Solution: Minimum Time Visiting All Points

Statement

You are given an array of $n$ points with integer coordinates on a 2D plane, points, where points[i] = [xi, yi]. Your task is to determine the minimum time in seconds required to visit all the points in the given order.

Movement rules:

1. In one second, you can perform any one of the following:

   1. Move vertically by one unit.

   2. Move horizontally by one unit.

   3. Move diagonally (1 unit vertically and 1 unit horizontally in 1 second).

2. You must visit the points in the exact sequence listed in the array.

3. You may pass through points that appear later in the order, but they will not count as visits.

Constraints:

• points.length $= n$

• $1 \leq n \leq 10^2$

• points[i].length $= 2$

• $-10^3 \leq$ points[i][0], points[i][1] $\leq 10^3$

Solution

The challenge lies in optimizing movement between points while considering the shortest path. As stated in the problem statement, movements can be made diagonally or along straight lines, with diagonal moves covering both $x$ and $y$ directions simultaneously. This makes the Math and Geometry pattern suitable for solving this problem as it focuses on coordinate differences and geometric properties to compute distances efficiently.

The key idea is that we iterate through consecutive points, calculating the absolute differences in their $x$ and $y$ coordinates. The time to move between two points is the larger of these differences, as diagonal movement optimizes travel. The total time is the sum of these values for all pairs.

Here are the detailed steps of the algorithm:

• Initialize a variable result to 0, storing the minimum time required to visit all points.

• Next, we start iterating through the points array from the first point to the last point, and in each iteration, we focus on two consecutive points:

  • The previous point, points[i - 1], represents the starting point.

  • The current point, points[i], represents the point to move to.

  • We calculate the absolute difference in the x-coordinates, xDiff, and y-coordinates, yDiff, using:

    • Horizontal difference: xDiff = abs(points[i][0] - points[i - 1][0])

    • Vertical difference: yDiff = abs(points[i][1] - points[i - 1][1])

    • These differences represent the distances between two points on a Cartesian plane.

  • Next, we determine the time to move between the points using max(xDiff, yDiff), as diagonal movement allows covering both dimensions (horizontal and vertical) simultaneously.

  • Add the calculated time to the result to maintain a running total.

• Once we’ve traversed over the points array, we return the accumulated result, representing the minimum time required to visit all points.

Let’s look at the following illustration to get a better understanding of the solution: