Generalized Bridge and Torch Problem with Non-Uniform Speeds

Scenario

A group of $N$ people must cross a dark, narrow bridge at night. The bridge can support at most 2 people simultaneously. The party possesses a single torch, which must be carried on every crossing (forward and return). When 2 people cross together, they must move at the pace of the slower individual: time taken is $\max(t_i, t_j)$. The sorted crossing times are $t_1 \le t_2 \le \dots \le t_N$.

Your Task

1. Formulate the dynamic programming recurrence relation that minimizes the total crossing time for $N$ people.
2. Detail the two candidate strategies (Strategy A vs Strategy B) for getting the two slowest individuals across.
3. Solve for the classic benchmark $T = [1, 2, 5, 10]$ and determine the exact minimum time.

Constraints

• Individual times $t_i > 0$.
• At most 2 people on bridge at any moment.

Examples For $[1, 2, 5, 10]$, Strategy B crosses 1 and 2, returns 1, crosses 5 and 10, returns 2, crosses 1 and 2, taking $2 + 1 + 10 + 2 + 2 = 17$ minutes.