Karen and Coffee
Contest: Codeforces Round 419 (Div. 2)
Difficulty rating: 1400
Tags: binary search, data structures, implementation

To stay woke and attentive during classes, Karen needs some coffee!

Karen, a coffee aficionado, wants to know the optimal temperature for brewing the perfect cup of coffee. Indeed, she has spent some time reading several recipe books, including the universally acclaimed "The Art of the Covfefe".

She knows n coffee recipes. The i-th recipe suggests that coffee should be brewed between li and ri degrees, inclusive, to achieve the optimal taste.

Karen thinks that a temperature is admissible if at least k recipes recommend it.

Karen has a rather fickle mind, and so she asks q questions. In each question, given that she only wants to prepare coffee with a temperature between a and b, inclusive, can you tell her how many admissible integer temperatures fall within the range?

Input format:
The first line of input contains three integers, n, k (1 ≤ k ≤ n ≤ 200000), and q (1 ≤ q ≤ 200000), the number of recipes, the minimum number of recipes a certain temperature must be recommended by to be admissible, and the number of questions Karen has, respectively.

The next n lines describe the recipes. Specifically, the i-th line among these contains two integers li and ri (1 ≤ li ≤ ri ≤ 200000), describing that the i-th recipe suggests that the coffee be brewed between li and ri degrees, inclusive.

The next q lines describe the questions. Each of these lines contains a and b, (1 ≤ a ≤ b ≤ 200000), describing that she wants to know the number of admissible integer temperatures between a and b degrees, inclusive.

Output format:
For each question, output a single integer on a line by itself, the number of admissible integer temperatures between a and b degrees, inclusive.

Constraints:
Time limit: 2.5 s
Memory limit: 512.0 MB

Note:
In the first test case, Karen knows 3 recipes.

1. The first one recommends brewing the coffee between 91 and 94 degrees, inclusive.
2. The second one recommends brewing the coffee between 92 and 97 degrees, inclusive.
3. The third one recommends brewing the coffee between 97 and 99 degrees, inclusive.

A temperature is admissible if at least 2 recipes recommend it.

She asks 4 questions.

In her first question, she wants to know the number of admissible integer temperatures between 92 and 94 degrees, inclusive. There are 3: 92, 93 and 94 degrees are all admissible.

In her second question, she wants to know the number of admissible integer temperatures between 93 and 97 degrees, inclusive. There are 3: 93, 94 and 97 degrees are all admissible.

In her third question, she wants to know the number of admissible integer temperatures between 95 and 96 degrees, inclusive. There are none.

In her final question, she wants to know the number of admissible integer temperatures between 90 and 100 degrees, inclusive. There are 4: 92, 93, 94 and 97 degrees are all admissible.

In the second test case, Karen knows 2 recipes.

1. The first one, "wikiHow to make Cold Brew Coffee", recommends brewing the coffee at exactly 1 degree.
2. The second one, "What good is coffee that isn't brewed at at least 36.3306 times the temperature of the surface of the sun?", recommends brewing the coffee at exactly 200000 degrees.

A temperature is admissible if at least 1 recipe recommends it.

In her first and only question, she wants to know the number of admissible integer temperatures that are actually reasonable. There are none.

Example input:
3 2 4
91 94
92 97
97 99
92 94
93 97
95 96
90 100

Example output:
3
3
0
4

Example input:
2 1 1
1 1
200000 200000
90 100

Example output:
0

Editorial:
There are two separate tasks here:
1. Efficiently generate an array c where ci is the number of recipes that recommend temperature i.
2. Efficiently answer queries "how many numbers ca, ca + 1, ca + 2, ..., cb" are at least k?" where k is fixed across all queries.
There are some solutions to this task using advanced data structures or algorithms. For example, a conceptually straightforward idea is the following: create a segment tree c. We can treat all recipes as range update queries which we can do efficiently in $${\mathcal {O}}(n\log m)$$ (where m is the largest ai or bi) time using lazy propagation.
After all recipes, we replace all ci by 1 if it's at least k, and 0 otherwise. Afterwards, each of the next q queries is a basic range sum query, which can be done simply in $${\mathcal O}(q\log m)$$ time.
Other solutions exist, too: Fenwick trees with range updates, event sorting, sqrt-decomposition with binary search, Mo's algorithm, and so on. These solutions all pass, but they are all overkill for this task.
A very simple solution is as follows. Initialize c with all zeroes. For recipe that recommends temperatures between li and ri, we should increment cli and decrement cri + 1.
Cumulate all values. That is, set ci to c1 + c2 + c3 + ... + ci. This can be done with one pass through the array.
Now, magically, ci is now the number of recipes that recommend temperature i. If ci is at least k, set it to 1, otherwise, set it to 0.
Cumulate all values again.
Now, every query that asks for the number of admissible temperatures between a and b can be answered simply as cb - ca - 1.
This runs in O(n + q + m), which is really fast.
Note that if your solution does this and still runs quite slow, chances are your solution is using slower input methods. We raised the time limit to 2.5 seconds in this problem in order to avoid failing slow input solutions.