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Maximum Height of a Triangle

Easy

Asked by:

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Topics:

Binary SearchGreedy Algorithms

You are given two integers red and blue representing the count of red and blue colored balls. You have to arrange these balls to form a triangle such that the 1^st row will have 1 ball, the 2^nd row will have 2 balls, the 3^rd row will have 3 balls, and so on.

All the balls in a particular row should be the same color, and adjacent rows should have different colors.

Return the maximum height of the triangle that can be achieved.

Example 1:

Input: red = 2, blue = 4

Output: 3

Explanation:

The only possible arrangement is shown above.

Example 2:

Input: red = 2, blue = 1

Output: 2

Explanation:

The only possible arrangement is shown above.

Example 3:

Input: red = 1, blue = 1

Output: 1

Example 4:

Input: red = 10, blue = 1

Output: 2

Explanation:

The only possible arrangement is shown above.

Constraints:

1 <= red, blue <= 100

Try coding it in LeetCode

Maximum Height of a Triangle

Easy

Asked by:

36 views

Topics:

Binary SearchGreedy Algorithms

All the balls in a particular row should be the same color, and adjacent rows should have different colors.

Return the maximum height of the triangle that can be achieved.

Example 1:

Input: red = 2, blue = 4

Output: 3

Explanation:

The only possible arrangement is shown above.

Example 2:

Input: red = 2, blue = 1

Output: 2

Explanation:

The only possible arrangement is shown above.

Example 3:

Input: red = 1, blue = 1

Output: 1

Example 4:

Input: red = 10, blue = 1

Output: 2

Explanation:

The only possible arrangement is shown above.

Constraints:

1 <= red, blue <= 100

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the `area` and `base` inputs? Can they be zero or negative?
Is the area guaranteed to be achievable with an integer height given the base?
If there is no possible integer height that satisfies the area requirement (i.e., the calculated height is less than 1), what should I return?
Should I round the height up or down to the nearest integer to ensure the area is *at least* the given area?
Are `area` and `base` provided as integers, or might they be floating-point numbers?

Brute Force Solution

Approach

The brute force approach to finding the maximum height of a triangle, given the side lengths and area, involves trying out different possible heights. We systematically check heights until we find the largest one that produces a valid triangle with the given area.

Here's how the algorithm would work step-by-step:

Start with a very small possible height.
Calculate the base length of the triangle using the given area and our chosen height.
Check if it's possible to form a triangle with the three sides (given in the question) and the calculated base length. We can do this using the triangle inequality rule: the sum of any two sides must be greater than the third side.
If it's a valid triangle, remember the current height.
If it's not a valid triangle, discard the current height.
Now, increase the height slightly and repeat the process from step two.
Keep doing this, each time checking if the calculated triangle is valid and updating the remembered height if it is.
Continue until we've tried a height that is clearly too large to form a valid triangle (e.g., a height larger than any of the side lengths).
The largest height we remembered during this process is the maximum possible height of the triangle.

Code Implementation

def maximum_height_of_triangle(sticks, area_limit):
    max_height = 0
    sticks_length = len(sticks)

    for i in range(sticks_length):
        for j in range(i + 1, sticks_length):
            for k in range(j + 1, sticks_length):
                side_a = sticks[i]
                side_b = sticks[j]
                side_c = sticks[k]

                # Check if the sides form a valid triangle
                if (side_a + side_b > side_c) and \
                   (side_a + side_c > side_b) and \
                   (side_b + side_c > side_a):

                    # Calculate the semi-perimeter
                    semi_perimeter = (side_a + side_b + side_c) / 2

                    # Calculate the area using Heron's formula
                    area = (semi_perimeter * (semi_perimeter - side_a) * \
                            (semi_perimeter - side_b) * (semi_perimeter - side_c))**0.5

                    # Proceed only if area is within limit
                    if area <= area_limit:

                        # Calculate height using side_a as base
                        height_a = (2 * area) / side_a
                        max_height = max(max_height, height_a)

                        height_b = (2 * area) / side_b
                        max_height = max(max_height, height_b)

                        height_c = (2 * area) / side_c
                        max_height = max(max_height, height_c)

    return max_height

Big(O) Analysis

Time Complexity

O(1) – The described algorithm iterates from a small height until a maximum height is reached that satisfies triangle inequality, incrementing the height in each step. The number of iterations depends on the precision required for the height and not on the size of any input array or list. The side lengths and area are fixed and given as direct inputs, therefore, the number of iterations can be seen as a constant or bounded by a constant, because the side lengths are constrained. Therefore, the time complexity is O(1).

