Minimum steps needed to cover a sequence of points on an infinite grid

Given an infinite grid, initial cell position (x, y) and a sequence of other cell position which needs to be covered in the given order. The task is to find the minimum number of steps needed to travel to all those cells.
Note: Movement can be done in any of the eight possible directions from a given cell i.e from cell (x, y) you can move to any of the following eight positions:(x-1, y+1), (x-1, y), (x-1, y-1), (x, y-1), (x+1, y-1), (x+1, y), (x+1, y+1), (x, y+1) is possible.

Examples:

Input: points[] = [(0, 0), (1, 1), (1, 2)]
Output: 2
Move from (0, 0) to (1, 1) in 1 step(diagonal) and
then from (1, 1) to (1, 2) in 1 step (rightwards)

Input: points[] = [{4, 6}, {1, 2}, {4, 5}, {10, 12}]
Output: 14
Move from (4, 6) -> (3, 5) -> (2, 4) -> (1, 3) ->
(1, 2) -> (2, 3) -> (3, 4) ->
(4, 5) -> (5, 6) -> (6, 7) ->
(7, 8) -> (8, 9) -> (9, 10) -> (10, 11) -> (10, 12)

Approach: Since all the given points are to be covered in the specified order. Find the minimum number of steps required to reach from a starting point to next point, then the sum of all such minimum steps for covering all the points would be the answer. One way to reach from a point (x1, y1) to (x2, y2) is to move abs(x2-x1) steps in the horizontal direction and abs(y2-y1) steps in the vertical direction, but this is not the shortest path to reach (x2, y2). The best way would be to cover the maximum possible distance in a diagonal direction and remaining in horizontal or vertical direction.
If we look closely this just reduces to the maximum of abs(x2-x1) and abs(y2-y1). Traverse for all points and summation of all diagonal distance will be the answer.

Below is the implementation of the above approach:

C++

// C++ program to cover a sequence of points
// in minimum steps in a given order.
#include <bits/stdc++.h>
using namespace std;

// cell structure denoted as point
struct point {
    int x, y;
};

// function to give minimum steps to
// move from point p1 to p2
int shortestPath(point p1, point p2)
{
    // dx is total horizontal
    // distance to be covered
    int dx = abs(p1.x - p2.x);

    // dy is total vertical
    // distance to be covered
    int dy = abs(p1.y - p2.y);

    // required answer is
    // maximum of these two
    return max(dx, dy);
}

// Function to return the minimum steps
int coverPoints(point sequence[], int size)
{
    int stepCount = 0;

    // finding steps for each
    // consecutive point in the sequence
    for (int i = 0; i < size - 1; i++) {
        stepCount += shortestPath(sequence[i],
                                  sequence[i + 1]);
    }

    return stepCount;
}

// Driver code
int main()
{
    // arr stores sequence of points
    // that are to be visited
    point arr[] = { { 4, 6 }, { 1, 2 }, { 4, 5 }, { 10, 12 } };

    int n = sizeof(arr) / sizeof(arr[0]);
    cout << coverPoints(arr, n);
}

Java

// Java program to cover a
// sequence of points in
// minimum steps in a given order.
import java.io.*;
import java.util.*;
import java.lang.*;

// class denoted as point
class point
{
    int x, y;
    point(int a, int b)
    {
        x = a;
        y = b;

    }
}

class GFG
{
// function to give minimum
// steps to move from point
// p1 to p2
static int shortestPath(point p1,
                        point p2)
{
    // dx is total horizontal
    // distance to be covered
    int dx = Math.abs(p1.x - p2.x);

    // dy is total vertical
    // distance to be covered
    int dy = Math.abs(p1.y - p2.y);

    // required answer is
    // maximum of these two
    return Math.max(dx, dy);
}

// Function to return
// the minimum steps
static int coverPoints(point sequence[],
                       int size)
{
    int stepCount = 0;

