Count islands with eight-direction adjacency

You are given an m×n grid of characters '1' (land) and '0' (water). Count the number of islands, where cells are connected if they touch in any of the eight directions (N, NE, E, SE, S, SW, W, NW). Implement an algorithm that returns the island count. Discuss and implement one approach (BFS or DFS), analyze time and space complexity, and explain how you would handle very large grids to avoid recursion limits. Optional: outline a Union-Find solution and compare trade-offs.