Took me longer than I'd like to admit to stop thinking about this as a graph problem.
Model the problem as a dynamic programming problem where you process cities from left to right, keeping track of whether the previous unit moved to the current city. Define states based on the position and the action of the unit at the previous city, then maximize the sum of populations of protected cities.
Pro tip: Clarify that units can only move left, so a unit at city i can protect i or i-1, but not i+1. This asymmetry simplifies the DP because decisions only affect the current and next city.
Restate the problem: n cities in a line, each with a population, and a binary string indicating initial unit positions. Each unit can stay or move one city left (except city 1). A city is protected if at least one unit ends there. Goal: maximize total protected population.
Let dp[i][moved] represent the maximum protected population considering cities 1..i, where 'moved' indicates whether the unit at city i (if any) has moved to city i-1. Alternatively, use dp[i][prev_action] where prev_action indicates if the unit from city i-1 moved to city i.
For each city i, consider possibilities based on whether there is a unit at i and whether it moves. Ensure city i is protected if either a unit stays at i or a unit from i+1 moves to i. Update DP accordingly, taking max of valid configurations.
Initialize DP for city 1: if there's a unit at city 1, it cannot move left, so it must stay (protecting city 1). If no unit, city 1 is unprotected unless a unit from city 2 moves to it (handled later).
Iterate through cities, filling DP table. The answer is the maximum value at city n considering all valid states. Discuss time and space complexity: O(n) time and O(1) space if optimized.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.