Minimum Numbers of cells that are connected with the smallest path between 3 given cells

Given coordinates of 3 cells (X1, Y1), (X2, Y2) and (X3, Y3) of a matrix. The task is to find the minimum path which connects all three of these cells and print the count of all the cells that are connected through this path. 

Note: Only possible moves are up, down, left and right.

Examples:

Input: X1 = 0, Y1 = 0, X2 = 1, Y2 = 1, X3 = 2 and Y3 = 2 
Output: 5 
(0, 0), (1, 0), (1, 1), (1, 2), (2, 2) are the required cells.

Input: X1 = 0, Y1 = 0, X2 = 2, Y2 = 0, X3 = 1 and Y3 = 1 
Output: 4 

Approach:

First sort the cells from the one with minimum row number at first to one with maximum row number at last. Also, store minimum column number and maximum column number among these three cells in variable MinCol and MaxCol respectively. 

After that, store row number of the middle cell(from sorted cells) in variable MidRow and mark all the cells of this MidRow from MinCol to MaxCol. 
Now our final step will be to mark all the column number of 1st and 3rd cell till they reach MidRow. 

Here, marking means we will store the required cells coordinate in a set. Thus, our answer will be size of this set.

Below is the implementation of the above approach: 

C++

// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;

// Function to return the minimum cells that are
// connected via the minimum length path
int Minimum_Cells(vector<pair<int, int> > v)
{
    int col[3], i, j;
    for (i = 0; i < 3; i++) {
        int column_number = v[i].second;

        // Array to store column number
        // of the given cells
        col[i] = column_number;
    }

    sort(col, col + 3);

    // Sort cells in ascending
    // order of row number
    sort(v.begin(), v.end());

    // Middle row number
    int MidRow = v[1].first;

    // Set pair to store required cells
    set<pair<int, int> > s;

    // Range of column number
    int Maxcol = col[2], MinCol = col[0];

    // Store all cells of middle row
    // within column number range
    for (i = MinCol; i <= Maxcol; i++) {
        s.insert({ MidRow, i });
    }

    for (i = 0; i < 3; i++) {
        if (v[i].first == MidRow)
            continue;

        // Final step to store all the column number
        // of 1st and 3rd cell upto MidRow
        for (j = min(v[i].first, MidRow);
             j <= max(v[i].first, MidRow); j++) {
            s.insert({ j, v[i].second });
        }
    }

    return s.size();
}

// Driver Function
int main()
{
    // vector pair to store X, Y, Z
    vector<pair<int, int> > v = { { 0, 0 }, { 1, 1 }, { 2, 2 } };

    cout << Minimum_Cells(v);

    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
import java.awt.Point;
public class Main
{
    // Function to return the minimum cells that are
    // connected via the minimum length path
    static int Minimum_Cells(Vector<Point> v)
    {
        int[] col = new int[3];
        int i, j;
        for (i = 0; i < 3; i++) {
            int column_number = v.get(i).y;

            // Array to store column number
            // of the given cells
            col[i] = column_number;
        }

        Arrays.sort(col);

        // Middle row number
        int MidRow = v.get(1).x;

        // Set pair to store required cells
        Set<Point> s = new HashSet<Point>();

        // Range of column number
        int Maxcol = col[2], MinCol = col[0];

        // Store all cells of middle row
        // within column number range
        for (i = MinCol; i <= Maxcol; i++) {
            s.add(new Point(MidRow, i));
        }

        for (i = 0; i < 3; i++) {
            if (v.get(i).x == MidRow)
                continue;

            // Final step to store all the column number
            // of 1st and 3rd cell upto MidRow
            for (j = Math.min(v.get(i).x, MidRow);
                 j <= Math.max(v.get(i).x, MidRow); j++) {
                s.add(new Point(j, v.get(i).x));
            }
        }

        return s.size();
    }

  // Driver code
    public static void main(String[] args)
    {

        // vector pair to store X, Y, Z
        Vector<Point> v = new Vector<Point>();
        v.add(new Point(0, 0));
        v.add(new Point(1, 1));
        v.add(new Point(2, 2));

        System.out.print(Minimum_Cells(v));
    }
}

// This code is contributed by mukesh07.

