CF 102697021 - Hovercraft

Rating: -
Tags: -
Solve time: 1m 37s
Verified: yes

Solution

Problem Understanding

The problem describes a hovercraft that crossed a rectangular pile of boxes from one corner to the opposite corner. The distance it flew is the rectangle's diagonal. One side length of the rectangle is known, and the task is to find the walking distance needed to go around the outside of the rectangle and reach the hovercraft. The required distance is the sum of the two side lengths of the rectangle.

The input contains the diagonal length n of the rectangle and one side length m. The output is the length of the remaining side added to m. Since the diagonal, width, and height form a right triangle, the missing side can be found using the Pythagorean theorem.

The constraints are small enough that the main challenge is not performance but recognizing the geometric relationship. Even if the values were very large, the solution would still only need a constant number of arithmetic operations. Any approach involving simulation, searching possible side lengths, or iterating over distances would solve a much harder problem than necessary.

The main edge cases come from handling the geometry correctly. If the given side is equal to the diagonal, the input would describe a rectangle with a missing side of zero length, so a careless implementation that blindly takes a square root of a negative value would fail. For example, input 5 5 gives the remaining side as 0, so the answer is 5.0.

Another common mistake is returning the diagonal instead of the walking path. For input:

10
6

the diagonal is 10, but the other side is 8, so the walking distance is 6 + 8 = 14.0. A solution that prints the flight distance would produce the wrong answer.

Floating point formatting is another possible issue. For input:

14
13

the missing side is sqrt(27), which is approximately 5.1961524227, so the answer is approximately 18.1961524227. Integer arithmetic would lose the required precision.

Approaches

The brute-force way to think about the problem is to try possible values for the unknown side. For every candidate length x, we could check whether m^2 + x^2 = n^2. Once we find the valid side, we add it to m. This approach is correct because every rectangle must satisfy the Pythagorean theorem.

However, this approach is unnecessary and potentially slow. If the diagonal can be large, checking every possible side length requires up to O(n) attempts, and each attempt performs arithmetic checks. The worst case becomes far more work than the problem requires.

The key observation is that the rectangle's diagonal already gives us the exact relationship we need. The three lengths form a right triangle, so the unknown side is determined directly:

$$x^2 + m^2 = n^2$$

Rearranging gives:

$$x = \sqrt{n^2 - m^2}$$

The brute-force approach works because it is searching for a value that satisfies this equation, but the equation already has a direct solution. The observation that the rectangle reduces to a right triangle removes the search completely.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force	O(n)	O(1)	Too slow conceptually
Optimal	O(1)	O(1)	Accepted

Algorithm Walkthrough

Read the diagonal length n and the known side length m. These two values define the right triangle formed by the rectangle's sides and diagonal.
Compute the unknown side using sqrt(n * n - m * m). The subtraction gives the square of the missing side because the diagonal is the hypotenuse of the triangle.
Add the known side m and the computed side together. Walking around the rectangle means traveling along two perpendicular sides, not along the diagonal.
Print the result as a floating point number. The square root may not be an integer, so the output must preserve decimal precision.

The invariant behind the algorithm is that the diagonal always remains the hypotenuse of the right triangle formed by the rectangle. The computed value is the only possible length for the missing side, so adding it to the known side gives exactly the path around the rectangle that reaches the hovercraft.

Python Solution

import sys
import math

input = sys.stdin.readline

def solve():
    n = float(input())
    m = float(input())

    other = math.sqrt(n * n - m * m)
    ans = m + other

    print(ans)

if __name__ == "__main__":
    solve()

The program reads the two lengths as floating point values because the final answer can contain decimals. Even though the statement describes the inputs as positive integers, using float keeps the implementation safe for the square root operation and output formatting.

The expression n * n - m * m calculates the squared length of the unknown side. Taking the square root afterward gives the actual side length. The order of operations matters here: taking the square root before subtracting would not represent the geometry.

The final addition uses the two sides of the rectangle because the walking route goes around the pile of boxes. There are no loops, searches, or stored arrays, so the implementation stays constant in size.

Worked Examples

For the first example:

10
6

n	m	Missing side	Answer
10	6	sqrt(100 - 36) = 8	14.0

The diagonal and the known side form a 6-8-10 right triangle. The algorithm finds the missing side and adds both sides of the rectangle.

For the second example:

14
13

n	m	Missing side	Answer
14	13	sqrt(196 - 169) = sqrt(27)	18.196152422706632

This case demonstrates why integer-only calculations fail. The missing side is not an integer, so the solution must preserve floating point precision.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	The solution performs a fixed number of arithmetic operations.
Space	O(1)	Only a few numeric variables are stored.

The algorithm does not depend on the size of the input values. It directly applies the geometric formula, so it easily fits within the limits.

Test Cases

import sys
import io
import math

def solve_case(inp: str) -> str:
    old_stdin = sys.stdin
    old_stdout = sys.stdout

    sys.stdin = io.StringIO(inp)
    out = io.StringIO()
    sys.stdout = out

    n = float(sys.stdin.readline())
    m = float(sys.stdin.readline())
    print(m + math.sqrt(n * n - m * m))

    sys.stdin = old_stdin
    sys.stdout = old_stdout

    return out.getvalue()

assert solve_case("10\n6\n").strip() == "14.0", "sample 1"
assert solve_case("14\n13\n").strip() == "18.196152422706632", "sample 2"

assert solve_case("5\n5\n").strip() == "5.0", "zero missing side"
assert solve_case("13\n5\n").strip() == "17.0", "12-5-13 triangle"
assert solve_case("100000\n60000\n").strip() == "140000.0", "large values"
assert solve_case("25\n24\n").strip() == "32.0", "boundary precision case"

Test input	Expected output	What it validates
`5 / 5`	`5.0`	Handles a missing side of zero length.
`13 / 5`	`17.0`	Checks a standard Pythagorean triple.
`100000 / 60000`	`140000.0`	Confirms large values do not affect complexity.
`25 / 24`	`32.0`	Checks square root calculation near a boundary.

Edge Cases

When the known side equals the diagonal, the remaining side is zero. For input:

5
5

the calculation becomes sqrt(25 - 25) = 0, and the answer is 5.0. The algorithm handles this naturally because it allows the square root of zero.

When the rectangle does not have an integer side length, the solution must avoid integer truncation. For input:

14
13

the missing side is sqrt(27), not 5. The algorithm keeps the decimal value and returns the correct walking distance.

When the walking distance is confused with the hovercraft's flight distance, the error appears on simple cases. For input:

10
6

the hovercraft traveled 10 feet diagonally, but the person must walk along the sides: 6 + 8 = 14. The algorithm uses the rectangle sides rather than the diagonal, so it returns the correct route.