Counting Triangles in a Rectangular space using BIT

Last Updated : 14 Jun, 2023

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

Insert x y - Insert a point (x, y) coordinate.
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) -            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# 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))

Counting Triangles in a Rectangular space using BIT

Saumye Malhotra

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Counting Triangles in a Rectangular space using BIT

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