would have to continue checking the numbers beyond the array until missing_counter is equal to k.

In the provided solution, we use the first termination condition. We iterate over positive integers starting from 1, not over the elements of the array. The loop ends when we find k missing positive integers, whether these are in the array or beyond it.

What are the basic building blocks of the solution?

The basic building blocks of the solution are:

  1. Iteration: A loop to iterate over the integers starting from 1.

  2. Checking for Membership : A test to check if the current integer is in the provided array.

  3. Counter Manipulation: Two counters - one for the current number (incremented at each step of the iteration) and one for the count of missing numbers (incremented each time the current number is found to be missing from the array).

  4. Conditional Termination: A check to end the loop when the count of missing numbers reaches k. This condition is the driving force for terminating the iteration and returning the result.

These building blocks come together to form a procedure that systematically checks each positive integer in order, determining if it’s in the array and if it contributes to the count of missing numbers, until the kth missing number is found.

What is the invariant ?

The invariant in this problem is the relationship between num_counter and missing_counter. At any point during the iteration, missing_counter equals the number of positive integers between 1 and num_counter (inclusive) that are not present in the array.

This property remains constant throughout the loop, making it the invariant. It’s what allows the algorithm to correctly identify the kth missing positive integer.

To put it simply, if you’ve iterated up to num_counter = x, then missing_counter will tell you how many positive integers less than or equal to x are missing from the array.

What is the loop invariant?

The loop invariant in this case would be the condition that at any given point in the loop, missing_counter represents the count of positive integers that are missing from the array in the range from 1 to num_counter.

This condition is the invariant because:

  1. Initialization: Before the loop starts, missing_counter is 0 and num_counter is 0, which correctly signifies that no positive integers in the range [1, 0] are missing from the array.

  2. Maintenance: During each iteration, num_counter is incremented, and if num_counter is not found in the array, missing_counter is also incremented. This maintains the condition that missing_counter equals the count of missing integers in the range [1, num_counter].

  3. Termination: When the loop terminates, missing_counter equals k, meaning that k positive integers are missing in the range [1, num_counter]. Thus, num_counter is the kth missing positive integer from the array. This is the desired outcome of the algorithm.

Therefore, the loop invariant holds at all stages of the loop: before the loop starts, during each iteration, and after the loop terminates.

In this context, the terms “invariant” and “loop invariant” are being used to describe the same concept. An invariant is a condition that remains true before, during, and after each iteration of a loop. In other words, it’s something that doesn’t change as your loop executes.

For this problem, the invariant or loop invariant is the relationship between num_counter and missing_counter. At any point during the loop, missing_counter represents the number of positive integers up to num_counter that are missing from the array. This condition holds true before the loop starts (when both counters are 0), during each iteration of the loop, and after the loop completes (when missing_counter equals k).