Space Complexity

O(1) – The described algorithm uses a few variables to store the current height, calculated base, and possibly the maximum valid height found so far. The number of these variables remains constant irrespective of the side lengths or area of the triangle provided as input. Thus, the auxiliary space used by the algorithm is constant, resulting in O(1) space complexity.

Optimal Solution

Approach

The optimal strategy focuses on maximizing the base of the triangle formed by the given lengths. By strategically arranging the lengths to form the widest possible base, we implicitly maximize the height. This avoids complex calculations and provides an efficient solution.

Here's how the algorithm would work step-by-step:

Sort the three given lengths from shortest to longest.
Recognize that the two shorter lengths must add up to be greater than the longest length in order to form a valid triangle.
If the sum of the two shortest lengths is not greater than the longest, it's impossible to form a triangle. Return a height of zero.
If a valid triangle can be formed, note the sides as a, b, and c, and calculate 's', which is half of the triangle's perimeter (a + b + c) / 2.
Use Heron's formula to find the area of the triangle: area = square root of (s * (s-a) * (s-b) * (s-c)).
To maximize height, consider the longest side 'c' as the base. The area of a triangle is also equal to (1/2) * base * height.
Calculate the height using the formula: height = (2 * area) / base, where base is the longest side 'c'.

Code Implementation

def maximum_height_of_triangle(triangle_area, triangle_base):
    # Height equals two times area divided by base.
    height = (2 * triangle_area) / triangle_base

    # Prevents returning negative or zero height, invalid cases.
    if triangle_area <= 0 or triangle_base <= 0:
        return 0

    # Return the calculated maximum possible height.
    return height

Big(O) Analysis

Time Complexity

O(1) – The algorithm sorts an array of fixed size 3, which takes constant time. It then performs a fixed number of arithmetic operations regardless of the input values. Heron's formula and height calculation involve a constant number of mathematical operations. Therefore, the time complexity is constant, O(1).

Space Complexity

O(1) – The algorithm sorts the three given lengths, which can be done in place with constant extra space or potentially using variables for swapping if not done in place which is still constant. It also uses a few variables (a, b, c, s, area, height) to store intermediate calculations. The number of variables used does not depend on the input size (N=3 in this case, the number of side lengths), therefore the auxiliary space complexity is constant.

Edge Cases

Case	How to Handle
Area is zero	Return 0 as the maximum height, as any positive height will result in a positive area.
Base is zero	Return 0, as any non-zero height would result in a zero area which does not meet the given condition.
Area is negative	Return 0, as the area of a triangle cannot be negative, implying an invalid input.
Base is negative	Return 0, as the base of a triangle cannot be negative, implying an invalid input.
Area is a very large number	Ensure the calculation of height (2 * area / base) doesn't cause integer overflow; using long or double could be necessary.
Base is a very large number	Ensure that when dividing a large number by the base it is handled accurately, especially if integer division truncates.
Area is a floating point number but is cast as int	Check for significant data loss upon conversion from float to int by rounding or casting.
Base is a floating point number but is cast as int	Check for significant data loss upon conversion from float to int by rounding or casting.

Area is zero

How to Handle:

Return 0 as the maximum height, as any positive height will result in a positive area.

Base is zero

How to Handle:

Return 0, as any non-zero height would result in a zero area which does not meet the given condition.

Area is negative

How to Handle:

Return 0, as the area of a triangle cannot be negative, implying an invalid input.

Base is negative

How to Handle:

Return 0, as the base of a triangle cannot be negative, implying an invalid input.

Area is a very large number

How to Handle:

Ensure the calculation of height (2 * area / base) doesn't cause integer overflow; using long or double could be necessary.

Base is a very large number

How to Handle:

Ensure that when dividing a large number by the base it is handled accurately, especially if integer division truncates.

Area is a floating point number but is cast as int

How to Handle:

Check for significant data loss upon conversion from float to int by rounding or casting.

Base is a floating point number but is cast as int

How to Handle:

Check for significant data loss upon conversion from float to int by rounding or casting.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the possible ranges for the `area` and `base` inputs? Can they be zero or negative?
Is the area guaranteed to be achievable with an integer height given the base?
If there is no possible integer height that satisfies the area requirement (i.e., the calculated height is less than 1), what should I return?
Should I round the height up or down to the nearest integer to ensure the area is *at least* the given area?
Are `area` and `base` provided as integers, or might they be floating-point numbers?

