| Oh boy, Sid's family has taken him to Pizza Planet today! Pizza Planet is a |
| fun family restaurant with lots of arcade games, but the highlight for Sid is |
| Space Crane, a crane game with cool toy prizes. He'd love to win some to add |
| to his collection! |
|
|
| Looking at Space Crane from the front, it can be represented as a 2D plane. At |
| the top, there's a horizontal crane track, a line segment running from (0, |
| **M**) to (1,000,000, **M**). There's a claw attached to this track by an |
| extendible wire. The claw is initially located at coordinates (0, **M**) and |
| may be moved anywhere within the inclusive range of x-coordinates [0, |
| 1,000,000] and the inclusive range of y-coordinates [0, **M**]. At all points |
| in time, the connecting wire runs vertically upwards from the claw's position |
| to the track. That is, when the claw is at some point (**x**, **y**), the wire |
| forms a line segment from (**x**, **y**) to (**x**, **M**). |
|
|
| Space Crane works differently than most crane games — rather than using the |
| claw to directly pick up prizes, the player's objective is to navigate the |
| claw to a series of targets. There are **N** targets, all with distinct |
| x-coordinates, with the _i_th target at coordinates (**Xi**, **Yi**). If Sid |
| manages to move the claw to touch the **N** targets in order from 1 to **N**, |
| and then return the claw to its original position at (0, **M**), he'll win a |
| prize! Targets are not collected along the way (they're only touched by the |
| claw), meaning that all **N** targets will remain in place for the duration of |
| the game. |
|
|
| The claw may never be anywhere directly underneath a target (at the same |
| x-coordinate but with a strictly smaller y-coordinate), as it would interfere |
| with the crane's wire. However, the claw may occupy exactly the same position |
| as a target, including passing directly through targets which Sid is not |
| currently trying to touch. |
|
|
| Before the game starts, Sid is given an opportunity to adjust each of the |
| **N** targets. There are two possible choices for each target: it may either |
| be left in its original position, or its y-coordinate may be increased by |
| exactly 1 unit. These adjustments may only be performed in advance, and the |
| targets must then all remain in their chosen positions for the duration of the |
| game. |
|
|
| Completing the game normally isn't much of a challenge for Sid, but he's heard |
| a rumour that Space Crane awards double prizes if completed as efficiently as |
| possible! The game measures efficiency based on how much the claw's wire |
| expands and contracts. As such, Sid would like to adjust the targets and then |
| move the claw around such that the total amount of vertical movement (changes |
| in y-coordinate) performed by the claw is minimized. Note that the claw's |
| horizontal movement (changes in x-coordinate) is ignored. Help Sid determine |
| the minimum total amount of vertical claw movement which might be required! |
|
|
| ### Input |
|
|
| Input begins with an integer **T**, the number of times Sid plays Space Crane. |
| For each game, there is first a line containing the space-separated integers |
| **N** and **M**. Then **N** lines follow, the _i_th of which contains the |
| space-separated integers **Xi** and **Yi**. |
|
|
| ### Output |
|
|
| For the _i_th game, output a line containing "Case #_i_: " followed by the the |
| minimum total amount of vertical claw movement, in units. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 100 |
| 1 ≤ **N** ≤ 1,000,000 |
| 3 ≤ **M** ≤ 1,000,000 |
| 0 ≤ **Xi** ≤ 1,000,000 |
| 1 ≤ **Yi** ≤ **M** \- 2 |
|
|
| ### Explanation of Sample |
|
|
| In the first case, the single target's height should be increased from 1 to 2. |
| Then, aside from moving right and left by 1,000,000 units, the crane will need |
| to move downwards by 8 units to reach the target and upwards by 8 units to |
| return to its original position, for a total of 16 units of vertical movement. |
|
|
| In the second case, if the targets are all left at their original heights, 10 |
| units of vertical movement will be required. If their heights are all |
| increased by 1, 8 units will be required. However, if just the first two |
| targets are raised, then only 6 units will be required, which is the minimum |
| achievable amount. |
|
|
|
|
