Question

Suppose you have access to a subroutine called IsWord that runs in constant time and checks if some string is a word in a dictionary. Consider the following variant of the text-splitting problem from class: instead of returning whether or not the string is splittable into words, have it return the number of ways. More formally, design a backtracking algorithm for solving the following problem: "Given a string S[1, . . . , n] and a variable i, return the total number of valid splittings of S[i, . . . , n] into words."

Answer 1

To design a backtracking algorithm for counting the number of valid splits of a string into words, one would implement a recursive function that utilizes a constant-time IsWord subroutine and potentially employs memoization to optimize performance by avoiding re-computation of subproblems.

Step-by-step explanation:

Designing a Backtracking Algorithm for Text-Splitting

To answer the question on designing a backtracking algorithm to count the number of ways a string S[i, ... , n] can be split into valid words using a subroutine IsWord, one can implement a recursive function. This function would iterate through the string, using IsWord to check if the substring from i to a point j is a valid word. If valid, the function would recursively call itself with the starting index as j+1, effectively moving to the next segment of the string. The algorithm would count all the valid ways the remainder of the string can be split, aggregating the count of splits as it backtracks from the recursive calls.

The complexity of such an algorithm would be influenced by the number of recursive calls, which depends on the number of ways the string can be split into valid dictionary words. Given a string that allows for multiple valid splittings, the recursion tree would grow exponentially. However, memoization can optimize the algorithm by storing the results of subproblems, thereby avoiding the re-computation of the number of ways a particular substring can be split.