CF1517B Morning Jogging

The 2050 volunteers are organizing the "Run! Chase the Rising Sun" activity. Starting on Apr 25 at 7:30 am, runners will complete the 6km trail around the Yunqi town. There are $ n+1 $ checkpoints on the trail. They are numbered by $ 0 $ , $ 1 $ , ..., $ n $ . A runner must start at checkpoint $ 0 $ and finish at checkpoint $ n $ . No checkpoint is skippable — he must run from checkpoint $ 0 $ to checkpoint $ 1 $ , then from checkpoint $ 1 $ to checkpoint $ 2 $ and so on. Look at the picture in notes section for clarification. Between any two adjacent checkpoints, there are $ m $ different paths to choose. For any $ 1\le i\le n $ , to run from checkpoint $ i-1 $ to checkpoint $ i $ , a runner can choose exactly one from the $ m $ possible paths. The length of the $ j $ -th path between checkpoint $ i-1 $ and $ i $ is $ b_{i,j} $ for any $ 1\le j\le m $ and $ 1\le i\le n $ . To test the trail, we have $ m $ runners. Each runner must run from the checkpoint $ 0 $ to the checkpoint $ n $ once, visiting all the checkpoints. Every path between every pair of adjacent checkpoints needs to be ran by exactly one runner. If a runner chooses the path of length $ l_i $ between checkpoint $ i-1 $ and $ i $ ( $ 1\le i\le n $ ), his tiredness is $ $$$\min_{i=1}^n l_i, $ $ i. e. the minimum length of the paths he takes.

Please arrange the paths of the $ m$$$ runners to minimize the sum of tiredness of them.

N/A

N/A

In the first case, the sum of tiredness is $ \min(2,5) + \min(3,3) + \min(4,1) = 6 $ . In the second case, the sum of tiredness is $ \min(2,4,3) + \min(3,1,5) = 3 $ .