Maximum Manhattan Distance After K Changes

Medium

Uber

0 views

Topics:

StringsDynamic Programming

You are given a string s consisting of the characters 'N', 'S', 'E', and 'W', where s[i] indicates movements in an infinite grid:

'N' : Move north by 1 unit.
'S' : Move south by 1 unit.
'E' : Move east by 1 unit.
'W' : Move west by 1 unit.

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.

The Manhattan Distance between two cells (xᵢ, yᵢ) and (xⱼ, yⱼ) is |xᵢ - xⱼ| + |yᵢ - yⱼ|.

Example 1:

Input: s = "NWSE", k = 1 Output: 3

Explanation: Change s[2] from 'S' to 'N'. The string s becomes "NWNE".

The maximum Manhattan distance from the origin that can be achieved is 3.

Example 2:

Input: s = "NSWWEW", k = 3 Output: 6

Explanation: Change s[1] from 'S' to 'N', and s[4] from 'E' to 'W'. The string s becomes "NNWWWW".

What is the most efficient algorithm to solve this problem? Explain the time and space complexity. Consider edge cases and show code where appropriate.

Maximum Manhattan Distance After K Changes

Medium

Uber

0 views

Topics:

StringsDynamic Programming

You are given a string s consisting of the characters 'N', 'S', 'E', and 'W', where s[i] indicates movements in an infinite grid:

'N' : Move north by 1 unit.
'S' : Move south by 1 unit.
'E' : Move east by 1 unit.
'W' : Move west by 1 unit.

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.

The Manhattan Distance between two cells (xᵢ, yᵢ) and (xⱼ, yⱼ) is |xᵢ - xⱼ| + |yᵢ - yⱼ|.

Example 1:

Input: s = "NWSE", k = 1 Output: 3

Explanation: Change s[2] from 'S' to 'N'. The string s becomes "NWNE".

The maximum Manhattan distance from the origin that can be achieved is 3.

Example 2:

Input: s = "NSWWEW", k = 3 Output: 6

Explanation: Change s[1] from 'S' to 'N', and s[4] from 'E' to 'W'. The string s becomes "NNWWWW".

What is the most efficient algorithm to solve this problem? Explain the time and space complexity. Consider edge cases and show code where appropriate.

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What are the ranges for the values of x, y, and K? Can K be greater than the number of points?
Can the coordinates x and y be negative or non-integer values?
If there are multiple pairs of points that achieve the maximum Manhattan distance after K changes, is any valid pair acceptable as output?
If making all K changes doesn't increase the maximum Manhattan distance, should the original maximum distance be returned, or should I apply as many changes as possible?
If the input array is empty or null, what should be returned?

Brute Force Solution

Approach

The core idea is to explore all possible scenarios of modifying our data points. Since we can make changes up to a certain limit, we consider every possible combination of changes for each point and calculate the Manhattan distance to find the maximum.

Here's how the algorithm would work step-by-step:

For each data point, consider all possible changes you can make within the given limit.
Because there are two coordinates per point and we can change each by adding or subtracting from it, we can treat the allowed changes as a budget.
Systematically go through all ways to spend that budget on the x and y coordinates of the point.
After changing each point, calculate the Manhattan distance between every possible pair of the modified points.
Remember the biggest distance you've found so far. Compare this value with the distances calculated with this modification.
Once you have tried all possible ways to change the coordinates of all data points, the maximum Manhattan distance you have remembered is the answer.

Code Implementation

def maximum_manhattan_distance_after_k_changes_brute_force(
    points, allowed_changes):
    maximum_distance = 0

    for first_point_index in range(len(points)):
        for second_point_index in range(len(points)):
            if first_point_index == second_point_index:
                continue

            # Iterate through all possible change combinations
            for first_point_x_change in range(-allowed_changes, allowed_changes + 1):
                for first_point_y_change in range(-allowed_changes, allowed_changes + 1):
                    if abs(first_point_x_change) + abs(first_point_y_change) > allowed_changes:
                        continue

                    for second_point_x_change in range(-allowed_changes, allowed_changes + 1):
                        for second_point_y_change in range(-allowed_changes, allowed_changes + 1):
                            if abs(second_point_x_change) + abs(second_point_y_change) > allowed_changes:
                                continue

                            first_point = points[first_point_index]
                            second_point = points[second_point_index]

                            modified_first_point_x = first_point[0] + first_point_x_change
                            modified_first_point_y = first_point[1] + first_point_y_change
                            modified_second_point_x = second_point[0] + second_point_x_change
                            modified_second_point_y = second_point[1] + second_point_y_change

                            current_distance = abs(modified_first_point_x - modified_second_point_x) + \
                                abs(modified_first_point_y - modified_second_point_y)

                            # Keep track of largest distance.
                            maximum_distance = max(maximum_distance, current_distance)

    return maximum_distance

Big(O) Analysis

Time Complexity

O(n² * 4^k) – Let n be the number of data points and k be the maximum change allowed. For each point, we iterate through all possible allocations of the change 'k' to the x and y coordinates by adding or subtracting. Since we can allocate the budget k to x and y in increments and decrements (adding or subtracting), the total number of possible changes for each point can be expressed as O(4^k), considering all possibilities (add to x, subtract from x, add to y, subtract from y). Then we iterate through all possible pairs of modified points (n * n). Hence the total time complexity would be O(n² * 4^k). In terms of approximation, there are n * n pairs with each pair needing up to 4^k operations to check for possible changes.

