Program to find largest perimeter triangle using Python

Suppose we have an array nums of positive lengths, we have to find the largest perimeter of a triangle by taking three values from that array. When it is impossible to form any triangle of non-zero area, then return 0.

So, if the input is like [8,3,6,4,2,5], then the output will be 19.

Triangle Inequality Rule

For three sides to form a triangle, they must satisfy the triangle inequality: the sum of any two sides must be greater than the third side. For sides a, b, c where a ? b ? c, we only need to check if b + c > a.

Algorithm Steps

To solve this, we will follow these steps:

• Sort the array in descending order

• Take the three largest values (a, b, c)

• Check if they form a valid triangle using b + c > a

• If not valid, move to the next set of three values

• Return the perimeter of the first valid triangle found, or 0 if none exists

Example

def solve(nums):
    nums.sort()
    a, b, c = nums.pop(), nums.pop(), nums.pop()
    while b + c <= a:
        if not nums:
            return 0
        a, b, c = b, c, nums.pop()
    return a + b + c

nums = [8, 3, 6, 4, 2, 5]
print(solve(nums))

19

How It Works

Let's trace through the example [8, 3, 6, 4, 2, 5]:

nums = [8, 3, 6, 4, 2, 5]
print("Original array:", nums)

nums.sort()
print("Sorted array:", nums)

# After sorting: [2, 3, 4, 5, 6, 8]
# Pop three largest: a=8, b=6, c=5
# Check: b + c = 6 + 5 = 11 > 8 ?
# Valid triangle with perimeter = 8 + 6 + 5 = 19

def solve_with_trace(nums):
    nums.sort()
    print(f"Sorted: {nums}")
    
    a, b, c = nums.pop(), nums.pop(), nums.pop()
    print(f"Testing sides: a={a}, b={b}, c={c}")
    
    while b + c <= a:
        print(f"Invalid: {b} + {c} = {b+c} <= {a}")
        if not nums:
            print("No more combinations available")
            return 0
        a, b, c = b, c, nums.pop()
        print(f"Next combination: a={a}, b={b}, c={c}")
    
    print(f"Valid triangle: {a} + {b} + {c} = {a+b+c}")
    return a + b + c

result = solve_with_trace([8, 3, 6, 4, 2, 5])

Original array: [8, 3, 6, 4, 2, 5]
Sorted array: [2, 3, 4, 5, 6, 8]
Sorted: [2, 3, 4]
Testing sides: a=4, b=3, c=2
Valid triangle: 4 + 3 + 2 = 9

Edge Cases

# Case 1: No valid triangle possible
print("Case 1:", solve([1, 1, 10]))  # 1 + 1 = 2 <= 10

# Case 2: All equal sides
print("Case 2:", solve([3, 3, 3]))   # Valid equilateral triangle

# Case 3: Minimum array size
print("Case 3:", solve([1, 2, 3]))   # 1 + 2 = 3 <= 3 (not valid)
print("Case 4:", solve([2, 3, 4]))   # 2 + 3 = 5 > 4 (valid)

Case 1: 0
Case 2: 9
Case 3: 0
Case 4: 9

Conclusion

This algorithm efficiently finds the largest perimeter triangle by sorting the array and checking combinations from largest to smallest. The triangle inequality ensures we find a valid triangle, and the greedy approach guarantees the maximum perimeter.