The spatial view (the grid of rectangles) and the tree view (the hierarchy of nodes) represent the same structure. Searching for a point means walking down the tree: at each node, you check which of the four children contains your target coordinate and recurse into that child. You never visit the other three.
Now that we can build the tree, let's use it to search. Finding a specific point means starting at the root and asking: which child quadrant contains this coordinate? Then you recurse into that child and ask again. Each level of the tree cuts the search space by roughly three-quarters.
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Is a Wi-Fi extender better than a mesh router?
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Step 2: Route on the Abstract Graph (The "Highway" Part):