Counting Triangles in a Rectangular space using BIT

Pre-requisite: BIT(Binary Indexed Tree or Fenwick Tree), 2D BIT
Given a 2D plane, respond to Q queries, each of the following type: 

1. Insert x y – Insert a point (x, y) coordinate.
2. Triangle x1 y1 x2 y2 – Print the number of triangles that can be formed, by joining the points inside the rectangle, described by two points (x1, y1) and (x2, y2), (x1, y1) is the lower-left corner while (x2, y2) is the upper right corner. We will represent it as {(x1, y1), (x2, y2)}.(Include the triangles with zero areas also in your answer)

Example:

In the red rectangle there are 6 points inserted, 
when each of them is joined with a line with every
other point, in all 20 triangles will be formed.

Let’s say we somehow have a mechanism to find the number of points inside a given rectangle, let the number of points be ‘m’ in an instance. 
The number of triangles that can be formed with it are ^mC₃ (every 3 points when joined will make a triangle); if we had to count only degenerate triangles, we would have to subtract the number of triangles with zero areas or formed from points on the same line.
Now we sneak into the mechanism to find the number of points in a rectangle; let’s assume we have a mechanism to find the number of points inside a rectangle.

Then number of points inside the rectangle 
{(x1, y1), (x2, y2)} are,

        P(x2, y2) - 
        P(x1 - 1, y2) - 
        P(x2, y1 - 1) + 
        P(x1 - 1, y1 - 1)

Where,
P(x, y) is number of triangles in rectangle
from (1, 1) to (x, y), i.e., {(1, 1), (x, y)}

Now once we find a way to find P(x, y), we are done.
If all the insert queries were made first, then all the triangle queries, it would have been an easy job, we could maintain a 2D table to store the points and table[x][y] would contain the total number of points in the rectangle {(1, 1), (x, y)}. 
It can be created using the following DP:
table[x][y] = table[x][y – 1] + table[x – 1][y] – table[x – 1][y – 1]
Or we can use a 2D BIT to insert a point and evaluate P(x, y). For visualization of a 2D BIT, refer Top Coder. The image shows a 16 * 8 2D BIT, and the state after an insertion at (5, 3), the blue nodes are the onces that are updated.
The horizontal BITs correspond to 3, and store 1 at index 3, 1 at index 4, and 1 at index 8; while the vertical representation corresponds to which horizontal BITs will receive that update, the ones that correspond to 5, so 5th BIT from the bottom gets an update, 6th BIT, then 8th BIT, and then 16th BIT.
Let’s consider update in a 1D BIT 

update(BIT, x)
  while ( x < maxn )
    BIT[x] += val
    x += x & -x

Update for 2D BIT goes like:  

update2DBIT(x, y)

// update BIT at index x (from bottom, in the image)
while ( x < maxn )
    update(BIT[x], y)
    x += x & -x

C++

// A C++ program implementing the above queries
#include<bits/stdc++.h>
#define maxn 2005
using namespace std;

// 2D Binary Indexed Tree. Note: global variable
// will have initially all elements zero
int bit[maxn][maxn];

// function to add a point at (x, y)
void update(int x, int y)
{
    int y1;
    while (x < maxn)
    {
        // x is the xth BIT that will be updated
        // while y is the indices where an update
        // will be made in xth BIT
        y1 = y;
        while ( y1 < maxn )
        {
            bit[x][y1]++;
            y1 += ( y1 & -y1 );
        }

        // next BIT that should be updated
        x += x & -x;
    }
}

// Function to return number of points in the
// rectangle (1, 1), (x, y)
int query(int x, int y)
{
    int res = 0, y1;
    while (x > 0)
    {
        // xth BIT's yth node must be added to the result
        y1 = y;
        while (y1 > 0)
        {
            res += bit[x][y1];
            y1 -= y1 & -y1;
        }

        // next BIT that will contribute to the result
        x -= x & -x;
    }
    return res;
}

// (x1, y1) is the lower left and (x2, y2) is the
// upper right corner of the rectangle
int pointsInRectangle(int x1, int y1, int x2, int y2)
{
    // Returns number of points in the rectangle
    // (x1, y1), (x2, y2) as described in text above
    return query(x2, y2) - query(x1 - 1, y2) -W
           query(x2, y1 - 1) + query(x1 - 1, y1 - 1);
}

// Returns count of triangles with n points, i.e.,
// it returns nC3
int findTriangles(int n)
{
    // returns pts choose 3
    return (n * (n - 1) * (n - 2)) / 6;
}

