Maximum Manhattan Distance After K Changes

Problem Description

You are given a string s consisting of the characters 'N', 'S', 'E', and 'W', where s[i] indicates movements in an infinite grid. Initially, you are at the origin (0, 0). You can change at most k characters to any of the four directions. Find the maximum Manhattan distance from the origin that can be achieved at any time while performing the movements in order.

Key Insights

• The Manhattan distance is defined as |x_i - x_j| + |y_i - y_j|.
• Changing a direction can potentially increase the distance from the origin, depending on the current movements.
• Movement in the cardinal directions affects the x and y coordinates, and changing moves can optimize the distance.
• We need to consider both the initial movements and the effect of changing up to k moves.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we will keep track of the current position as we iterate through the movement string. For each movement, we calculate the Manhattan distance from the origin. We will also determine the number of moves that could be changed to maximize the distance. The algorithm involves:

1. Iterating through the string and updating the position based on the movements.
2. Calculating the current Manhattan distance after each move.
3. Considering the effect of changing up to k moves to achieve the maximum distance.

Code Solutions