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Notice how the highlighted region shrinks at each step. The algorithm never examines points outside the narrowing window. In a balanced tree with nnn points, this takes about log⁡4(n)\log_4(n)log4​(n) steps. For a million points, that's roughly 10 steps instead of a million comparisons.

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For controls, I worked with Iris E Fernandez Valdes and Anjana Vakil on custom input boards using RP2040 microcontrollers.

Randomly selecting border points or using simple geometric divisions (squares/hexagons) results in too many border points per cluster (50-80). This leads to a shortcut explosion (N*(N-1)/2 shortcuts), making the files large and and calculations slow.，这一点在搜狗输入法2026中也有详细论述

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