1437. Check If All 1's Are at Least Length K Places Away

Problem Description

The problem provides us with a binary array called nums, which means it only contains 0s and 1s. We are also given an integer k. The task is to determine if all the 1s in the array are separated by at least k places from each other. Put differently, after any 1 in the array, the next 1 should not appear before we have seen at least k number of 0s. If this condition is met for all 1s in the array, we should return true. Otherwise, the function should return false.

For instance, if we have nums = [1,0,0,0,1] and k = 2, the function should return true since the two 1s are separated by three 0s which is at least k places apart. Conversely, if k were 3, the function should return false since the 1s are not separated by at least 3 places.

Intuition

The intuition behind the solution is to scan through the given array while keeping track of the position of the last 1 we encountered. As we iterate over the array, whenever we find a 1, we check if there was any previous 1. If there was, we calculate the distance between the current 1 and the last 1. This distance must be greater than or equal to k+1 places (since we start counting from 0), indicating at least k 0s are in between them. If this condition is not met, then we return false immediately, as we have found two 1s that are too close to each other. If no such pair of 1s violating our separation condition is found by the end of the array, we can safely return true.

To implement this, we initialize a variable j with a very small number (-inf, negative infinity) to represent the position of the last 1 we've seen. We use this extreme initial number to handle the case where the first 1 appears at the start of the array; since there is no other 1 before it, the algorithm should not incorrectly flag it as being too close to a previous 1. As we iterate through the array with index i, if we find a 1, we check if the distance i - j - 1 is less than k. If it is, it means that 1s are not k places apart, and we should return false. If the condition is not violated, we update j to the current index i. If we reach the end of the array without finding poorly spaced 1s, we return true.

Solution Approach

The solution uses a simple linear scan approach to traverse the array, which is a common algorithm pattern when we need to check each element in a sequence. There are no complex data structures used in this solution; it relies on a single integer j to keep track of the index of the previous 1. The choice of j being initialized to -inf is a deliberate one to ensure that the first 1 encountered does not falsely trigger our condition for being too close to another 1.

Following is a step-by-step explanation of the algorithm as implemented in the provided code:

We initialize j to -inf to represent the index of the last 1 seen. This is purposely a very small number to ensure that the first 1 in the array does not compare against a non-existent previous 1.
We iterate through the array using a for loop, utilizing enumerate to get both the index i and the value x at that index.
Within the loop, we check if the current value x is 1. If it's not, we do nothing and continue to the next iteration.
If x is 1, we enter our if-statement where the condition is i - j - 1 < k. This condition checks the distance between the current 1 and the last 1. Since distance is index-based and starts from 0, we subtract an additional 1 from i - j, effectively ensuring that there are at least k zeros in-between. If the condition is True, it means the 1s are too close, and we immediately return False.
If the condition in step 4 is not met, meaning the 1s are sufficiently spaced apart, we update j to be the current index i, marking it as the new position of the last 1.
After the loop, if we have not returned False, it means all 1s were appropriately spaced apart according to the given k value, and we can safely return True to indicate the array satisfies the condition.

The algorithm runs in O(n) time since it only needs to traverse the array once, where n is the number of elements in nums. The space complexity is O(1) as we only use a fixed number of variables regardless of the size of the input.

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Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we have the binary array nums and integer k given as follows:

nums = [1, 0, 0, 1, 0, 1]
k = 2

We need to determine if all 1s in this array are separated by at least k places. Here's a step-by-step walkthrough following the solution approach:

Initialize j to -inf. For the sake of the example, let's consider -inf to be -1 since array indices are zero-based. This is to handle the case where the first 1 is correctly positioned regardless of k.
Start iterating over nums. Our iteration will go through indices 0 to 5.
At i = 0, x is 1. Since this is the first 1, there is no previous 1 to compare against, so no action is necessary apart from updating j to 0.
At i = 1 and i = 2, x is 0. Nothing is done.
At i = 3, x is 1 again. We now check if i - j - 1 < k, which is 3 - 0 - 1 < 2. The actual comparison is 2 < 2, which is false. Since there are 2 zeroes between the 1s, they are at least k places apart, which is our requirement. So, we update j to 3.
At i = 4, x is 0. Nothing is done.
At i = 5, x is 1. We check i - j - 1 < k, which is 5 - 3 - 1 < 2. This simplifies to 1 < 2, which is true. The 1s are not at least k places apart because there is only 1 zero between the 1 at index 3 and the 1 at index 5.
Since the condition is true, the algorithm returns false as the 1s are too close to each other based on the k value provided.

