| There are **N** dots on a 2D grid, the _i_th of which is a point at | |
| coordinates (**Xi**, **Yi**). All coordinates are positive integers, and all | |
| **N** dots' positions are distinct. | |
| You'd like to draw **N** line segments, each of which is either horizontal or | |
| vertical, to "connect" each of the dots to one of the grid's axes. In | |
| particular, for each dot _i_, you'll draw either a horizontal line segment | |
| connecting it to the y-axis (with endpoints (0, **Yi**) and (**Xi**, **Yi**)), | |
| or a vertical line segment connecting it to the x-axis (with endpoints | |
| (**Xi**, 0) and (**Xi**, **Yi**)). Each line segment only counts as | |
| "connecting" the single dot located at its endpoint, even if it happens to | |
| pass through other dots along the way. | |
| No horizontal line segment is allowed to intersect with any vertical line | |
| segment. Line segments are **not** considered to intersect at either of their | |
| endpoints — for example, it's permitted for a horizontal line segment to pass | |
| through the endpoint of a vertical one, or vice versa. Horizontal line | |
| segments are allowed to overlap with other horizontal ones, as are vertical | |
| line segments with other vertical ones. | |
| The cost of drawing a non-empty set of horizontal line segments is equal to | |
| the length of the longest one (in dollars), while the cost of drawing no | |
| horizontal line segments is $0. The cost of drawing a set of vertical line | |
| segments is similarly equal to the length of the longest one (if any), and the | |
| total cost of drawing all **N** line segments is equal to the cost of drawing | |
| the set of horizontal ones plus the cost of drawing the set of vertical ones. | |
| You can choose to draw at most **H** horizontal line segments, and at most | |
| **V** vertical ones. What's the minimum total cost required to connect all | |
| **N** dots to the grid's axes, without using too many of either type of line | |
| segment or causing any horizontal line segments to intersect with vertical | |
| ones, if that can be done at all? | |
| In order to reduce the size of the input, the dots' coordinates will not all | |
| be provided explicitly. Instead, you'll be given **X1**, **X2**, **Y1**, | |
| **Y2**, as well as 8 constants **Ax**, **Bx**, **Cx**, **Dx**, **Ay**, **By**, | |
| **Cy**, and **Dy**, and you must then compute **X3..N** and **Y3..N** as | |
| follows (bearing in mind that intermediate values may not fit within 32-bit | |
| integers): | |
| **Xi** = ((**Ax** * **Xi-2** \+ **Bx** * **Xi-1** \+ **Cx**) modulo **Dx**) + 1, for _i_ = 3..**N** | |
| **Yi** = ((**Ay** * **Yi-2** \+ **By** * **Yi-1** \+ **Cy**) modulo **Dy**) + 1, for _i_ = 3..**N** | |
| ### Input | |
| Input begins with an integer **T**, the number of grids. For each room, there | |
| are three lines. The first line contains the space-separated integers **N**, | |
| **H**, and **V**. The second line contains the space-separated integers | |
| **X1**, **X2**, **Ax**, **Bx**, **Cx**, and **Dx**. The third line contains | |
| the space-separated integers **Y1**, **Y2**, **Ay**, **By**, **Cy**, and | |
| **Dy**. | |
| ### Output | |
| For the _i_th grid, print a line containing "Case #_i_: " followed by the | |
| minimum total cost (in dollars) required to validly connect all **N** dots to | |
| the grid's axes, or -1 if it's impossible to do so. | |
| ### Constraints | |
| 1 ≤ **T** ≤ 160 | |
| 2 ≤ **N** ≤ 800,000 | |
| 0 ≤ **H**, **V** ≤ **N** | |
| 0 ≤ **Ax**, **Bx**, **Cx** **Ay**, **By**, **Cy** ≤ 1,000,000,000 | |
| 1 ≤ **Dx**, **Dy** ≤ 1,000,000,000 | |
| 1 ≤ **Xi** ≤ **Dx** | |
| 1 ≤ **Yi** ≤ **Dy** | |
| In the first case, the dots are at coordinates (6, 2) and (3, 4). The cheapest | |
| option is to connect both dots using vertical line segments, having lengths 2 | |
| and 4 and altogether costing $4 to draw. The lack of horizontal line segments | |
| costs an additional $0, bringing the total to $4 + $0 = $4. | |
|  | |
| The second case is the same as the first, except that at most one vertical | |
| line may be drawn. The cheapest valid option is now to connect the second dot | |
| using a horizontal line segment (of length 3) while still connecting the first | |
| dot using a vertical one (of length 2). These two line segments do not | |
| intersect, and cost a total of $3 + $2 = $5 to draw. | |
|  | |
| In the third case, not all of the dots can be connected. | |
| In the fourth case, the dots are at coordinates (1, 1), (1, 2), (2, 1), and | |
| (2, 2). You can connect the first dot using a horizontal line segment (of | |
| length 1), and the other dots with vertical ones (of lengths at most 2), for a | |
| total cost of $1 + $2 = $3. Note that this causes two vertical line segments | |
| to overlap (the ones connecting the third and fourth points). | |
|  | |
| In the fifth case, the dots are at coordinates (15, 34), (19, 3), (2, 38), | |
| (13, 17), (18, 14), (25, 15), (42, 18), (9, 11), (26, 34), and (41, 19). | |