Find minimum time to cross bridge with flashlight

Problem Statement

There are n people standing on one side of a narrow bridge at night. They need to cross to the other side. The bridge can hold at most two people at a time, and they must carry a single flashlight to navigate safely. Since the bridge is dark, the flashlight must be passed back and forth between the people who have already crossed to those who have yet to cross.

Each person walks at a different speed. When two people cross together, they move at the speed of the slower person. Given an array speeds where speeds[i] is the time it takes for person i to cross the bridge, calculate the minimum total time required for everyone to reach the other side.

Function Signature

def min_bridge_time(speeds: List[int]) -> int:

Examples

Example 1: Input: speeds = [1, 2, 5, 10] Output: 17 Explanation:

1. 1 and 2 cross (2 mins), 1 returns (1 min) [Total: 3]
2. 5 and 10 cross (10 mins), 2 returns (2 mins) [Total: 15]
3. 1 and 2 cross (2 mins) [Total: 17]

Example 2: Input: speeds = [1, 3, 4] Output: 8 Explanation: 1 and 3 cross (3 mins), 1 returns (1 min), 1 and 4 cross (4 mins). Total = 3 + 1 + 4 = 8.

Constraints

• 1 <= speeds.length <= 1000
• 1 <= speeds[i] <= 1000
• The input array is not necessarily sorted.

Edge Cases

• One or two people: They simply cross together or individually.
• Large values in speeds: Ensure integer overflow is not an issue (though not applicable here due to constraints).