Problem Description

In this problem, we are given an integer array nums. We need to find a particular type of subsequence known as a "square streak." A square streak has the following characteristics:

• It has a length of at least 2 elements.
• When sorted, each element of the subsequence, except for the first, is the square of the previous element.

Our goal is to figure out the length of the longest square streak in nums. If no square streak exists, we return -1.

A subsequence can be understood as a sequence that can be obtained from the original array by removing some elements (possibly, none) without changing the order of the remaining elements.

Intuition

The solution approach involves iterating through each element in the integer array nums and attempting to construct the longest square streak starting from that element. To achieve this, we use the following intuition and steps:

• First, we convert the list of numbers into a set s for faster lookups. Checking if an item is in a set is faster than checking if it is in a list because sets in Python are implemented as hash tables.

• For each number v in the array, we initialize a temporary counter t to zero. This counter will be used to track the length of the square streak starting with v.

• We then enter a while loop that continuously squares v and checks whether the new v is in the set s. If it is found, it means we can continue the square streak, and we increment the counter t by one.

• The loop stops when the squared number is no longer in the set, indicating the streak cannot be extended further.

• If the counter t is greater than 1, meaning we have found a square streak, we compare it with our current answer ans and update ans with the maximum value.

• Finally, after we have checked all numbers, we return the value of ans.

This brute-force approach works because it tries to extend a square streak from each number in the array. However, it may not be optimal for very large arrays, as the time complexity is proportional to the number of elements times the length of the longest streak.

Learn more about Binary Search, Dynamic Programming and Sorting patterns.

Solution Approach

The solution approach involves a few key steps, using a set for fast element lookup and a simple iteration pattern to build potential square streaks.

Here's a detailed walk-through of the implementation:

1. Create a Set for Fast Lookups: The algorithm begins by converting the list nums into a set s. This transformation is done to take advantage of the constant time complexity O(1) for checking if an element is present in the set.

2. Iterate Over the Array: For each element v in nums, we try to find the longest square streak starting with that number.

3. Initialize a Counter: For each v, we set a counter t to zero. This counter will track the length of the streak.

4. While Loop to Extend the Streak: We then enter a loop that continues as long as v is in the set s. Within this loop, we do two things:

   • Square v by setting v = v * v. This is based on the rule for the square streak where each following number must be the square of its predecessor.
   • Increment the counter t by 1 to denote that the streak has been extended by another element.

5. Check Streak Validity: After the loop ends, we check if t > 1 to ensure that the streak constitutes at least two elements (as single-element streaks are not valid).

6. Update Maximum Streak Length: If a valid streak is found, we compare t with the current maximum streak length ans, updating ans with max(ans, t) if necessary.

7. Return the Result: After iterating through all elements in nums, if any square streak was found, ans would be greater than -1. If no streak was found, ans would be -1, which is then returned as the result.

class Solution:
    def longestSquareStreak(self, nums: List[int]) -> int:
        s = set(nums)
        ans = -1
        for v in nums:
            t = 0
            while v in s:
                v *= v # Square the current number
                t += 1 # Increment the streak length
            if t > 1: # If the streak contains at least 2 elements
                ans = max(ans, t) # Update the maximum streak length
        return ans

By applying these steps, the solution manages to identify the longest square streak effectively without the need for sorting or additional complex data structures.

Example Walkthrough

Let's go through an example to illustrate the solution approach described above using the following array of integers:

nums = [1, 16, 4, 2, 64]

Following the algorithm steps:

1. Create a Set for Fast Lookups: We convert nums into a set s for fast element lookup. So, s = {1, 16, 4, 2, 64}.

2. Iterate Over the Array: We begin iterating with v = 1.

3. Initialize a Counter: We set a temporary counter t = 0. This will track the length of a potential square streak.

4. While Loop to Extend the Streak: We check if v = 1 is in s. It is, so we enter the loop.

   • We square v to get 1 * 1 = 1 and since 1 is still in the set s, we increment t to 1.

   Since squaring v = 1 yields the same number, we end up in an infinite loop. To mitigate this, our approach should check if the squared number is different from the current number before looping. Here, we would need to add a condition to the loop to continue only if the squared v is different from v.

   • Now, we should have v = 1, and t does not increase further as we are trapped in a loop where the value of v doesn't change.

   • To correct the algorithm and continue the example, we should stop incrementing t once we notice that v*v does not produce a new number.

5. Check Streak Validity: We observe that after correcting the loop, t = 1, which means no streak found, as the streak must be at least 2 numbers long.

6. Iterate Over the Next Array Element: v = 16

   • Initialize t = 0.
   • 16 is in s, square v to get 256. Since 256 is not in s, we stop the loop.
   • t remains 0 since 16 is not the beginning of a square streak within the set.

7. Repeat the Process:

   • For v = 4, we initialize t = 0. Squaring v yields 16, which is in s. Increment t to 1. Squaring again yields 256, which is not in s, so we stop here. t is 1, so no valid streak.

   • With v = 2, square it to get 4, which is in s. Increment t to 1. Square 4 to get 16, which is in s. Increment t to 2. Square 16 to get 256, which is not in s, break. t is now 2, so we have found a valid streak of length 2. We set ans to 2 as it's the maximum so far.

   • For v = 64, we square it to get 4096, which is not in s, so the streak does not start here.

8. Return the Result: After completing the iteration, the longest square streak found has a length of 2 (starting with v = 2), and hence we return ans = 2.

Please note that the provided Python code in the original problem description had an issue where it could lead to an infinite loop if v squared did not yield a different number. An additional loop condition is necessary to prevent that, and the corrected example assumes that this change has been made.

Solution Implementation

Time and Space Complexity

Time Complexity:

The given Python code iterates through each element in the input list nums with a for loop. For each element v, it enters a while loop to square v repeatedly until v is no longer present in the set s, incrementing t each time a square is encountered in s.

Let n be the number of elements in nums. The worst-case time complexity occurs when the squaring sequence for an element remains within the set s for a prolonged series of operations. However, the squaring operation rapidly increases the value of v, making it unlikely to have a long sequence unless the input values are specifically chosen to create a long chain (e.g., powers of 2). Nevertheless, in a pathological case, this repetition could occur O(m) times for each element, where m represents the maximum possible length of the squaring sequence within the set. However, m could be considered constant in this context, as the numbers tend to grow very large very quickly and thus exit the set.

The in operation within the while loop has an average-case time complexity of O(1) due to the set data structure.

Therefore, the total average-case time complexity is O(n), assuming m doesn't depend on n and is considered constant.

Space Complexity:

The space complexity of the code is dominated by the creation of the set s, which contains unique elements from the list nums. This set can have at most n elements if all elements in nums are unique.

Therefore, the space complexity is O(n).

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