Python3

# Python3 implementation of the approach

# Function to return the minimum cells that
# are connected via the minimum length path
def Minimum_Cells(v) :

    col = [0] * 3
    for i in range(3) :
        column_number = v[i][1]

        # Array to store column number
        # of the given cells
        col[i] = column_number

    col.sort()

    # Sort cells in ascending order
    # of row number
    v.sort()

    # Middle row number
    MidRow = v[1][0]

    # Set pair to store required cells
    s = set()

    # Range of column number
    Maxcol = col[2]
    MinCol = col[0]

    # Store all cells of middle row
    # within column number range
    for i in range(MinCol, int(Maxcol) + 1) :
        s.add((MidRow, i))

    for i in range(3) :
        if (v[i][0] == MidRow) :
            continue;

        # Final step to store all the column
        # number of 1st and 3rd cell upto MidRow
        for j in range(min(v[i][0], MidRow),
                       max(v[i][0], MidRow) + 1) :
            s.add((j, v[i][1]));

    return len(s)

# Driver Code
if __name__ == "__main__" :

    # vector pair to store X, Y, Z
    v = [(0, 0 ), ( 1, 1 ), ( 2, 2 )]

    print(Minimum_Cells(v))

# This code is contributed by Ryuga

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;
class GFG {

    // Function to return the minimum cells that are
    // connected via the minimum length path
    static int Minimum_Cells(List<Tuple<int, int>> v)
    {
        int[] col = new int[3];
        int i, j;
        for (i = 0; i < 3; i++) {
            int column_number = v[i].Item2;

            // Array to store column number
            // of the given cells
            col[i] = column_number;
        }

        Array.Sort(col);

        // Sort cells in ascending
        // order of row number
        v.Sort();

        // Middle row number
        int MidRow = v[1].Item1;

        // Set pair to store required cells
        HashSet<Tuple<int, int>> s = new HashSet<Tuple<int, int>>();

        // Range of column number
        int Maxcol = col[2], MinCol = col[0];

        // Store all cells of middle row
        // within column number range
        for (i = MinCol; i <= Maxcol; i++) {
            s.Add(new Tuple<int,int>(MidRow, i));
        }

        for (i = 0; i < 3; i++) {
            if (v[i].Item1 == MidRow)
                continue;

            // Final step to store all the column number
            // of 1st and 3rd cell upto MidRow
            for (j = Math.Min(v[i].Item1, MidRow);
                 j <= Math.Max(v[i].Item1, MidRow); j++) {
                s.Add(new Tuple<int,int>(j, v[i].Item1));
            }
        }

        return s.Count;
    }

  static void Main()
  {

    // vector pair to store X, Y, Z
    List<Tuple<int, int>> v = new List<Tuple<int, int>>();
    v.Add(new Tuple<int,int>(0, 0));
    v.Add(new Tuple<int,int>(1, 1));
    v.Add(new Tuple<int,int>(2, 2));

    Console.Write(Minimum_Cells(v));
  }
}

// This code is contributed by divyeshrabadiya07.

Javascript

<script>
    // Javascript implementation of the approach

    // Function to return the minimum cells that are
    // connected via the minimum length path
    function Minimum_Cells(v)
    {
        let col = new Array(3);
        let i, j;
        for (i = 0; i < 3; i++) {
            let column_number = v[i][1];

            // Array to store column number
            // of the given cells
            col[i] = column_number;
        }

        col.sort(function(a, b){return a - b});

        // Sort cells in ascending
        // order of row number
        v.sort();

        // Middle row number
        let MidRow = v[1][0];

        // Set pair to store required cells
        let s = new Set();

        // Range of column number
        let Maxcol = col[2], MinCol = col[0];

        // Store all cells of middle row
        // within column number range
        for (i = MinCol; i <= Maxcol; i++) {
            s.add([MidRow, i]);
        }

        for (i = 0; i < 3; i++) {
            if (v[i][0] == MidRow)
                continue;

            // Final step to store all the column number
            // of 1st and 3rd cell upto MidRow
            for (j = Math.min(v[i][0], MidRow);
                 j <= Math.max(v[i][0], MidRow); j++) {
                s.add([j, v[i][0]]);
            }
        }

        return s.size-2;
    }

    // vector pair to store X, Y, Z
    let v = [];
    v.push([0, 0]);
    v.push([1, 1]);
    v.push([2, 2]);

    document.write(Minimum_Cells(v));

    // This code is contributed by decode2207.
</script>

Output

5

Time complexity: O(nlogn) as it uses the sort function 

Space complexity: O(n) as it uses a set.