    // finding steps for
    // each consecutive
    // point in the sequence
    for (int i = 0; i < size - 1; i++)
    {
        stepCount += shortestPath(sequence[i],
                                  sequence[i + 1]);
    }

    return stepCount;
}

// Driver code
public static void main(String args[])
{
    // arr stores sequence of points
    // that are to be visited
    point arr[] = new point[4];
    arr[0] = new point(4, 6);
    arr[1] = new point(1, 2);
    arr[2] = new point(4, 5);
    arr[3] = new point(10, 12);

    int n = arr.length;
    System.out.print(coverPoints(arr, n));
}
}

Python3

# Python program to cover a sequence of points
# in minimum steps in a given order.

# function to give minimum steps to
# move from pop1 to p2
def shortestPath(p1, p2):

    # dx is total horizontal
    # distance to be covered
    dx = abs(p1[0] - p2[0])

    # dy is total vertical
    # distance to be covered
    dy = abs(p1[1] - p2[1])

    # required answer is
    # maximum of these two
    return max(dx, dy)

# Function to return the minimum steps
def coverPoints(sequence, size):

    stepCount = 0

    # finding steps for each
    # consecutive poin the sequence
    for i in range(size-1):
        stepCount += shortestPath(sequence[i],sequence[i + 1])

    return stepCount

# Driver code
# arr stores sequence of points
# that are to be visited
arr =  [[4, 6] ,[ 1, 2 ], [ 4, 5] , [ 10, 12]]

n = len(arr)
print(coverPoints(arr, n))

# This code is contributed by shivanisinghss2110.

C#

// C# program to cover a
// sequence of points in
// minimum steps in a given order.

using System;
// class denoted as point
public class point
{
    public int x, y;
    public point(int a, int b)
    {
        x = a;
        y = b;

    }
}

public class GFG
{
    // function to give minimum
    // steps to move from point
    // p1 to p2[]
    static int shortestPath(point p1,
                            point p2)
    {
        // dx is total horizontal
        // distance to be covered
        int dx = Math.Abs(p1.x - p2.x);

        // dy is total vertical
        // distance to be covered
        int dy = Math.Abs(p1.y - p2.y);

        // required answer is
        // maximum of these two
        return Math.Max(dx, dy);
    }

    // Function to return
    // the minimum steps
    static int coverPoints(point []sequence,
                           int size)
    {
        int stepCount = 0;

        // finding steps for
        // each consecutive
        // point in the sequence
        for (int i = 0; i < size - 1; i++)
        {
            stepCount += shortestPath(sequence[i],
                                      sequence[i + 1]);
        }

        return stepCount;
    }

    // Driver code
    public static void Main()
    {
        // arr stores sequence of points
        // that are to be visited
        point []arr = new point[4];
        arr[0] = new point(4, 6);
        arr[1] = new point(1, 2);
        arr[2] = new point(4, 5);
        arr[3] = new point(10, 12);

        int n = arr.Length;
        Console.WriteLine(coverPoints(arr, n));
    }
}
// This code is contributed by Rajput-Ji

# Traversing from one point to another point
# storing the minimum number of steps
def traversal_steps(points):
    minSteps = 0
    for p in range(len(points)-1):

        # taking the manhattan distance between
        # x and y-coordinates
        d1 = abs(points[p][0] - points[p+1][0])
        d2 = abs(points[p][1] - points[p+1][1])

        # adding the maximum among the two to the
        # running steps parameter
        minSteps += max(d1,d2)
    return (minSteps)

# Main Driver Code
if __name__ == '__main__':
    points = [(0,0),(1,1),(1,2)]
    print (traversal_steps(points))
    points = [(4,6),(1,2),(4,5),(10,12)]
    print (traversal_steps(points))

<script>
// JacaScript program to cover a sequence of points
// in minimum steps in a given order.

// function to give minimum steps to
// move from pop1 to p2
function shortestPath(p1, p2){

    // dx is total horizontal
    // distance to be covered
    let dx = Math.abs(p1[0] - p2[0])

    // dy is total vertical
    // distance to be covered
    let dy = Math.abs(p1[1] - p2[1])

    // required answer is
    // maximum of these two
    return Math.max(dx, dy)

}

// Function to return the minimum steps
function coverPoints(sequence, size){

    let stepCount = 0

    // finding steps for each
    // consecutive poin the sequence
    for(let i=0;i<(size-1);i++)
        stepCount += shortestPath(sequence[i],sequence[i + 1])

    return stepCount

}

// Driver code
// arr stores sequence of points
// that are to be visited
let arr = [[4, 6] ,[ 1, 2 ], [ 4, 5] , [ 10, 12]]

let n = arr.length
document.write(coverPoints(arr, n))

// This code is contributed by shivanisinghss2110.
</script>

Output:

14

Time Complexity: O(N)