# Number of intersections between two ranges

Given **N** ranges of type_{1} ranges and **M** ranges of type_{2}.The task is to find the total number of intersections between all possible type_{1} and type_{2} range pairs. All start and end points of type_{1} and type_{2} ranges are given.

**Examples:**

Input :N = 2, M = 2

type1[ ] = { { 1, 3 }, { 5, 9 } }

type2[ ] = { { 2, 8 }, { 9, 12 } }

Output :3

Range {2, 8} intersects with type_{1}ranges {1, 3} and {5, 9}

Range {9, 12} intersects with {5, 9} only.

So the total number of intersections is 3.

Input :N = 3, M = 1

type1[ ] = { { 1, 8 }, { 5, 10 }, { 14, 28 }

type2[ ] = { { 2, 8 } }

Output :2

**Approach:**

- Idea is to use inclusion-exclusion method to determine the total number of intersections.
- Total possible number of intersections are
**n * m**. Now subtract those count of type_{1}ranges which do not intersect with i^{th}type_{2}range. - Those type
_{1}ranges will not intersect with i^{th}type_{2}range which ends before starts of i^{th}type_{2}range and starts after the end of i^{th}type_{2}range. - This count can be determined by using binary search . The C++ inbuilt function
**upper_bound**can be used directly.

Below is the implementation of above approach:

## C++

`// C++ implementation of above approach ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return total ` `// number of intersections ` `int` `FindIntersection(pair<` `int` `, ` `int` `> type1[], ` `int` `n, ` ` ` `pair<` `int` `, ` `int` `> type2[], ` `int` `m) ` `{ ` ` ` ` ` `// Maximum possible number ` ` ` `// of intersections ` ` ` `int` `ans = n * m; ` ` ` ` ` `vector<` `int` `> start, end; ` ` ` `for` `(` `int` `i = 0; i < n; i++) { ` ` ` ` ` `// Store all starting ` ` ` `// points of type1 ranges ` ` ` `start.push_back(type1[i].first); ` ` ` ` ` `// Store all endpoints ` ` ` `// of type1 ranges ` ` ` `end.push_back(type1[i].second); ` ` ` `} ` ` ` ` ` `sort(start.begin(), start.end()); ` ` ` `sort(end.begin(), end.end()); ` ` ` ` ` `for` `(` `int` `i = 0; i < m; i++) { ` ` ` ` ` `// Starting point of type2 ranges ` ` ` `int` `L = type2[i].first; ` ` ` ` ` `// Ending point of type2 ranges ` ` ` `int` `R = type2[i].second; ` ` ` ` ` `// Subtract those ranges which ` ` ` `// are starting after R ` ` ` `ans -= (start.end() - ` ` ` `upper_bound(start.begin(), start.end(), R)); ` ` ` ` ` `// Subtract those ranges which ` ` ` `// are ending before L ` ` ` `ans -= ` ` ` `(upper_bound(end.begin(), end.end(), L - 1) ` ` ` `- end.begin()); ` ` ` `} ` ` ` ` ` `return` `ans; ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` ` ` `pair<` `int` `, ` `int` `> type1[] = ` ` ` `{ { 1, 2 }, { 2, 3 }, { 4, 5 }, { 6, 7 } }; ` ` ` ` ` `pair<` `int` `, ` `int` `> type2[] = ` ` ` `{ { 1, 5 }, { 2, 3 }, { 4, 7 }, { 5, 7 } }; ` ` ` ` ` `int` `n = ` `sizeof` `(type1) / (` `sizeof` `(type1[0])); ` ` ` `int` `m = ` `sizeof` `(type2) / ` `sizeof` `(type2[0]); ` ` ` ` ` `cout << FindIntersection(type1, n, type2, m); ` ` ` ` ` `return` `0; ` `} ` |

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## Python3

`# Python3 implementation of above approach ` `from` `bisect ` `import` `bisect as upper_bound ` ` ` `# Function to return total ` `# number of intersections ` `def` `FindIntersection(type1, n, type2, m): ` ` ` ` ` `# Maximum possible number ` ` ` `# of intersections ` ` ` `ans ` `=` `n ` `*` `m ` ` ` ` ` `start ` `=` `[] ` ` ` `end ` `=` `[] ` ` ` `for` `i ` `in` `range` `(n): ` ` ` ` ` `# Store all starting ` ` ` `# points of type1 ranges ` ` ` `start.append(type1[i][` `0` `]) ` ` ` ` ` `# Store all endpoints ` ` ` `# of type1 ranges ` ` ` `end.append(type1[i][` `1` `]) ` ` ` ` ` `start ` `=` `sorted` `(start) ` ` ` `start ` `=` `sorted` `(end) ` ` ` ` ` `for` `i ` `in` `range` `(m): ` ` ` ` ` `# Starting poof type2 ranges ` ` ` `L ` `=` `type2[i][` `0` `] ` ` ` ` ` `# Ending poof type2 ranges ` ` ` `R ` `=` `type2[i][` `1` `] ` ` ` ` ` `# Subtract those ranges which ` ` ` `# are starting after R ` ` ` `ans ` `-` `=` `(` `len` `(start)` `-` `upper_bound(start, R)) ` ` ` ` ` `# Subtract those ranges which ` ` ` `# are ending before L ` ` ` `ans ` `-` `=` `(upper_bound(end, L ` `-` `1` `)) ` ` ` ` ` `return` `ans ` ` ` `# Driver Code ` `type1 ` `=` `[ [ ` `1` `, ` `2` `], [ ` `2` `, ` `3` `], ` ` ` `[ ` `4` `, ` `5` `], [ ` `6` `, ` `7` `] ] ` ` ` `type2 ` `=` `[ [ ` `1` `, ` `5` `], [ ` `2` `, ` `3` `], ` ` ` `[ ` `4` `, ` `7` `], [ ` `5` `, ` `7` `] ] ` ` ` `n ` `=` `len` `(type1) ` `m ` `=` `len` `(type2) ` ` ` `print` `(FindIntersection(type1, n, type2, m)) ` ` ` `# This code is contributed by Mohit Kumar ` |

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**Output:**

9

**Time Complexity:** O(M*log(N))

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