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Find if there exists a direction for ranges such that no two range intersect
• Last Updated : 07 Dec, 2020

Given N ranges [L, R] with velocities vel[]. The task is to assign each range a direction i.e. either left or right. All the ranges will start moving in the assigned direction at time t = 0. Find, if there is an assignment of directions possible given that no two ranges overlap at infinite time.
Examples:

Input: range[][] = {{1, 2}, {2, 5}, {3, 10}, {4, 4}, {5, 7}},
vel[] = {3, 1, 1, 1, 10}
Output: No
Intervals {2, 5}, {3, 10} and {4, 4} share a common point 4
and have the same velocity.
Input: range[][] = {{1, 2}, {2, 5}, {3, 10}, {4, 4}, {5, 7}},
vel[] = {3, 1, 11, 1, 10}
Output: Yes

Approach:

• At infinite time, if there are two ranges with different velocities then they can never overlap irrespective of their directions since range with greater velocity will always be ahead of that with slower velocity.
• Now, consider ranges with same velocity. If there exist a point lying in at least 3 ranges with same velocity then the assignment of directions is not possible because even if two of these ranges are assigned a different direction but the third range will definitely intersect with the any one of the other ranges with the same direction.
• So, for all the velocities, find whether there exists at least 3 ranges such that they share a common point, if yes then assignment is not possible. Otherwise it is possible.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#define MAX 100001``using` `namespace` `std;` `// Structure to hold details of``// each interval``typedef` `struct``{``    ``int` `l, r, v;``} interval;` `// Comparator to sort intervals``// based on velocity``bool` `cmp(interval a, interval b)``{``    ``return` `a.v < b.v;``}` `// Function that returns true if the``// assignment of directions is possible``bool` `isPossible(``int` `range[][3], ``int` `N)``{``    ``interval test[N];``    ``for` `(``int` `i = 0; i < N; i++) {``        ``test[i].l = range[i][0];``        ``test[i].r = range[i][1];``        ``test[i].v = range[i][2];``    ``}` `    ``// Sort the intervals based on velocity``    ``sort(test, test + N, cmp);` `    ``for` `(``int` `i = 0; i < N; i++) {``        ``int` `count[MAX] = { 0 };``        ``int` `current_velocity = test[i].v;` `        ``int` `j = i;` `        ``// Test the condition for all intervals``        ``// with same velocity``        ``while` `(j < N && test[j].v == current_velocity) {``            ``for` `(``int` `k = test[j].l; k <= test[j].r; k++) {``                ``count[k]++;` `                ``// If for any velocity, 3 or more intervals``                ``// share a common point return false``                ``if` `(count[k] >= 3)``                    ``return` `false``;``            ``}``            ``j++;``        ``}` `        ``i = j - 1;``    ``}` `    ``return` `true``;``}` `// Driver code``int` `main()``{``    ``int` `range[][3] = { { 1, 2, 3 },``                       ``{ 2, 5, 1 },``                       ``{ 3, 10, 1 },``                       ``{ 4, 4, 1 },``                       ``{ 5, 7, 10 } };``    ``int` `n = ``sizeof``(range) / ``sizeof``(range[0]);` `    ``if` `(isPossible(range, n))``        ``cout << ``"Yes"``;``    ``else``        ``cout << ``"No"``;` `    ``return` `0;``}`

## Java

 `// Java implementation of the approach``import` `java.util.*;` `class` `GFG``{``static` `int` `MAX = ``100001``;` `// Structure to hold details of``// each interval``static` `class` `interval``{``    ``int` `l, r, v;``}` `static` `class` `Sort ``implements` `Comparator``{``    ``// Comparator to sort intervals``    ``// based on velocity``    ``public` `int` `compare(interval a, interval b)``    ``{``        ``return` `(a.v < b.v ? ``1` `: ``0``);``    ``}``}` `// Function that returns true if the``// assignment of directions is possible``static` `boolean` `isPossible(``int` `range[][], ``int` `N)``{``    ``interval test[] = ``new` `interval[N];``    ``for` `(``int` `i = ``0``; i < N; i++)``    ``{``        ``test[i] = ``new` `interval();``        ``test[i].l = range[i][``0``];``        ``test[i].r = range[i][``1``];``        ``test[i].v = range[i][``2``];``    ``}` `    ``// Sort the intervals based on velocity``    ``Arrays.sort(test, ``new` `Sort());` `    ``for` `(``int` `i = ``0``; i < N; i++)``    ``{``        ``int` `count[] = ``new` `int``[MAX];``        ``int` `current_velocity = test[i].v;` `        ``int` `j = i;` `        ``// Test the condition for all intervals``        ``// with same velocity``        ``while` `(j < N && test[j].v == current_velocity)``        ``{``            ``for` `(``int` `k = test[j].l; k <= test[j].r; k++)``            ``{``                ``count[k]++;` `                ``// If for any velocity, 3 or more intervals``                ``// share a common point return false``                ``if` `(count[k] >= ``3``)``                    ``return` `false``;``            ``}``            ``j++;``        ``}``        ``i = j - ``1``;``    ``}``    ``return` `true``;``}` `// Driver code``public` `static` `void` `main(String args[])``{``    ``int` `range[][] = {{ ``1``, ``2``, ``3` `},``                     ``{ ``2``, ``5``, ``1` `},``                     ``{ ``3``, ``10``, ``1` `},``                     ``{ ``4``, ``4``, ``1` `},``                     ``{ ``5``, ``7``, ``10` `}};``    ``int` `n = range.length;` `    ``if` `(isPossible(range, n))``        ``System.out.println(``"Yes"``);``    ``else``        ``System.out.println(``"No"``);``}``}` `// This code is contributed by Arnab Kundu`