//driver code
int main()
{
    //inserting points
    update(2, 2);
    update(3, 5);
    update(4, 2);
    update(4, 5);
    update(5, 4);

    cout << "No. of triangles in the rectangle (1, 1)"
            " (6, 6) are: "
         << findTriangles(pointsInRectangle(1, 1, 6, 6));

    update(3, 3);

    cout << "\nNo. of triangles in the rectangle (1, 1)"
            " (6, 6) are: "
         << findTriangles( pointsInRectangle(1, 1, 6, 6));

    return 0;
}

Java

// Java program implementing the above queries
import java.util.*;
class GFG
{
static int maxn = 2005;

// 2D Binary Indexed Tree. Note: global variable
// will have initially all elements zero
static int [][]bit = new int[maxn][maxn];

// function to add a point at (x, y)
static void update(int x, int y)
{
    int y1;
    while (x < maxn)
    {
        // x is the xth BIT that will be updated
        // while y is the indices where an update
        // will be made in xth BIT
        y1 = y;
        while ( y1 < maxn )
        {
            bit[x][y1]++;
            y1 += (y1 & -y1);
        }

        // next BIT that should be updated
        x += x & -x;
    }
}

// Function to return number of points in the
// rectangle (1, 1), (x, y)
static int query(int x, int y)
{
    int res = 0, y1;
    while (x > 0)
    {
        // xth BIT's yth node
        // must be added to the result
        y1 = y;
        while (y1 > 0)
        {
            res += bit[x][y1];
            y1 -= y1 & -y1;
        }

        // next BIT that will contribute to the result
        x -= x & -x;
    }
    return res;
}

// (x1, y1) is the lower left and (x2, y2) is the
// upper right corner of the rectangle
static int pointsInRectangle(int x1, int y1,
                             int x2, int y2)
{
    // Returns number of points in the rectangle
    // (x1, y1), (x2, y2) as described in text above
    return query(x2, y2) - query(x1 - 1, y2) -
           query(x2, y1 - 1) +
           query(x1 - 1, y1 - 1);
}

// Returns count of triangles with n points, i.e.,
// it returns nC3
static int findTriangles(int n)
{
    // returns pts choose 3
    return (n * (n - 1) * (n - 2)) / 6;
}

// Driver Code
public static void main(String[] args)
{
    // inserting points
    update(2, 2);
    update(3, 5);
    update(4, 2);
    update(4, 5);
    update(5, 4);

    System.out.print("No. of triangles in the " +
                     "rectangle (1, 1) (6, 6) are: " +
                      findTriangles(pointsInRectangle(1, 1, 6, 6)));

    update(3, 3);

    System.out.print("\nNo. of triangles in the " +
                     "rectangle (1, 1) (6, 6) are: " +
                     findTriangles( pointsInRectangle(1, 1, 6, 6)));
}
}

// This code is contributed by Rajput-Ji

Python3

# A Python3 program implementing
# the above queries
maxn = 2005

# 2D Binary Indexed Tree.
# Note: global variable
# will have initially all
# elements zero
bit = [[0 for j in range(maxn)]
          for i in range(maxn)]

# function to add a point
# at(x, y)
def update(x, y):

    y1 = 0
    while (x < maxn):

        # x is the xth BIT that will
        # be updated while y is the
        # indices where an update
        # will be made in xth BIT
        y1 = y

        while (y1 < maxn):
            bit[x][y1] += 1
            y1 += (y1 & -y1)

        # next BIT that should
        # be updated
        x += x & -x

# Function to return number of
# points in the rectangle(1, 1),
# (x, y)
def query(x, y):

    res = 0
    y1 = 0

    while (x > 0):

        # xth BIT's yth node must
        # be added to the result
        y1 = y

        while (y1 > 0):
            res += bit[x][y1]
            y1 -= y1 & -y1

        # next BIT that will contribute
        # to the result
        x -= x & -x

    return res

# (x1, y1) is the lower left
# and (x2, y2) is the upper
# right corner of the rectangle
def pointsInRectangle(x1, y1,
                      x2, y2):

    # Returns number of points
    # in the rectangle (x1, y1),
    # (x2, y2) as described in
    # text above
    return (query(x2, y2) - query(x1 - 1, y2) -
            query(x2, y1 - 1) + query(x1 - 1, y1 - 1))

# Returns count of triangles with
# n points, i.e., it returns nC3
def findTriangles(n):

    # returns pts choose 3
    return ((n * (n - 1) *
            (n - 2)) // 6)

# Driver code
if __name__ == "__main__":

    # inserting points
    update(2, 2)
    update(3, 5)
    update(4, 2)
    update(4, 5)
    update(5, 4)

    print("No. of triangles in the " +
          "rectangle (1, 1)  (6, 6) are: ",
           findTriangles(pointsInRectangle(1, 1,
                                           6, 6)))
    update(3, 3)
    print("No. of triangles in the rectangle " +
          "(1, 1) (6, 6) are:", findTriangles(
            pointsInRectangle(1, 1, 6, 6)))

# This code is contributed by Rutvik_56

C#

// C# program implementing the above queries
using System;

class GFG
{
static int maxn = 2005;