Space Complexity

O(1) – The described algorithm iterates through possible changes to coordinates and computes Manhattan distances. It keeps track of the maximum distance found so far using a single variable. No auxiliary data structures, such as lists or hash maps, that scale with the input size (N, the number of data points) are used. Therefore, the auxiliary space used is constant, independent of the input size.

Optimal Solution

Approach

The key is to realize Manhattan distance can be maximized independently along each dimension. We transform the points to work with these independent max/min calculations, making the problem simpler and avoiding direct distance calculations.

Here's how the algorithm would work step-by-step:

Transform the original coordinates by adding and subtracting them to create two new coordinate sets. Think of this as rotating and scaling the coordinate system to simplify the distance calculation.
For each transformed coordinate set, find the maximum and minimum values among all the points. These represent the extremes along the new axes.
Consider how many 'moves' we can apply to increase the difference between the maximum and minimum. For each of the four extreme points (max x, min x, max y, min y), determine the change in location, either increasing the max or decreasing the min.
Apply the allowed 'moves' in a greedy fashion. Prioritize the coordinate where shifting the point will increase the difference between the max and min the most per move. Keep track of the moves you've used.
After applying all possible moves, the maximum Manhattan distance will be the sum of the difference between the maximum and minimum values in both transformed coordinate sets. This result is the answer.

Code Implementation

def max_manhattan_distance(points, allowed_moves):
    transformed_x = [point[0] + point[1] for point in points]
    transformed_y = [point[0] - point[1] for point in points]

    max_x = max(transformed_x)
    min_x = min(transformed_x)
    max_y = max(transformed_y)
    min_y = min(transformed_y)

    # We greedily apply changes to the most beneficial coordinate
    while allowed_moves > 0:
        increase_max_x = max_x - max(point[0] + point[1] for point in points) + 1
        decrease_min_x = min(point[0] + point[1] for point in points) - min_x + 1
        increase_max_y = max_y - max(point[0] - point[1] for point in points) + 1
        decrease_min_y = min(point[0] - point[1] for point in points) - min_y + 1

        best_improvement = 0
        best_coordinate = None

        if increase_max_x > best_improvement:
            best_improvement = increase_max_x
            best_coordinate = 'max_x'
        if decrease_min_x > best_improvement:
            best_improvement = decrease_min_x
            best_coordinate = 'min_x'
        if increase_max_y > best_improvement:
            best_improvement = increase_max_y
            best_coordinate = 'max_y'
        if decrease_min_y > best_improvement:
            best_improvement = decrease_min_y
            best_coordinate = 'min_y'

        if best_coordinate == 'max_x':
            max_x += 1
        elif best_coordinate == 'min_x':
            min_x -= 1
        elif best_coordinate == 'max_y':
            max_y += 1
        elif best_coordinate == 'min_y':
            min_y -= 1

        allowed_moves -= 1

    # The final answer is the sum of the ranges in each dimension
    return (max_x - min_x) + (max_y - min_y)

Big(O) Analysis

Time Complexity

O(n) – The algorithm first transforms the n input points, taking O(n) time. Then, finding the maximum and minimum values for each transformed coordinate set takes another O(n) time, as it requires iterating through all points. Finally, determining the optimal moves involves iterating through the four extreme points and updating them based on K. The overall time complexity is therefore dominated by the initial transformations and finding the max/min, resulting in O(n) + O(n) which simplifies to O(n).

Space Complexity

O(1) – The algorithm transforms the original coordinates, but these transformations are done in place or using a fixed number of variables to store the maximum and minimum x and y transformed coordinates. It then greedily applies 'moves' using only a few extra variables to track the used moves and update the max/min values. Therefore, the auxiliary space used is constant and independent of the number of points N, resulting in O(1) space complexity.

Edge Cases

Case	How to Handle
Null input points array	Return 0 or throw an IllegalArgumentException, depending on the specifications and expected behavior.
Empty input points array	Return 0, as there are no points to compute a distance from.
Points array with only one point	Return 0, as there's no other point to calculate the Manhattan distance with.
K is larger than the maximum possible change for any coordinate	Cap K to the maximum change allowed to avoid unnecessary computations.
Points with large coordinate values that could lead to integer overflow after adding K	Use long data type to store intermediate calculations of the coordinates and distances to prevent integer overflow.
K is negative	Throw IllegalArgumentException or return an error value as negative changes are invalid.
All points are the same, and K is zero	The maximum Manhattan distance will be zero, return 0.
Large input size, potential for time complexity issues	Optimize the approach, possibly using a quadtree or other spatial indexing technique to reduce distance calculations.