Shift Distance Between Two Strings - Solution & Explanation

Problem Overview: You are given two strings s and t of the same length. Each character can be shifted forward or backward in the alphabet with different costs. The goal is to compute the minimum total cost required to transform s into t character by character.

Approach 1: Greedy with Prefix Sum Cost Lookup (O(n) time, O(1) space)

Each position in the string can be solved independently. For a character s[i], you can either shift forward through the alphabet until reaching t[i], or shift backward. The key observation is that the alphabet is circular, so forward movement may wrap from 'z' to 'a', and backward movement may wrap in the opposite direction.

To compute shift costs quickly, precompute prefix sums for the forward and backward cost arrays of size 26. A prefix sum lets you calculate the total cost of shifting across a range of characters in constant time. For each index i, calculate the forward cost from s[i] to t[i] and the backward cost using the prefix arrays, then choose the minimum. Iterate through the entire string once and accumulate the result. This greedy decision works because shifts for different indices do not interact.

This approach relies heavily on efficient cumulative cost queries, which makes prefix sum preprocessing ideal. The overall scan of the strings keeps the runtime linear while memory usage remains constant because the alphabet size is fixed.

Approach 2: Dynamic Programming for Cost Optimization (O(n + 26^2) time, O(26^2) space)

A dynamic programming formulation can precompute the minimum cost to shift between every pair of letters. Treat each alphabet character as a node and transitions as shifting forward or backward with the given costs. A DP table dp[a][b] stores the minimum cost to convert character a into character b.

Build this table by simulating forward and backward movements around the alphabet while accumulating costs. Since the alphabet size is fixed (26), the preprocessing is small and predictable. Once the table is built, iterate through the strings and sum dp[s[i]][t[i]] for each position.

This approach is slightly more general and easier to reason about when modeling the alphabet as a transition graph. It still runs efficiently because the preprocessing work is bounded by the constant alphabet size. The strings themselves are processed in linear time.

Recommended for interviews: The greedy + prefix sum solution is what interviewers usually expect. It shows you recognize that each character transformation is independent and that prefix sums allow constant‑time range cost queries. Explaining the dynamic programming formulation also demonstrates strong problem modeling skills with string transformations and cost transitions over a small array domain.