Brute Force Solution

Approach

Here's how the algorithm would work step-by-step:

Start with a very small possible height.
Calculate the base length of the triangle using the given area and our chosen height.
Check if it's possible to form a triangle with the three sides (given in the question) and the calculated base length. We can do this using the triangle inequality rule: the sum of any two sides must be greater than the third side.
If it's a valid triangle, remember the current height.
If it's not a valid triangle, discard the current height.
Now, increase the height slightly and repeat the process from step two.
Keep doing this, each time checking if the calculated triangle is valid and updating the remembered height if it is.
Continue until we've tried a height that is clearly too large to form a valid triangle (e.g., a height larger than any of the side lengths).
The largest height we remembered during this process is the maximum possible height of the triangle.

Code Implementation

def maximum_height_of_triangle(sticks, area_limit):
    max_height = 0
    sticks_length = len(sticks)

    for i in range(sticks_length):
        for j in range(i + 1, sticks_length):
            for k in range(j + 1, sticks_length):
                side_a = sticks[i]
                side_b = sticks[j]
                side_c = sticks[k]

                # Check if the sides form a valid triangle
                if (side_a + side_b > side_c) and \
                   (side_a + side_c > side_b) and \
                   (side_b + side_c > side_a):

                    # Calculate the semi-perimeter
                    semi_perimeter = (side_a + side_b + side_c) / 2

                    # Calculate the area using Heron's formula
                    area = (semi_perimeter * (semi_perimeter - side_a) * \
                            (semi_perimeter - side_b) * (semi_perimeter - side_c))**0.5

                    # Proceed only if area is within limit
                    if area <= area_limit:

                        # Calculate height using side_a as base
                        height_a = (2 * area) / side_a
                        max_height = max(max_height, height_a)

                        height_b = (2 * area) / side_b
                        max_height = max(max_height, height_b)

                        height_c = (2 * area) / side_c
                        max_height = max(max_height, height_c)

    return max_height

Big(O) Analysis

Time Complexity

Space Complexity

Optimal Solution

Approach

Here's how the algorithm would work step-by-step:

Sort the three given lengths from shortest to longest.
Recognize that the two shorter lengths must add up to be greater than the longest length in order to form a valid triangle.
If the sum of the two shortest lengths is not greater than the longest, it's impossible to form a triangle. Return a height of zero.
If a valid triangle can be formed, note the sides as a, b, and c, and calculate 's', which is half of the triangle's perimeter (a + b + c) / 2.
Use Heron's formula to find the area of the triangle: area = square root of (s * (s-a) * (s-b) * (s-c)).
To maximize height, consider the longest side 'c' as the base. The area of a triangle is also equal to (1/2) * base * height.
Calculate the height using the formula: height = (2 * area) / base, where base is the longest side 'c'.

Code Implementation

def maximum_height_of_triangle(triangle_area, triangle_base):
    # Height equals two times area divided by base.
    height = (2 * triangle_area) / triangle_base

    # Prevents returning negative or zero height, invalid cases.
    if triangle_area <= 0 or triangle_base <= 0:
        return 0

    # Return the calculated maximum possible height.
    return height

Big(O) Analysis

Time Complexity

Space Complexity

Edge Cases

Case	How to Handle
Area is zero	Return 0 as the maximum height, as any positive height will result in a positive area.
Base is zero	Return 0, as any non-zero height would result in a zero area which does not meet the given condition.
Area is negative	Return 0, as the area of a triangle cannot be negative, implying an invalid input.
Base is negative	Return 0, as the base of a triangle cannot be negative, implying an invalid input.
Area is a very large number	Ensure the calculation of height (2 * area / base) doesn't cause integer overflow; using long or double could be necessary.
Base is a very large number	Ensure that when dividing a large number by the base it is handled accurately, especially if integer division truncates.
Area is a floating point number but is cast as int	Check for significant data loss upon conversion from float to int by rounding or casting.
Base is a floating point number but is cast as int	Check for significant data loss upon conversion from float to int by rounding or casting.

Area is zero

How to Handle:

Return 0 as the maximum height, as any positive height will result in a positive area.

Base is zero

How to Handle:

Return 0, as any non-zero height would result in a zero area which does not meet the given condition.

Area is negative

How to Handle:

Return 0, as the area of a triangle cannot be negative, implying an invalid input.

Base is negative

How to Handle:

Return 0, as the base of a triangle cannot be negative, implying an invalid input.

Area is a very large number

How to Handle:

Ensure the calculation of height (2 * area / base) doesn't cause integer overflow; using long or double could be necessary.

Base is a very large number

How to Handle:

Ensure that when dividing a large number by the base it is handled accurately, especially if integer division truncates.

Area is a floating point number but is cast as int

How to Handle:

Check for significant data loss upon conversion from float to int by rounding or casting.

Base is a floating point number but is cast as int

How to Handle:

Check for significant data loss upon conversion from float to int by rounding or casting.