Minimum number of Straight Lines to connect all the given Points

Given 2d integer array arr[][] containing N coordinates each of type {X, Y}, the task is to find the minimum number of straight lines required to connect all the points of the array.

Examples:

Input: arr[][] = {{1, 7}, {2, 6}, {3, 5}, {4, 4}, {5, 4}, {6, 3}, {7, 2}, {8, 1}}
Output: 3
Explanation: The diagram represents the input points and the minimum straight lines required.
The following 3 lines can be drawn to represent the line chart:
=> Line 1 (in red) from (1, 7) to (4, 4) passing through (1, 7), (2, 6), (3, 5), and (4, 4).
=> Line 2 (in blue) from (4, 4) to (5, 4).
=> Line 3 (in green) from (5, 4) to (8, 1) passing through (5, 4), (6, 3), (7, 2), and (8, 1).
It can be shown that it is not possible to represent the line chart using less than 3 lines.

Input: arr[][] = {{3, 4}, {1, 2}, {7, 8}, {2, 3}}
Output: 1
Explanation: A single line passing through all the points is enough to connect them

Approach: The problem can be solved based on the following geometrical idea:

If the points are sorted according to their X axis value then to calculate the number of lines check for three consecutive points. 
If the slope of lines formed by first two points and the last two points are same then they can be connected by a single line.
Otherwise, we will need two different lines.

To calculate the slope we will use the formulae: 
    => slope 1 = (y2 - y1) / (x2 - x1)
    => slope 2 = (y3 - y2) / (x3 - x2)

If slope1 = slope2
(y2 - y1) / (x2 - x1) = (y3 - y2) / (x3 - x2)
(y3 - y2) * (x2 - x1) = (y2 - y1) * (x3 - x2).

Follow the steps mentioned below to implement the idea:

• Firstly sort the array based on the value of the X coordinate.
• Traverse the array from i = 0 to N-2:
  • If the above condition of slope1 = slope2 is satisfied then continue without incrementing the count of lines.
  • Otherwise, increment the count of straight lines required.
• Return the final count of straight lines as the answer. 

Below is the implementation of the above approach:

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to find minimum number of lines
int minimumLines(vector<vector<int> >& arr)
{
    int n = arr.size();

    // Base case when there is only one point,
    // then min lines = 0
    if (n == 1)
        return 0;

    // Sorting in ascending order of X coordinate
    sort(arr.begin(), arr.end());

    int numoflines = 1;

    // Traverse through points and check
    // whether the slopes matches or not.
    // If they does not match
    // increment the count of lines
    for (int i = 2; i < n; i++) {
        int x1 = arr[i][0];
        int x2 = arr[i - 1][0];
        int x3 = arr[i - 2][0];
        int y1 = arr[i][1];
        int y2 = arr[i - 1][1];
        int y3 = arr[i - 2][1];
        int slope1 = (y3 - y2) * (x2 - x1);
        int slope2 = (y2 - y1) * (x3 - x2);

        if (slope1 != slope2)
            numoflines++;
    }

    // Return the num of lines
    return numoflines;
}

// Driver Code
int main()
{
    vector<vector<int> > vect{ { 1, 7 }, { 2, 6 }, { 3, 5 }, { 4, 4 }, { 5, 4 }, { 6, 3 }, { 7, 2 }, { 8, 1 } };

    // Function call
    cout << minimumLines(vect);
    return 0;
}

// Java code to implement the approach
import java.io.*;
import java.util.*;

class GFG {
    // Function to find minimum number of lines
    public static int minimumLines(int arr[][])
    {
        int n = arr.length;

        // Base case when there is only one point,
        // then min lines = 0
        if (n == 1)
            return 0;

        // Sorting in ascending order of X coordinate
        Arrays.sort(arr,
                    (a, b) -> Integer.compare(a[0], b[0]));

        int numoflines = 1;

        // Traverse through points and check
        // whether the slopes matches or not.
        // If they does not match
        // increment the count of lines
        for (int i = 2; i < n; i++) {
            int x1 = arr[i][0];
            int x2 = arr[i - 1][0];
            int x3 = arr[i - 2][0];
            int y1 = arr[i][1];
            int y2 = arr[i - 1][1];
            int y3 = arr[i - 2][1];
            int slope1 = (y3 - y2) * (x2 - x1);
            int slope2 = (y2 - y1) * (x3 - x2);

            if (slope1 != slope2)
                numoflines++;
        }

        // Return the num of lines
        return numoflines;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int vect[][]
            = { { 1, 7 }, { 2, 6 }, { 3, 5 }, { 4, 4 },
                { 5, 4 }, { 6, 3 }, { 7, 2 }, { 8, 1 } };

