What is the time complexity of Manhattan Distances of All Arrangements of Pieces?

The optimal solution for Manhattan Distances of All Arrangements of Pieces runs in O(m * n * m * n) time and uses O(1) space. The complexity is due to iterating through all pairs of positions on the grid, which is manageable given the constraints.

What is the brute force solution for Manhattan Distances of All Arrangements of Pieces?

The brute force approach for Manhattan Distances of All Arrangements of Pieces is Brute Force, with O(n²) time complexity and O(1) space complexity. We can calculate the Manhattan distance for every possible arrangement of k pieces on an m x n grid. This approach is straightforward but inefficient, as it involves generating all combinations of placements. The optimal Optimal Solution approach improves this to O(m * n * m * n).

What algorithm or data structure is used to solve Manhattan Distances of All Arrangements of Pieces?

Manhattan Distances of All Arrangements of Pieces uses the following concepts: Math, Combinatorics. The recommended patterns to study are: Combinatorial Counting, Mathematical Optimization.

What are the interview tips for Manhattan Distances of All Arrangements of Pieces?

Always clarify constraints and edge cases before diving into the solution. Discuss your thought process and approach before coding. Be prepared to optimize your solution if the brute force approach is too slow.

What are common mistakes when solving Manhattan Distances of All Arrangements of Pieces?

Not considering all pairs of positions correctly. Overlooking the need to use combinatorial counting for arrangements.

#3426

Manhattan Distances of All Arrangements of Pieces

Hard

MathCombinatoricsCombinatorial CountingMathematical Optimization

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(m * n * m * n)
Space	O(1)	O(1)

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Intuition

Time O(m * n * m * n)Space O(1)

Instead of calculating distances for every arrangement, we can fix pairs of pieces and calculate how many arrangements can be formed with them fixed. This reduces the number of calculations significantly.

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Algorithm

4 steps

1Step 1: Iterate through all pairs of positions on the grid.
2Step 2: For each pair, calculate the Manhattan distance.
3Step 3: Count how many valid arrangements can be formed with the remaining k-2 pieces using combinations.
4Step 4: Multiply the distance by the number of arrangements and sum it up.

solution.py16 lines

1# Full working Python code
2from math import comb
3
4MOD = 10**9 + 7
5def manhattan_distance_optimal(m, n, k):
6    total_distance = 0
7    for x1 in range(m):
8        for y1 in range(n):
9            for x2 in range(m):
10                for y2 in range(n):
11                    if (x1, y1) != (x2, y2):
12                        distance = abs(x1 - x2) + abs(y1 - y2)
13                        arrangements = comb(m * n - 2, k - 2)
14                        total_distance += distance * arrangements
15                        total_distance %= MOD
16    return total_distance

ℹ

Complexity note: The complexity is due to iterating through all pairs of positions on the grid, which is manageable given the constraints.

1Fixing pairs of positions allows us to count arrangements efficiently.
2The Manhattan distance can be calculated directly from the coordinates.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.