F. Triangular Paths
constructive algorithms; graphs; math; shortest paths; sortings
Consider an infinite triangle made up of layers. Let's number the layers, starting from one, from the top of the triangle (from top to bottom). The \(k\)-th layer of the triangle contains \(k\) points, numbered from left to right. Each point of an infinite triangle is described by a pair of numbers \((r, c)\) (\(1 \le c \le r\)), where \(r\) is the number of the layer, and \(c\) is the number of the point in the layer. From each point \((r, c)\) there are two directed edges to the points \((r+1, c)\) and \((r+1, c+1)\), but only one of the edges is activated. If \(r + c\) is even, then the edge to the point \((r+1, c)\) is activated, otherwise the edge to the point \((r+1, c+1)\) is activated. Look at the picture for a better understanding. Activated edges are colored in black. Non-activated edges are colored in gray. From the point \((r_1, c_1)\) it is possible to reach the point \((r_2, c_2)\), if there is a path between them only from activated edges. For example, in the picture above, there is a path from \((1, 1)\) to \((3, 2)\), but there is no path from \((2, 1)\) to \((1, 1)\).Initially, you are at the point \((1, 1)\). For each turn, you can: Replace activated edge for point \((r, c)\). That is if the edge to the point \((r+1, c)\) is activated, then instead of it, the edge to the point \((r+1, c+1)\) becomes activated, otherwise if the edge to the point \((r+1, c+1)\), then instead if it, the edge to the point \((r+1, c)\) becomes activated. This action increases the cost of the path by \(1\); Move from the current point to another by following the activated edge. This action does not increase the cost of the path. You are given a sequence of \(n\) points of an infinite triangle \((r_1, c_1), (r_2, c_2), \ldots, (r_n, c_n)\). Find the minimum cost path from \((1, 1)\), passing through all \(n\) points in arbitrary order.
The first line contains one integer \(t\) (\(1 \le t \le 10^4\)) is the number of test cases. Then \(t\) test cases follow.Each test case begins with a line containing one integer \(n\) (\(1 \le n \le 2 \cdot 10^5\)) is the number of points to visit.The second line contains \(n\) numbers \(r_1, r_2, \ldots, r_n\) (\(1 \le r_i \le 10^9\)), where \(r_i\) is the number of the layer in which \(i\)-th point is located.The third line contains \(n\) numbers \(c_1, c_2, \ldots, c_n\) (\(1 \le c_i \le r_i\)), where \(c_i\) is the number of the \(i\)-th point in the \(r_i\) layer.It is guaranteed that all \(n\) points are distinct.It is guaranteed that there is always at least one way to traverse all \(n\) points.It is guaranteed that the sum of \(n\) over all test cases does not exceed \(2 \cdot 10^5\).
For each test case, output the minimum cost of a path passing through all points in the corresponding test case.