367. Valid Perfect Square

Problem Description

The problem is straightforward: given a positive integer num, our task is to determine whether it is a perfect square without using any built-in library functions, such as sqrt. A perfect square is a number that can be expressed as the product of an integer with itself. For example, the number 16 is a perfect square because it can be expressed as 4 x 4.

Intuition

To solve this problem, we can use two different methods - Binary Search and the Math Trick.

Method 1: Binary Search

The Binary Search approach involves setting two pointers left and right, where left starts at 1 (the smallest perfect square) and right starts at num (as the largest possible perfect square our input could be). We iteratively narrow down the search range by finding the midpoint of left and right and squaring it. If the square of this midpoint is larger than or equal to num, we know our perfect square root, if it exists, is at or before this midpoint, so we move the right pointer to the midpoint. Otherwise, we move left up to mid + 1. The moment left and right converge, we check if the square of left is equal to num to conclude whether num is a perfect square.

Method 2: Math Trick

This method uses the observation that every perfect square is the sum of a sequence of odd numbers starting from 1. We keep adding sequentially larger odd numbers to a sum. This sum starts at 0, and we increase the odd number to add by 2 each time. Whenever the sum equals the number num, we confirm that num is a perfect square. The underlying math of this trick is that the sum of the first n odd numbers is n^2, which is exactly the definition of a perfect square.

Both methods have an upper time complexity of O(log n), however, the math trick can sometimes conclude that a number isn't a perfect square more quickly since not all numbers are perfect squares and the sum can exceed num before we've added n terms.

Learn more about Math and Binary Search patterns.

Solution Approach

The solution implements two approaches to determine if a number is a perfect square without using the built-in sqrt function.

Method 1: Binary Search Approach

In the binary search approach, we use the concept of a binary search algorithm to efficiently find the target perfect square, if it exists. We start by initializing two pointers: left at 1 (since 1 is the smallest square) and right at num (since a number cannot be a perfect square of any number larger than itself).

Here’s the step-by-step binary search algorithm applied to this problem:

While left is less than right, perform the following steps to narrow down the search space:
- Calculate the midpoint mid by averaging left and right (using bitwise shifting >> 1 to divide by 2 for efficiency).
- Multiply mid by itself to check if it gives num.
- If mid * mid is greater than or equal to num, we set right to mid. This is because if mid squared is larger than or equal to num, the number we're looking for, if it exists, cannot be greater than mid.
- If mid * mid is less than num, we set left to mid + 1. This is because the number we’re looking for must be larger than mid.
After the loop, we check if left * left equals num to determine if num is indeed a perfect square.

The binary search approach ensures that we can quickly zone in on the potential candidate for the square root of the number and confirm if it's a perfect square in O(log(n)) time complexity.

Method 2: Math Trick

The math trick approach makes use of a mathematical pattern where every perfect square can be represented as a sum of odd numbers sequentially. For instance:

1 = 1
4 = 1 + 3
9 = 1 + 3 + 5
...

This pattern can be continued indefinitely, and each time the resulting sum will be a perfect square.

The algorithm for this approach is as follows:

Initialize a variable sum as 0 to keep track of the sum of odd numbers, and another variable i representing the current odd number to add to the sum, starting at 1.
While sum is less than num, add i to sum and check if sum equals num:
- If sum becomes equal to num, then num is a perfect square and we return true.
- If sum is still less than num, increment i by 2 to get to the next odd number.
If the loop ends without returning true, then num isn't a perfect square, return false.

This method runs in O(sqrt(n)) time complexity since in the worst case, it adds up the sequence of odd numbers up to the square root of the number.

Both of these approaches efficiently determine whether a number is a perfect square without using any built-in square root function, providing a reliable way to solve the problem with a guaranteed logarithmic or square root time complexity.

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Example Walkthrough

Let's illustrate the solution using the number num = 16 to determine whether it is a perfect square through both methods.

Binary Search Approach Example with num = 16:

Set left to 1 and right to num (16).
While left < right:
- Calculate mid which is (left + right) / 2, so initially (1 + 16) / 2 = 8.5, we take the integer part and consider mid = 8.
- Now, check mid * mid which equals 8 * 8 = 64. This is greater than num (16), so set right = mid to 8.
Update the while loop, and calculate the new mid as (1 + 8) / 2 = 4.5, consider mid = 4.
- Check mid * mid which is 4 * 4 = 16. This equals num (16), break the loop and confirm that num is a perfect square.

In a real run, the loop would continue until left and right converge to the point where left == right. However, in this case, we've found that 16 is a perfect square of 4 during the process. The other steps are not performed because we already found a match.

