2016/round2/snakes_and_ladders.html

<p>
As the owner of a vast collection of fine ladders, you would like to display them to potential onlookers. 
You've decided to stand your <strong>N</strong> ladders up vertically, with their bases at distinct points on a horizontal number line. 
The <strong>i</strong>th ladder's base is at position <strong>X<sub>i</sub></strong>, and it has height <strong>H<sub>i</sub></strong>.
</p>

<p>
The local snake population has taken an interest in your ladders. As everyone knows, snakes love nothing more than to suspend themselves horizontally above the ground! 
In particular, a snake of length <strong>L</strong> is able to suspend itself between the tops of two ladders 
<strong>a</strong> and <strong>b</strong> if and only if they meet the following conditions:
</p>

<ul>
<li> - the ladders are exactly <strong>L</strong> units apart (|<strong>X<sub>a</sub></strong> - <strong>X<sub>b</sub></strong>| = <strong>L</strong>) </li>
<li> - the ladders are of equal height (<strong>H<sub>a</sub></strong> = <strong>H<sub>b</sub></strong>) </li>
<li> - there are no taller ladders in between them (there exists no ladder <strong>c</strong> such that 
        min{<strong>X<sub>a</sub></strong>, <strong>X<sub>b</sub></strong>} &lt; <strong>X<sub>c</sub></strong> &lt; 
        max{<strong>X<sub>a</sub></strong>, <strong>X<sub>b</sub></strong>} 
        and <strong>H<sub>c</sub></strong> &gt; <strong>H<sub>a</sub></strong>)
</li>
</ul>

<p>
A number of snakes are planning to take up residence amongst your ladders. In particular, for every position in which a snake could suspend itself 
(in other words, for every distinct, unordered pair of ladders <strong>a</strong> and <strong>b</strong>), 
one snake will move in of the appropriate length.
</p>

<p>
You'll have no choice but to take care of these snakes, of course. To feed a snake of length <strong>L</strong>, it'll cost you <strong>L</strong><sup>2</sup> dollars daily. 
To prepare your budget, you'd like to calculate how many dollars you'll be spending each day on your new pets!
</p>


<h3>Input</h3>
<p>
Input begins with an integer <strong>T</strong>, the number of sets of ladders you own.
For each set, there is first a line containing the integer <strong>N</strong>.
Then, <strong>N</strong> lines follow, the <strong>i</strong>th of which contains two space-separated integers,
<strong>X<sub>i</sub></strong> and <strong>H<sub>i</sub></strong>.
</p>


<h3>Output</h3>
<p>
For the <strong>i</strong>th set of ladders, print a line containing "Case #<strong>i</strong>: " followed by the daily feeding cost of all the snakes that move in,
modulo 10<sup>9</sup> + 7.
</p>


<h3>Constraints</h3>
<p>
1 &le; <strong>T</strong> &le; 50 <br />
1 &le; <strong>N</strong> &le; 200,000 <br />
0 &le; <strong>X<sub>i</sub></strong>, <strong>H<sub>i</sub></strong> &le; 1,000,000,000
</p>


<h3>Explanation of Sample</h3>
<p>
In the first case, one snake will move in between your two ladders. It will have length 20, so it will cost 20<sup>2</sup> = 400 dollars a day to feed.

In the third case, one snake will move in between the ladders of height 3, and another will move in between the ladders at X = 2 and X = 3. The first snake has length 3, and the second has length 1. The total cost is therefore 3<sup>2</sup> + 1<sup>2</sup> = 10.
</p>