Maximizing Force on Bridge

Question

Le Juste Pont – Qualification 2025
Joseph is two steps away from the fortress! He only needs to cross a bridge to reach the entrance. Unfortunately, some parts of the bridge are threatening to collapse due to its age. Joseph's spacesuit is very heavy, and this fatigues him. He wants to avoid getting exhausted to arrive at the fortress in good condition.

The bridge is N meters long and alternates between safe and fragile zones. The bridge is described by an array of 0s and 1s, one digit per meter. A 0 describes a position in a safe zone, and a 1 describes a position in a fragile zone. A safe zone is represented by one or more consecutive 0s, and a fragile zone by one or more consecutive 1s.

During his traversal, at each meter, Joseph Nageant can either move by walking or by swimming. Joseph Nageant must traverse the tunnel by swimming in all fragile zones to avoid causing the bridge to collapse. In safe zones, he can either pass by swimming or by walking.

Unlike a previous exercise, Joseph is not obliged to walk in every safe zone. He can choose to traverse a safe zone entirely by swimming.

However, Joseph has a certain endurance E ≥ 0 which allows him to swim for a maximum of E meters before having to start walking again. After swimming, he is obliged to walk for at least R meters. (Joseph cannot start a walking phase during the last R - 1 meters of the bridge. Furthermore, if Joseph starts to walk, he must walk for at least R meters before swimming again.)

The force of Joseph is the value of R - E. Help Joseph Nageant find a movement strategy that maximizes his force.

Input:

The input contains:

On the first line, an integer N, the length in meters of the bridge.

On the next line, a list of N integers separated by spaces, pont, the array representing the bridge. A 0 corresponds to a position in a safe zone, and a 1 corresponds to a position in a fragile zone.

Output:

Display the maximum force value that Joseph can obtain while crossing the bridge.

Constraints:

1 ≤ N ≤ 1000

pont<sub>i</sub> ∈ {0, 1}

There is always at least one safe zone.

Performance Constraints:

1 ≤ N ≤ 500 000

Execution Constraints:

Maximum memory usage: 100000 kilobytes

Maximum execution time: 500 milliseconds

Answer 1

To solve the problem of maximizing Joseph Nageant's force $F = R - E$ while crossing the bridge, we need to analyze the bridge's structure and determine the optimal strategy based on the constraints provided.

Understanding the Problem

Bridge Representation:
- The bridge is represented as a sequence of 0s and 1s.
- 0: Safe zone where Joseph can choose to walk or swim.
- 1: Fragile zone where Joseph must swim to avoid collapsing the bridge.
Constraints:
- Endurance $E$ : Maximum number of consecutive meters Joseph can swim before he must start walking.
- Recovery $R$ : Minimum number of meters Joseph must walk after any swimming segment.
- After any swimming segment, Joseph must walk at least $R$ meters.
- Joseph cannot start a walking phase during the last $R - 1$ meters of the bridge.
Objective:
- Maximize the force $F = R - E$ .

Key Observations

Minimum Endurance $E$ :
- Joseph must swim through all fragile zones. Therefore, $E$ must be at least the length of the longest consecutive 1s (fragile zone) on the bridge.
Maximum Recovery $R$ :
- To maximize $R$ , Joseph should maximize the distance he walks after any swimming segment. The maximum possible $R$ is limited by the smallest safe zone (sequence of 0s) between fragile zones.
Force Calculation:
- The force $F = R - E$ will be maximized when $R$ is as large as possible and $E$ is as small as possible.
- Thus, $F$ can be calculated as the difference between the smallest safe zone and the largest fragile zone.

Solution Steps

Identify Safe and Fragile Zones:
- Traverse the bridge and identify all consecutive sequences of 0s (safe zones) and 1s (fragile zones).
- Keep track of the length of each safe and fragile zone.
Determine $E$ and $R$ :
- $E$ : The maximum length of any fragile zone.
- $R$ : The minimum length of any safe zone.
Compute the Force $F$ :
- $F = R - E$

Implementation

Below is the Python code that implements the above strategy:

python
def max_force(N, pont):
    min_safe = float('inf')
    max_fragile = 0
    i = 0
    while i < N:
        current = pont[i]
        start = i
        while i < N and pont[i] == current:
            i += 1
        length = i - start
        if current == 0:
            if length < min_safe:
                min_safe = length
        else:
            if length > max_fragile:
                max_fragile = length
    return min_safe - max_fragile

# Read input
import sys

def main():
    import sys
    input = sys.stdin.read
    data = input().split()
    N = int(data[0])
    pont = list(map(int, data[1:N+1]))
    result = max_force(N, pont)
    print(result)

if __name__ == "__main__":
    main()

Explanation of the Code

Function max_force:
- Parameters:
  - N: Length of the bridge.
  - pont: List representing the bridge.
- Variables:
  - min_safe: Stores the length of the smallest safe zone.
  - max_fragile: Stores the length of the largest fragile zone.
- Logic:
  - Iterate through the bridge to identify and measure all safe and fragile zones.
  - Update min_safe and max_fragile accordingly.
- Return:
  - The force $F = R - E$ .
Reading Input:
- The input is read from standard input.
- The first integer is N, followed by N integers representing the bridge.
Output:
- The computed force is printed as the output.

Sample Input and Output

Example 1:

text
5
0 1 0 1 0

Explanation:
- Safe zones: [0], [0], [0] (lengths 1 each)
- Fragile zones: [1], [1] (lengths 1 each)
- $R = 1$ , $E = 1$
- $F = 1 - 1 = 0$
Output:
```
0
```

Example 2:

text
6
0 0 1 1 0 0

Explanation:
- Safe zones: [0,0], [0,0] (lengths 2 each)
- Fragile zones: [1,1] (length 2)
- $R = 2$ , $E = 2$
- $F = 2 - 2 = 0$
Output:
```
0
```

Example 3:

text
9
0 1 1 1 0 0 1 1 0

Explanation:
- Safe zones: [0], [0,0], [0] (minimum length 1)
- Fragile zones: [1,1,1], [1,1] (maximum length 3)
- $R = 1$ , $E = 3$
- $F = 1 - 3 = -2$
Output:
```
-2
```

Conclusion

By identifying the smallest safe zone and the largest fragile zone, we can efficiently compute the maximum possible force $F = R - E$ that Joseph can achieve while crossing the bridge. This approach ensures that Joseph adheres to the constraints of swimming through fragile zones and managing his endurance and recovery optimally.