如何找到最便宜的地铁订阅组合

Question

Lets say we have the following fares available:

• 1 trip
• 2 trips
• 10 trips
• unlimited weekend (saturday to sunday, not 2 days)
• 1 day
• 3 days
• unlimited week (monday to sunday, not 7 days)
• unlimited month (1st to last day of month)

... with a price for every one of them.

The problem is: **How to determine what set of subscriptions to chose given a date of arrival and a date of departure? **

Lets say we want the solution for n between 1 and 8, n being the number of time we take the metro daily (so we assume we take the metro the same number of time every day)

For example it would say something like:

n = 1 Arriving on Friday 19th and leaving Thursday 23th, the best is taking the 1 trip, then the weekend, then the 2 trips (didnt calculate but you see the point)

n = 2...

I have found examples with only 1 day, 2 days, 7 days fares with dynamic programming, but it looks a lot harder when you considerate the days of the week.

Thanks:)

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Answer 1

I like to view this kind of dynamic program as finding a shortest path in a directed acyclic graph.

Each node of the graph encodes

• what the current day is (either during the travel period or the day after), and
• how many trips remain on trip-limited passes (at most n + 9).

Each arc represents either

• purchasing a specific pass at a specific time (the length of the arc is the cost of this pass), or
• using trip-limited passes to cover the day's trips (the length of the arc is zero).

The time-limited passes advance the day to the first day they no longer work. The trip-limited passes increase the number of remaining trips. The zero-cost arcs advance the day by one while decreasing the number of remaining trips by n.

Given the shortest path, it is easy to decode it to a plan for purchasing passes.

(PS I don't know what the rules are on, eg, purchasing a week pass on a Tuesday for the rest of the week. Even if this is not allowed, you're going to want to put arcs for the time-limited passes that could have been purchased on a previous day during the travel period.)