Throughout this example, we can see how the algorithm determines the correct spacing between 1s. By the end of the array, we have successfully identified an instance where 1s were not separated by at least k places, determining the correct output of the function which, in this case, is false.

Solution Implementation

1from math import inf
2
3class Solution:
4    def kLengthApart(self, nums: List[int], k: int) -> bool:
5        # Initialize the index of the previous '1' found to negative infinity
6        # to handle the case where the first '1' appears at index 0.
7        previous_one_index = -inf
8      
9        # Iterate over the indices and values in the nums array.
10        for index, value in enumerate(nums):
11            # Check if the current value is '1'
12            if value == 1:
13                # If the gap between the current and previous '1' is 
14                # less than k, return False.
15                if index - previous_one_index - 1 < k:
16                    return False
17                # Update the position of the last found '1' to the current index.
18                previous_one_index = index
19      
20        # If all '1's are at least k positions apart, return True.
21        return True
22

1class Solution {
2
3    // Method to check if all '1's in the array are at least k length apart
4    public boolean kLengthApart(int[] nums, int k) {
5        // Initialize previous index of '1' found to a position that is 
6        // k positions before the start of the array
7        int lastOneIndex = -(k + 1);
8
9        // Iterate over all elements in the array
10        for (int currentIndex = 0; currentIndex < nums.length; ++currentIndex) {
11            // Check if we have found a '1'
12            if (nums[currentIndex] == 1) {
13                // If the distance between this '1' and the previous '1'
14                // is less than k, return false, as the requirement is not met
15                if (currentIndex - lastOneIndex - 1 < k) {
16                    return false;
17                }
18                // Update the index of the last found '1'
19                lastOneIndex = currentIndex;
20            }
21        }
22
23        // If we finish looping through the array without returning false,
24        // it means all '1's are at least k length apart, so return true
25        return true;
26    }
27}
28

1class Solution {
2public:
3    // Function to check if all 1's are at least 'k' distance apart
4    bool kLengthApart(vector<int>& nums, int k) {
5        // Initialize the variable to store the index of the last seen 1.
6        // We start with an index that is smaller by more than k. So, the first comparison is guaranteed to succeed.
7        int lastOneIndex = -(k + 1);
8      
9        // Iterate through the array of nums
10        for (int i = 0; i < nums.size(); ++i) {
11            // Check if we have found a 1
12            if (nums[i] == 1) {
13                // Check if the distance from the last seen 1 is less than k
14                if (i - lastOneIndex - 1 < k) {
15                    // If 'k' constraint is not satisfied, return false
16                    return false;
17                }
18                // Update the index of the last seen 1
19                lastOneIndex = i;
20            }
21        }
22        // If all 1's are at least 'k' distance apart, return true
23        return true;
24    }
25};
26

1// This function checks whether all 1's in the array are at least 'k' distance apart.
2// @param nums - array of numbers, consisting of 0's and 1's
3// @param k - minimum distance required between two 1's
4// @returns true if all 1's are 'k' or more apart, otherwise false
5function kLengthApart(nums: number[], k: number): boolean {
6    // Initialize the previous index of 1 to a value such that the first comparison succeeds
7    let previousOneIndex = -(k + 1);
8  
9    // Loop through the array to find the 1's and check their distances.
10    for (let currentIndex = 0; currentIndex < nums.length; ++currentIndex) {
11      
12        // Check if the current element is 1
13        if (nums[currentIndex] === 1) {
14          
15            // Check if the distance from the current 1 to the previous 1 is less than k.
16            // If it is, return false.
17            if (currentIndex - previousOneIndex - 1 < k) {
18                return false;
19            }
20          
21            // Update the index of the most recently found 1.
22            previousOneIndex = currentIndex;
23        }
24    }
25  
26    // Return true if no pairs of 1's are found that are less than 'k' distance apart.
27    return true;
28}
29

Time and Space Complexity

Time Complexity: The function iterates over each element in the nums list once. The number of iterations is therefore linearly proportional to the length of nums, denoted as N. There are no nested loops, and operations within the loop are constant time operations. Therefore, the time complexity of the code is O(N).
Space Complexity: The space complexity of the code is constant, O(1). This is because only a fixed number of variables (j, i, x) are used, regardless of the input size. The space used by the input nums and the integer k are not counted towards the space complexity since they are part of the input.

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