## Python

 `# Python implementation of the approach``MAX` `=` `100001` `# Function that returns true if the``# assignment of directions is possible``def` `isPossible(rangee, N):``    ` `    ``# Structure to hold details of``    ``# each interval``    ``test ``=` `[[``0` `for` `x ``in` `range``(``3``)] ``for` `x ``in` `range``(N)]``    ``for` `i ``in` `range``(N):``        ``test[i][``0``] ``=` `rangee[i][``0``]``        ``test[i][``1``] ``=` `rangee[i][``1``]``        ``test[i][``2``] ``=` `rangee[i][``2``]``        ` `    ``# Sort the intervals based on velocity``    ``test.sort(key ``=` `lambda` `x: x[``2``])``    ``for` `i ``in` `range``(N):``        ``count ``=` `[``0``] ``*` `MAX``        ``current_velocity ``=` `test[i][``2``]``        ``j ``=` `i``        ` `        ``# Test the condition for all intervals``        ``# with same velocity``        ``while` `(j < N ``and` `test[j][``2``] ``=``=` `current_velocity):``            ``for` `k ``in` `range``(test[j][``0``], test[j][``1``] ``+` `1``):``                ``count[k] ``+``=` `1``                ` `                ``# If for any velocity, 3 or more intervals``                ``# share a common poreturn false``                ``if` `(count[k] >``=` `3``):``                    ``return` `False``            ``j ``+``=` `1``        ``i ``=` `j ``-` `1``    ` `    ``return` `True``    ` `# Driver code``rangee ``=` `[[``1``, ``2``, ``3``] ,[``2``, ``5``, ``1``] ,[``3``, ``10``, ``1``],``        ``[``4``, ``4``, ``1` `],[``5``, ``7``, ``10` `]]``n ``=` `len``(rangee)``if` `(isPossible(rangee, n)):``    ``print``(``"Yes"``)``else``:``    ``print``(``"No"``)``    ` `# This code is contributed by SHUBHAMSINGH10`

## C#

 `// C# implementation of the approach``using` `System;``using` `System.Collections;``using` `System.Collections.Generic;` `class` `GFG``{``    ` `// Structure to hold details of``// each interval``public` `class` `interval``{``    ``public` `int` `l, r, v;``}` `    ` `public` `class` `sortHelper : IComparer``{``   ``public` `int` `Compare(interval a, interval b)``   ``{``      ``return` `(a.v < b.v ? 1 : 0);``   ``}``}``    ` `static` `int` `MAX = 100001;` `// Function that returns true if the``// assignment of directions is possible``static` `bool` `isPossible(``int` `[,]range, ``int` `N)``{``    ``interval []test = ``new` `interval[N];``    ``for` `(``int` `i = 0; i < N; i++)``    ``{``        ``test[i] = ``new` `interval();``        ``test[i].l = range[i, 0];``        ``test[i].r = range[i, 1];``        ``test[i].v = range[i, 2];``    ``}` `    ``// Sort the intervals based on velocity``    ``Array.Sort(test, ``new` `sortHelper());` `    ``for` `(``int` `i = 0; i < N; i++)``    ``{``        ``int` `[]count = ``new` `int``[MAX];``        ``int` `current_velocity = test[i].v;` `        ``int` `j = i;` `        ``// Test the condition for all intervals``        ``// with same velocity``        ``while` `(j < N && test[j].v == current_velocity)``        ``{``            ``for` `(``int` `k = test[j].l; k <= test[j].r; k++)``            ``{``                ``count[k]++;` `                ``// If for any velocity, 3 or more intervals``                ``// share a common point return false``                ``if` `(count[k] >= 3)``                    ``return` `false``;``            ``}``            ``j++;``        ``}``        ``i = j - 1;``    ``}``    ``return` `true``;``}` `// Driver code``public` `static` `void` `Main(``string` `[]args)``{``    ``int` `[,]range = {{ 1, 2, 3 },``                     ``{ 2, 5, 1 },``                     ``{ 3, 10, 1 },``                     ``{ 4, 4, 1 },``                     ``{ 5, 7, 10 }};``                     ` `    ``int` `n = range.GetLength(0);` `    ``if` `(isPossible(range, n))``        ``Console.WriteLine(``"Yes"``);``    ``else``        ``Console.WriteLine(``"No"``);``}``}` `// This code is contributed by rutvik_56`
Output:
`No`

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