// 2D Binary Indexed Tree. Note: global variable
// will have initially all elements zero
static int [,]bit = new int[maxn, maxn];

// function to add a point at (x, y)
static void update(int x, int y)
{
    int y1;
    while (x < maxn)
    {
        // x is the xth BIT that will be updated
        // while y is the indices where an update
        // will be made in xth BIT
        y1 = y;
        while (y1 < maxn)
        {
            bit[x, y1]++;
            y1 += (y1 & -y1);
        }

        // next BIT that should be updated
        x += x & -x;
    }
}

// Function to return number of points in the
// rectangle (1, 1), (x, y)
static int query(int x, int y)
{
    int res = 0, y1;
    while (x > 0)
    {
        // xth BIT's yth node
        // must be added to the result
        y1 = y;
        while (y1 > 0)
        {
            res += bit[x, y1];
            y1 -= y1 & -y1;
        }

        // next BIT that will
        // contribute to the result
        x -= x & -x;
    }
    return res;
}

// (x1, y1) is the lower left and (x2, y2) is the
// upper right corner of the rectangle
static int pointsInRectangle(int x1, int y1,
                             int x2, int y2)
{
    // Returns number of points in the rectangle
    // (x1, y1), (x2, y2) as described in text above
    return query(x2, y2) - query(x1 - 1, y2) -
           query(x2, y1 - 1) +
            query(x1 - 1, y1 - 1);
}

// Returns count of triangles with n points, i.e.,
// it returns nC3
static int findTriangles(int n)
{
    // returns pts choose 3
    return (n * (n - 1) * (n - 2)) / 6;
}

// Driver Code
public static void Main(String[] args)
{
    // inserting points
    update(2, 2);
    update(3, 5);
    update(4, 2);
    update(4, 5);
    update(5, 4);

    Console.Write("No. of triangles in the " +
             "rectangle (1, 1) (6, 6) are: " +
              findTriangles(pointsInRectangle(1, 1, 6, 6)));

    update(3, 3);

    Console.Write("\nNo. of triangles in the " +
               "rectangle (1, 1) (6, 6) are: " +
                findTriangles( pointsInRectangle(1, 1, 6, 6)));
}
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

// JavaScript program implementing the above queries
    var maxn = 2005;

    // 2D Binary Indexed Tree. Note: global variable
    // will have initially all elements zero
     var bit = Array(maxn).fill().map(()=>Array(maxn).fill(0));

    // function to add a point at (x, y)
    function update(x , y) {
        var y1;
        while (x < maxn) {
            // x is the xth BIT that will be updated
            // while y is the indices where an update
            // will be made in xth BIT
            y1 = y;
            while (y1 < maxn) {
                bit[x][y1]++;
                y1 += (y1 & -y1);
            }

            // next BIT that should be updated
            x += x & -x;
        }
    }

    // Function to return number of points in the
    // rectangle (1, 1), (x, y)
    function query(x , y) {
        var res = 0, y1;
        while (x > 0) {
            // xth BIT's yth node
            // must be added to the result
            y1 = y;
            while (y1 > 0) {
                res += bit[x][y1];
                y1 -= y1 & -y1;
            }

            // next BIT that will contribute to the result
            x -= x & -x;
        }
        return res;
    }

    // (x1, y1) is the lower left and (x2, y2) is the
    // upper right corner of the rectangle
    function pointsInRectangle(x1 , y1 , x2 , y2) {
        // Returns number of points in the rectangle
        // (x1, y1), (x2, y2) as described in text above
        return query(x2, y2) - query(x1 - 1, y2) -
        query(x2, y1 - 1) + query(x1 - 1, y1 - 1);
    }

    // Returns count of triangles with n points, i.e.,
    // it returns nC3
    function findTriangles(n) {
        // returns pts choose 3
        return (n * (n - 1) * (n - 2)) / 6;
    }

    // Driver Code

        // inserting points
        update(2, 2);
        update(3, 5);
        update(4, 2);
        update(4, 5);
        update(5, 4);

        document.write("No. of triangles in the " +
        "rectangle (1, 1) (6, 6) are: "
       + findTriangles(pointsInRectangle(1, 1, 6, 6)));

        update(3, 3);

        document.write("<br/>No. of triangles in the " +
        "rectangle (1, 1) (6, 6) are: "
        + findTriangles(pointsInRectangle(1, 1, 6, 6)));

// This code contributed by gauravrajput1

</script>

Output:

No of triangles in the rectangle (1, 1) (6, 6) are: 10
No of triangles in the rectangle (1, 1) (6, 6) are: 20

Time Complexity: For each query of either type: O(log(x) * log(y)) Or infact: O( number of ones in binary representation of x * number of ones in a binary representation of y )
Total time complexity: O(Q * log(maxX) * log(maxY))
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