Problem Description

You are given an array of positive integers, named nums, and a target number represented by the integer k. Your goal is to collect a list of elements labeled from 1 to k through a specific operation. The operation is defined as removing the last element from the array nums and placing that element in your collection. You must determine the fewest number of operations needed to collect all elements from 1 to k. If an element is not present in nums or cannot be obtained by performing the allowed operations, it implies that it is impossible to collect all elements from 1 to k.

For example, if your array nums is [1,3,2,4,3] and k is 3, you can perform the following operations:

• Remove 3 from nums and add it to the collection. Now your collection has [3] and nums is [1,3,2,4].
• Remove 4 from nums but do not add to the collection since 4 is not needed (we want elements from 1 to k=3).
• Remove 2 from nums and it goes to the collection. Your collection is now [3,2] and nums is [1,3].
• Finally, remove 3 from nums and since it’s already in the collection, you can ignore it.
• Remove 1 from nums and add it to your collection.

After these 4 operations, your collection has the elements [1, 2, 3], and thus, the minimum number of operations is 4. The task is to figure out this minimum number of operations for any given array nums and integer k.

Intuition

The intuition behind the solution is to address the problem efficiently by working backwards, starting from the end of the array - since it's the only place where we can remove elements - and moving towards the front. By doing this, we keep a check on what elements are getting added to the collection and ensure the following:

• We only care about elements that are less than or equal to k, because we want to collect elements 1 to k.
• We avoid adding duplicates to our collection because each number from 1 to k should only be collected once.

We use a list called is_added that keeps track of whether an element has been added to the collection. This array is of size k, where each index represents an element from 1 to k, and the value at each index represents whether the corresponding element has been added to the collection.

If we encounter an element, while traversing from the end, that is less than or equal to k and has not been added (is_added[element - 1] is False), we mark it as added and increase our count of unique elements. We continue this process until our count reaches k, which means we have all the elements from 1 to k.

The number of operations is then simply the total number of elements in nums minus the index of the last added element because all elements from the end of the array to this index have to be removed in order to collect all elements from 1 to k. Essentially, we're tracking how many removals we made from the end of the array to get all elements from 1 to k.

Solution Approach

The solution involves a single pass through the given array in reverse order, beginning from the last element and moving towards the first. This strategy is chosen because elements can only be removed from the end of the array. The language of choice for the implementation is Python.

Here's a step-by-step explanation of the solution with reference to the provided code snippet:

• An array is_added of size k is created to keep track of elements from 1 to k that have been added to our collection. Initially, all values in is_added are set to False, indicating that no elements have been collected yet.

• We define a count variable count, which keeps the count of unique elements that we have collected so far, starting the count from 0.

• We iterate through the array nums from the last element to the first, using a reverse loop. This is implemented by the loop for i in range(n - 1, -1, -1):, where n is the length of the nums array.

• For each element nums[i] encountered during the traversal:

  • We check if nums[i] is greater than k or if is_added[nums[i] - 1] is True (the element has already been added to the collection). If either condition is true, we continue to the next iteration without performing any operations since we either don't need the element or have already collected it.

  • If nums[i] is needed and has not been added to the collection yet, we set is_added[nums[i] - 1] to True and increment our count.

  • As soon as our count equals k, we know we have collected all elements from 1 up to k. The minimum number of operations required is then the total length of the array minus the current index i, which gives us n - i.

This approach efficiently ensures that we do not collect unnecessary elements and simultaneously avoid duplication in our collection. Once we've collected all required elements, we immediately return the result.

Using this algorithm, we leverage simple data structures such as an array (is_added) and a counter variable to reach an optimal solution with a time complexity of O(n), where n is the number of elements in nums.

Example Walkthrough

Let's consider a small example to illustrate the solution approach with the array nums = [5,4,2,3,1,2] and target k = 3.

• The is_added array will be initialized with values [False, False, False] which signifies that none of the numbers 1, 2, or 3 have been added to our collection yet.

• We iterate over nums starting from the last element, so our loop begins at nums[5] which is 2.

  • Since 2 is less than or equal to k and hasn't been added to the collection yet (is_added[2 - 1] is False), we add it by setting is_added[1] to True. Our collection becomes [False, True, False], indicating that 2 has been added.

• Moving to nums[4] which is 1:

  • Following our rules, we add 1 to the collection and is_added becomes [True, True, False].

• Next, we look at nums[3] which is 3:

• 3 is less than or equal to k and is not present in the collection (is_added[3 - 1] is False), so we add it. Our is_added array now becomes [True, True, True]. At this point, we have all elements from 1 to k in our collection, and we stop the iteration.

Since we have collected all the numbers from 1 to k after reaching the 3rd index from the right (inclusive), we have done a total of 6 (length of nums) - 3 (current index) = 3 operations to collect all necessary elements. Therefore, the fewest number of operations required in this example is 3.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is O(n), where n is the length of the input array nums. This is because there is a single for loop that iterates backwards over the array nums, and in each iteration, it performs a constant time check and assignment operation. Since these operations do not depend on the size of k and there are no nested loops, the iteration will occur n times, leading to a linear time complexity with respect to the size of the array.

Space Complexity

The space complexity of the given code is O(k). The is_added list is the only additional data structure whose size scales with the input parameter k. Since it is initialized to have k boolean values, the amount of memory used by this list is directly proportional to the value of k. The rest of the variables used within the function (like count, n, and i) use a constant amount of space, and therefore do not contribute to the space complexity beyond a constant factor.

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