Understanding the Problem
The problem involves calculating the minimum total distance needed to type a given word using two fingers on a QWERTY keyboard. Each letter on the keyboard is mapped to specific coordinates on an XY plane. For instance, the letter A is positioned at (0, 0), while the letter Z is located at (4, 1). The objective is to minimize the total distance traveled by the two fingers as they type the word.
Key elements of this problem include considering the distance function, which calculates the Manhattan distance between two points, and the freedom to start with both fingers at any position. This flexibility in starting positions is an essential factor in optimizing the solution.
Dynamic Programming Formulation
The solution employs a dynamic programming (DP) approach to minimize the total typing distance. The DP state is represented as dp[i][f], where i is the index of the character currently being typed, and f is the position of the free finger. The value stored in dp[i][f] represents the minimum total distance required to type all characters up to index i.
The initial state assumes that both fingers are free and can start at any position. The first character typed by either finger incurs no cost. For each subsequent character, the algorithm decides which finger should move to minimize the total distance. This decision is based on precomputed distances between all pairs of letters on the keyboard.
Precomputing Distances
To facilitate quick calculations, the distances between all 26 letters are precomputed. This involves determining the row and column indices of each letter based on their positions on the QWERTY keyboard. The Manhattan distance is calculated using the formula: |x1 - x2| + |y1 - y2|.
By precomputing these distances, the algorithm can efficiently determine the cost of moving a finger from one letter to another. This step significantly reduces the computational overhead during the DP state transitions.
State Transitions and Cost Calculation
At each step, the algorithm evaluates two possible transitions: moving the finger that typed the previous character to the current character, or moving the free finger to the current character. The cost of each transition is calculated using the precomputed distances.
The new DP state is updated by taking the minimum of the two possible costs. This ensures that the overall distance is minimized as the word is typed. The algorithm iteratively updates the DP table until all characters in the word have been processed.
Implementation Details
The solution can be implemented in various programming languages, including PHP. After initializing the DP table and precomputing the distances, the main loop iterates through the characters of the word. At each iteration, the DP state is updated based on the transition costs.
Test cases, such as typing the words CAKE and HAPPY, demonstrate the algorithm's effectiveness. For example, the minimum distance for typing CAKE is calculated as 3, while typing HAPPY results in a minimum distance of 6. These results highlight the efficiency of the DP approach in solving this problem.