4989 - [Usaco2017 Feb]Why Did the Cow Cross the Road

Why did the cow cross the road? We may never know the full reason, but it is certain that Farmer Joh n's cows do end up crossing the road quite frequently. In fact, they end up crossing the road so oft en that they often bump into each-other when their paths cross, a situation Farmer John would like t o remedy.Farmer John raises N breeds of cows (1≤N≤100,000), and each of his fields is dedicated to grazing for one specific breed; for example, a field dedicated to breed 12 can only be used for cow s of breed 12 and not of any other breed. A long road runs through his farm. There is a sequence of NN fields on one side of the road (one for each breed), and a sequence of N fields on the other side of the road (also one for each breed). When a cow crosses the road, she therefore crosses between t he two fields designated for her specific breed.Had Farmer John planned more carefully, he would hav e ordered the fields by breed the same way on both sides of the road, so the two fields for each bre ed would be directly across the road from each-other. This would have allowed cows to cross the road without any cows from different breeds bumping into one-another. Alas, the orderings on both sides of the road might be different, so Farmer John observes that there might be pairs of breeds that cro ss. A pair of different breeds (a,b) is "crossing" if any path across the road for breed aa must int ersect any path across the road for breed bb.Farmer John would like to minimize the number of crossi ng pairs of breeds. For logistical reasons, he figures he can move cows around on one side of the ro ad so the fields on that side undergo a "cyclic shift". That is, for some 0≤k<N, every cow re-locat es to the field kk fields ahead of it, with the cows in the last kk fields moving so they now popula te the first kk fields. For example, if the fields on one side of the road start out ordered by bree d as 3, 7, 1, 2, 5, 4, 6 and undergo a cyclic shift by k=2, the new order will be 4, 6, 3, 7, 1, 2,

1. Please determine the minimum possible number of crossing pairs of breeds that can exist after an appropriate cyclic shift of the fields on one side of the road.上下有两个位置分别对应的序列A、B，长度为n， 两序列为n的一个排列。当Ai == Bj时，上下会连一条边。 你可以选择序列A或者序列B进行旋转任意K步， 如 3 4 1 5 2 旋转两步为 5 2 3 4 1。 求旋转后最小的相交的线段的对数。

The first line of input contains N. The next N lines describe the order, by breed ID, of fields on one side of the road; each breed ID is an integer in the range 1…N. The last N lines describe the order, by breed ID, of the fields on the other side of the road.

Please output the minimum number of crossing pairs of breeds after a cyclic shift of the fields on one side of the road (either side can be shifted).