Math Trick Approach Example with num = 16:

Initially, sum = 0 and i = 1 (the first odd number).
Add i to sum, sum becomes 1. Then increment i by 2 to get 3.
Now, sum is 1 and i is 3. Add i to sum, sum becomes 1 + 3 = 4. Increment i by 2 to get 5.
Continuing, add the new i to sum to get 4 + 5 = 9. Increment i by 2 to get 7.
Add 7 to sum to get 9 + 7 = 16, which equals num.
Since the sum now equals num, we can assert that num is indeed a perfect square.

Through both methods, we have confirmed that 16 is a perfect square. The binary search approached the conclusion more directly by halving the possible range, while the math trick added sequential odd numbers until the sum matched the input (num).

Solution Implementation

1class Solution:
2    def isPerfectSquare(self, num: int) -> bool:
3        # Initialize the binary search boundaries.
4        left, right = 1, num
5
6        # Use binary search to find the potential square root of the number.
7        while left < right:
8            # Calculate the middle point of the current search boundary.
9            mid = (left + right) // 2
10          
11            # If the square of mid is greater than or equal to num, we narrow the search space
12            # to the left half including mid.
13            if mid * mid >= num:
14                right = mid
15            # Otherwise, we narrow the search space to the right half excluding mid.
16            else:
17                left = mid + 1
18      
19        # After the loop, left will be equal to right and should be the smallest number
20        # whose square is greater than or equal to num.
21        # Check if it's a perfect square of num.
22        return left * left == num
23

1class Solution {
2    // Method to check if a given number is a perfect square
3    public boolean isPerfectSquare(int num) {
4        long left = 1;              // Set the lower bound of the search range
5        long right = num;           // Set the upper bound of the search range
6
7        // Binary search to find the square root of num
8        while (left < right) {
9            // Calculate the midpoint to avoid overflow
10            long mid = (left + right) >>> 1;
11
12            // If mid squared is greater than or equal to num, it could be the root
13            if (mid * mid >= num) {
14                right = mid;        // Adjust the upper bound for the next iteration
15            } else {
16                left = mid + 1;     // Adjust the lower bound if mid squared is less than num
17            }
18        }
19
20        // Check if the final left value squared equals the original number to confirm if it's a perfect square
21        return left * left == num;
22    }
23}
24

1class Solution {
2public:
3    // Function to check if a given number is a perfect square
4    bool isPerfectSquare(int num) {
5        long left = 1;            // Initializing the lower boundary of the search space
6        long right = num;         // Initializing the upper boundary of the search space
7
8        // Using binary search to find the square root of the number
9        while (left < right) {    
10            long mid = left + (right - left) / 2; // Calculating the mid-value to prevent overflow
11            // If mid squared is greater than or equal to num, we narrow down the upper boundary
12            if (mid * mid >= num) {
13                right = mid;
14            } else {
15                // If mid squared is less than num, we narrow down the lower boundary
16                left = mid + 1;
17            }
18        }
19
20        // Once left and right converge, we verify if the number is indeed a perfect square
21        return left * left == num;
22    }
23};
24

1/**
2 * Checks whether a given number is a perfect square or not.
3 * 
4 * @param {number} num - The number to check.
5 * @returns {boolean} - True if num is a perfect square, false otherwise.
6 */
7function isPerfectSquare(num: number): boolean {
8    // Initialize the search range
9    let left: number = 1;
10    let right: number = num >> 1; // Equivalent to Math.floor(num / 2)
11
12    // Perform binary search to find the square root of num
13    while (left < right) {
14        // Calculate the midpoint of the current search range, using bitwise shift for division by 2
15        const mid: number = (left + right) >>> 1;
16
17        // Compare the square of the mid value with num
18        if (mid * mid < num) {
19            left = mid + 1; // If mid^2 is less than num, narrow the range to the upper half
20        } else {
21            right = mid; // If mid^2 is greater or equal to num, narrow the range to the lower half, including mid
22        }
23    }
24
25    // After the loop, left should be the integer part of the square root if it exists.
26    // Check if the square of 'left' is exactly num to conclude if num is a perfect square.
27    return left * left === num;
28}
29

Time and Space Complexity

The time complexity of the given binary search algorithm is O(log n), where n is the value of the input num. This is because the algorithm effectively halves the search space with each iteration by updating either the left or right variable to the mid value.

The space complexity of the algorithm is O(1) since it uses a fixed amount of extra space - variables left, right, mid, and the space needed for a few calculations do not depend on the size of the input num.

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

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