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Partitioned Peace

Partitioned Peace, illustration

Credit: Getty Images

In a mythical land of rhetorically encouraged antagonism, different factions manage to co-exist, though poorly. Imagine a set of red and blue hill towns connected by a network of roads. People in the red towns deal well with one another. People in the blue towns deal well with one another. But when a person from a red town travels through a blue town or vice versa, things can get unpleasant. The leaders of the red and the blue towns get together and decide the best way to resolve their differences is to perform a series of swaps in which the inhabitants of k red towns swap towns with the inhabitants of k blue towns with the end result that a person from a blue town can visit any other blue town without passing through a red town and likewise for a person from a red town. We call such a desirable state "partitioned peace." The goal is to make k as small as possible.

Warm-Up 1. Given the configuration in the figure here, what is the minimum number of swaps needed to achieve partitioned peace?


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