Assignment #9 Solution

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Description

This is a simple algorithmic problem using set/multiset/map/multimap. You are

given 2n points in the plane, not necessarily distinct, n of which are

coloured red and n blue. You have to form disjoint pairs of points such that

each pair contains one red and one blue point. Also, each coordinate of a red

point in a pair must be <= the corresponding coordinate of the blue point in

the pair. In other words, the blue point must not be below or to the left

of the red point. Find the maximum number of such pairs that can be formed

from the given points.

Input Format

The first line of input specifies n, the number of red and blue points,

where 1 <= n <= 10^6. The next n lines specify the coordinates of the red

points. Each line contains two numbers the x-coordinate followed by the

y-coordinate. The next n lines specify the blue points in the same way.

Each coordinate is at most 10^9 in magnitude.

Output Format

Output the maximum possible number of disjoint pairs that can be formed

satisfying the required properties.

Sample Input1

2

0 0

1 1

0 1

1 0

Sample Output 1

1

Sample Input 2

8

0 0

0 2

1 2

1 3

2 1

2 3

3 1

3 2

0 1

0 3

1 0

1 1

2 0

2 2

3 0

3 3

Sample Output 2

4

The code for this is very short. Think carefully about the algorithm and how

to implement it before starting coding. There will be NO resubmission for

this assignment. Some test cases are at www.cse.iitb.ac.in/~aad/cs293/Assign9

(While I think they are correct, no guarantees!).

Submit a single file named RollNo_9.cpp.

Homework: Suppose there are no colours and any point can be paired with any

other, subject to the constraint on the coordinates. Find the maximum number

of disjoint pairs that can be formed in this case. Can be done in O(n log n)

time, but is not easy. What happens for points in 3 dimensions?