Find grid cell minimizing sum distances

Q: Find grid cell minimizing sum distances

This question evaluates proficiency in grid-based graph traversal and shortest-path computation using BFS, reachability analysis from multiple marked points, and algorithmic time/space complexity reasoning.

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Question

You are given an m x n 2D grid with k marked points. Movement is allowed in four directions (up, down, left, right) with unit cost per step. Find a grid cell whose sum of shortest-path distances to all marked points is minimized; if any marked point is unreachable from every cell, return -1. Implement a brute-force BFS-based solution that computes these distances, return both the minimum total distance and one optimal cell, and analyze time and space complexity. Discuss when multi-source BFS or using coordinate medians (if there are no obstacles) can reduce complexity.

Find grid cell minimizing sum distances

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