How to Solve Minimum Time to Transport All Individuals Problem
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Minimum Time to Transport All Individuals
You are given n individuals at a base camp who need to cross a river to reach a destination using a single boat. The boat can carry at most k people at a time. The trip is affected by environmental co...
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Understanding the Minimum Time to Transport All Individuals Problem
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Problem Statement
You are given n individuals at a base camp who need to cross a river to reach a destination using a single boat. The boat can carry at most k people at a time. The trip is affected by environmental conditions that vary cyclically over m stages. Each stage j has a speed multiplier mul[j]: If mul[j] > 1, the trip slows down. If mul[j] < 1, the trip speeds up. Each individual i has a rowing strength represented by time[i], the time (in minutes) it takes them to cross alone in neutral conditions. Rules: A group g departing at stage j takes time equal to the maximum time[i] among its members, multiplied by mul[j] minutes to reach the destination. After the group crosses the river in time d, the stage advances by floor(d) % m steps. If individuals are left behind, one person must return with the boat. Let r be the index of the returning person, the return takes time[r] × mul[current_stage], defined as return_time, and the stage advances by floor(return_time) % m. Return the minimum total time required to transport all individuals. If it is not possible to transport all individuals to the destination, return -1.
Minimum Time to Transport All Individuals
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Examples
n = 1, k = 1, m = 2, time = [5], mul = [1.0,1.3]5.00000Individual 0 departs from stage 0, so crossing time = 5 × 1.00 = 5.00 minutes. All team members are now at the destination. Thus, the total time taken is 5.00 minutes.
n = 3, k = 2, m = 3, time = [2,5,8], mul = [1.0,1.5,0.75]14.50000The optimal strategy is: Send individuals 0 and 2 from the base camp to the destination from stage 0. The crossing time is max(2, 8) × mul[0] = 8 × 1.00 = 8.00 minutes. The stage advances by floor(8.00) % 3 = 2, so the next stage is (0 + 2) % 3 = 2. Individual 0 returns alone from the destination to the base camp from stage 2. The return time is 2 × mul[2] = 2 × 0.75 = 1.50 minutes. The stage advances by floor(1.50) % 3 = 1, so the next stage is (2 + 1) % 3 = 0. Send individuals 0 and 1 from the base c
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