CF343E Pumping Stations

Mad scientist Mike has applied for a job. His task is to manage a system of water pumping stations. The system consists of $ n $ pumping stations, which are numbered by integers from 1 to $ n $ . Some pairs of stations are connected by bidirectional pipes through which water can flow in either direction (but only in one at a time). For each pipe you know its bandwidth — the maximum number of liters of water that can flow through it in one hour. Each pumping station can pump incoming water from some stations to other stations through the pipes, provided that in one hour the total influx of water to the station is equal to the total outflux of water from the station. It is Mike's responsibility to pump water between stations. From station $ a $ to station $ b $ through the pipes (possibly through other stations) within one hour one can transmit a certain number of liters of water according to the rules described above. During this time, water from other stations can not flow into station $ a $ , and can not flow out of the station $ b $ . However, any amount of water can flow out of station $ a $ or in station $ b $ . If a total of $ x $ litres of water flows out of the station $ a $ in an hour, then Mike gets $ x $ bollars more to his salary. To get paid, Mike needs to work for $ n-1 $ days, according to the contract. On the first day he selects two stations $ v_{1} $ and $ v_{2} $ , and within one hour he pumps a certain amount of water from $ v_{1} $ to $ v_{2} $ . Next, on the $ i $ -th day Mike chooses a station $ v_{i+1} $ that has been never selected before, and pumps a certain amount of water out of the station $ v_{i} $ to station $ v_{i+1} $ for one hour. The quantity of water he pumps on the $ i $ -th day does not depend on the amount of water pumped on the $ (i-1) $ -th day. Mike needs to earn as much bollars as he can for his projects. Help Mike find such a permutation of station numbers $ v_{1} $ , $ v_{2} $ , $ ... $ , $ v_{n} $ so Mike will be able to earn the highest possible salary.

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