Join us as we dissect the art gallery problem for simple polygons: triangulate the shape, color the vertices with three colors, and pick guards from the smallest color class to cover every spot. We trace the logic from the floor(n/3) bound to efficient algorithms like Jarvis's march and Chan's O(n log H), and explore trapezoidal maps and randomized incremental construction for fast point location. Along the way we connect the theory to real-world spatial problems and touch on the challenges and opportunities of extending these ideas to higher dimensions.
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