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Asked: 2024-09-25T03:03:57+05:30

Minimum Steps in Infinite Grid

Problem Description

You are in an infinite 2D grid where you can move in any of the 8 directions

 (x,y) to 
    (x-1, y-1), 
    (x-1, y)  , 
    (x-1, y+1), 
    (x  , y-1),
    (x  , y+1), 
    (x+1, y-1), 
    (x+1, y)  , 
    (x+1, y+1)

You are given a sequence of points and the order in which you need to cover the points.. Give the minimum number of steps in which you can achieve it. You start from the first point.

Problem Constraints

1 <= |A| <= 106
– 2 * 103 <= Ai, Bi <= 2 * 103
|A| == |B|

Input Format

Given two integer arrays A and B, where A[i] is x coordinate and B[i] is y coordinate of ith point respectively.

Output Format

Return an Integer, i.e minimum number of steps.

Example Input

Input 1:

A = [0, 1, 1] B = [0, 1, 2]

Example Output

Output 1:

2

Example Explanation

Explanation 1:

 Given three points are: (0, 0), (1, 1) and (1, 2).
 It takes 1 step to move from (0, 0) to (1, 1). It takes one more step to move from (1, 1) to (1, 2).
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    1 Answer

    1. To solve the problem of finding the minimum number of steps required to move through a sequence of points in an infinite 2D grid, we need to take into account the movement capabilities in 8 directions (diagonal, horizontal, and vertical). The key aspect to note is that the maximum of the absolute differences in x and y coordinates gives us the required number of steps to move from one point to another.

      Here’s a step-by-step breakdown of how to compute the minimum number of steps:

      1. **Understanding the Movement**: In an 8-directional movement, the distance (in terms of steps) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
      \[
      \text{steps} = \max(|x_2 – x_1|, |y_2 – y_1|)
      \]
      This formula works because you can cover the distance in either the x or y direction simultaneously by moving diagonally.

      2. **Iterating Through the Points**: Starting from the first point, we will calculate the number of steps to each subsequent point using the formula above and sum these steps to get the total.

      3. **Implementation**: Here’s how you can implement this logic in Python:

      “`python
      def min_steps(A, B):
      total_steps = 0
      # Start from the first point
      x_prev, y_prev = A[0], B[0]

      # Iterate through the points from the second to the last
      for i in range(1, len(A)):
      x_curr, y_curr = A[i], B[i]
      # Calculate steps from previous point to current point
      steps = max(abs(x_curr – x_prev), abs(y_curr – y_prev))
      total_steps += steps
      # Update previous point to current point
      x_prev, y_prev = x_curr, y_curr

      return total_steps

      # Example usage
      A = [0, 1, 1]
      B = [0, 1, 2]
      print(min_steps(A, B)) # Output: 2
      “`

      Explanation of the Code

      – We start by initializing `total_steps` to 0 and setting the previous coordinates to the first point in the lists \(A\) and \(B\).
      – We loop through the indices of the points starting from the second point (index 1).
      – For each point, we calculate the number of steps needed to reach it from the previous point using the `max` function with the absolute differences of their coordinates.
      – We accumulate these steps in `total_steps`.
      – Finally, we return the total number of steps.

      This implementation efficiently computes the result in \(O(n)\) time, where \(n\) is the number of points, making it suitable even for the maximum constraints given.