#1320

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Distance to Type a Word Using Two Fingers

Calculate the minimum total distance to type a word using two fingers on a keyboard, applying dynamic programming.

Problem Statement

You are given a 2D keyboard layout with each English uppercase letter placed at specific coordinates. Your goal is to determine the minimum total distance needed to type a given string using two fingers, each starting on any letter.

The distance between two points (x1, y1) and (x2, y2) is the Manhattan distance, computed as |x1 - x2| + |y1 - y2|. The problem asks you to use dynamic programming to calculate the minimum cost of typing the word efficiently using both fingers.

Examples

Example 1

Input: word = "CAKE"

Output: 3

Using two fingers, one optimal way to type "CAKE" is: Finger 1 on letter 'C' -> cost = 0 Finger 1 on letter 'A' -> cost = Distance from letter 'C' to letter 'A' = 2 Finger 2 on letter 'K' -> cost = 0 Finger 2 on letter 'E' -> cost = Distance from letter 'K' to letter 'E' = 1 Total distance = 3

Example 2

Input: word = "HAPPY"

Output: 6

Using two fingers, one optimal way to type "HAPPY" is: Finger 1 on letter 'H' -> cost = 0 Finger 1 on letter 'A' -> cost = Distance from letter 'H' to letter 'A' = 2 Finger 2 on letter 'P' -> cost = 0 Finger 2 on letter 'P' -> cost = Distance from letter 'P' to letter 'P' = 0 Finger 1 on letter 'Y' -> cost = Distance from letter 'A' to letter 'Y' = 4 Total distance = 6

Constraints

2 <= word.length <= 300
word consists of uppercase English letters.

Solution Approach

Dynamic Programming State Transition

Use dynamic programming to model the problem with two states representing the positions of each finger. At each step, you transition between these states to minimize the total distance by considering both possible finger moves.

State Space Representation

Define the state as a tuple of positions for both fingers and the index of the character in the word being typed. You will then calculate the cost of moving either finger to the next character and choose the optimal move.

Minimization of Distance

At each step, the optimal cost is determined by considering the distance of both fingers from the current character. The state transition ensures that you select the minimal distance by using both fingers efficiently.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexities depend on the size of the state space and the word length. With a state space of size O(26 * 26) (for each finger’s position), the time complexity is O(n * 26 * 26), where n is the length of the word. Space complexity is also O(n * 26 * 26).

What Interviewers Usually Probe

Look for familiarity with dynamic programming and state transitions.
Assess whether the candidate can optimize the finger movements to minimize the total distance.
Evaluate how well the candidate understands the structure of the problem and the importance of considering both fingers simultaneously.

Common Pitfalls or Variants

Common pitfalls

Ignoring the need to minimize the total distance for both fingers at every step.
Failing to optimize the state transitions between different letter positions.
Not considering edge cases with small word lengths or repeated characters.

Follow-up variants

Allowing more than two fingers to type, which changes the state space.
Increasing the size of the keyboard to include lowercase letters, affecting the state space.
Changing the grid layout of the keyboard to test different spatial arrangements.

FAQ

What is the optimal approach to solve the 'Minimum Distance to Type a Word Using Two Fingers' problem?

The optimal solution involves dynamic programming where you track the positions of both fingers and minimize the distance between the current and next letter in the word.

How does dynamic programming apply to this problem?

Dynamic programming helps by modeling the finger positions as states and minimizing the movement cost at each step while transitioning through different positions.

What is the complexity of the solution for the 'Minimum Distance to Type a Word Using Two Fingers' problem?

The time complexity is O(n * 26 * 26), where n is the word length, and the space complexity is O(n * 26 * 26), due to the state space for both fingers.

How can I optimize my solution for the minimum distance typing problem?

Focus on minimizing the total distance by efficiently managing finger placements at each step and considering all possible transitions between letters.

Are there any variants of the 'Minimum Distance to Type a Word Using Two Fingers' problem?

Variants include increasing the number of fingers, expanding the keyboard layout, or changing the grid arrangement, which affects the state space and solution complexity.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i][j][k]$ to represent the minimum distance after typing $\textit{word}[i]$, with finger 1 at position $j$ and finger 2 at position $k$. Here, positions $j$ and $k$ represent the numbers corresponding to the letters, ranging from $[0,..25]$. Initially, $f[i][j][k] = \infty$.

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class Solution:
    def minimumDistance(self, word: str) -> int:
        def dist(a: int, b: int) -> int:
            x1, y1 = divmod(a, 6)
            x2, y2 = divmod(b, 6)
            return abs(x1 - x2) + abs(y1 - y2)

        n = len(word)
        f = [[[inf] * 26 for _ in range(26)] for _ in range(n)]
        for j in range(26):
            f[0][ord(word[0]) - ord('A')][j] = 0
            f[0][j][ord(word[0]) - ord('A')] = 0
        for i in range(1, n):
            a, b = ord(word[i - 1]) - ord('A'), ord(word[i]) - ord('A')
            d = dist(a, b)
            for j in range(26):
                f[i][b][j] = min(f[i][b][j], f[i - 1][a][j] + d)
                f[i][j][b] = min(f[i][j][b], f[i - 1][j][a] + d)
                if j == a:
                    for k in range(26):
                        t = dist(k, b)
                        f[i][b][j] = min(f[i][b][j], f[i - 1][k][a] + t)
                        f[i][j][b] = min(f[i][j][b], f[i - 1][a][k] + t)
        a = min(f[n - 1][ord(word[-1]) - ord('A')])
        b = min(f[n - 1][j][ord(word[-1]) - ord('A')] for j in range(26))
        return int(min(a, b))

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