        // Function call
        System.out.print(minimumLines(vect));
    }
}

// This code is contributed by Rohit Pradhan

# Python3 code to implement the approach

# Function to find minimum number of lines
def minimumLines(arr) :

    n = len(arr);

    # Base case when there is only one point,
    # then min lines = 0
    if (n == 1) :
        return 0;

    # Sorting in ascending order of X coordinate
    arr.sort();
    
    numoflines = 1;

    # Traverse through points and check
    # whether the slopes matches or not.
    # If they does not match
    # increment the count of lines
    for i in range(2, n) :
        x1 = arr[i][0];
        x2 = arr[i - 1][0];
        x3 = arr[i - 2][0];
        y1 = arr[i][1];
        y2 = arr[i - 1][1];
        y3 = arr[i - 2][1];
        slope1 = (y3 - y2) * (x2 - x1);
        slope2 = (y2 - y1) * (x3 - x2);

        if (slope1 != slope2) :
            numoflines += 1;

    # Return the num of lines
    return numoflines;

# Driver Code
if __name__ == "__main__" :

    vect = [ [ 1, 7 ], [ 2, 6 ], [ 3, 5 ], [ 4, 4 ], [ 5, 4 ], [ 6, 3 ], [ 7, 2 ], [ 8, 1 ] ];

    # Function call
    print(minimumLines(vect));
    
    # This code is contributed by AnkThon

// C# code to implement the approach

using System;
using System.Collections.Generic;

// Comparer class implementation
class Comparer : IComparer<int[]> {

    public int Compare(int[] x, int[] y)
    {
        return x[0].CompareTo(y[0]);
    }
}

public class GFG {
    // Function to find minimum number of lines
    public static int minimumLines(int[][] arr)
    {
        int n = arr.Length;

        // Base case when there is only one point,
        // then min lines = 0
        if (n == 1)
            return 0;

        // Sorting in ascending order of X coordinate
        Array.Sort<int[]>(arr, new Comparer());

        int numoflines = 1;

        // Traverse through points and check
        // whether the slopes matches or not.
        // If they does not match
        // increment the count of lines
        for (int i = 2; i < n; i++) {
            int x1 = arr[i][0];
            int x2 = arr[i - 1][0];
            int x3 = arr[i - 2][0];
            int y1 = arr[i][1];
            int y2 = arr[i - 1][1];
            int y3 = arr[i - 2][1];
            int slope1 = (y3 - y2) * (x2 - x1);
            int slope2 = (y2 - y1) * (x3 - x2);

            if (slope1 != slope2)
                numoflines++;
        }

        // Return the num of lines
        return numoflines;
    }
    public static void Main(string[] args)
    {
        int[][] vect = new int[][] {
            new int[] { 1, 7 }, new int[] { 2, 6 },
            new int[] { 3, 5 }, new int[] { 4, 4 },
            new int[] { 5, 4 }, new int[] { 6, 3 },
            new int[] { 7, 2 }, new int[] { 8, 1 }
        };

        // Function call
        Console.Write(minimumLines(vect));
    }
}

// This code is contributed by phasing17

 <script>
        // JavaScript code for the above approach

        // Function to find minimum number of lines
        function minimumLines(arr) {
            let n = arr.length;

            // Base case when there is only one point,
            // then min lines = 0
            if (n == 1)
                return 0;

            // Sorting in ascending order of X coordinate
            arr.sort(function (a, b) { return a[0] - b[0] });

            let numoflines = 1;

            // Traverse through points and check
            // whether the slopes matches or not.
            // If they does not match
            // increment the count of lines
            for (let i = 2; i < n; i++) {
                let x1 = arr[i][0];
                let x2 = arr[i - 1][0];
                let x3 = arr[i - 2][0];
                let y1 = arr[i][1];
                let y2 = arr[i - 1][1];
                let y3 = arr[i - 2][1];
                let slope1 = (y3 - y2) * (x2 - x1);
                let slope2 = (y2 - y1) * (x3 - x2);

                if (slope1 != slope2)
                    numoflines++;
            }

            // Return the num of lines
            return numoflines;
        }

        // Driver Code

        let vect = [[1, 7], [2, 6], [3, 5], [4, 4], [5, 4], [6, 3], [7, 2], [8, 1]];

        // Function call
        document.write(minimumLines(vect));

    // This code is contributed by Potta Lokesh
    </script>

3

Time complexity: O(N * logN) 
Auxiliary Space: O(1)