Geometry & Intervals: Overview
Geometry & Intervals: The Architecture of Space
The Curse of Dimensionality
Sorting numbers is easy. Is 5 bigger than 3? Yes. But sorting space is hard. Is (5, 3) bigger than (4, 8)? It depends.
When data has multiple dimensions (Latitude/Longitude, Start Time/End Time, X/Y/Z), a simple list is no longer enough. If you want to find āthe nearest gas stationā or āall meetings that overlap with 2 PM,ā you cannot simply scan a sorted array.
This section is about Spatial Indexing. It is the art of partitioning space (and time) into manageable hierarchies, allowing us to query infinite planes with finite speed.
The Strategy of Partition
In this section, we move from 1D lines to N-D worlds:
| Concept | The Soul / Metaphor | Representative | Best Forā¦ |
|---|---|---|---|
| Time Management | The Scheduler Detects conflicts in a sea of overlapping events. | Interval Tree | Calendars Finding free time slots. |
| Zooming | The Zoom Lens Recursively divides a 2D map into 4 smaller quadrants. | Quadtree | Game Physics Collision detection & Image compression. |
| Cutting | The Dimensional Cutter Alternates between cutting X and Y axes to organize points. | K-D Tree | Nearest Neighbor āFind the 5 closest stars to Earth.ā |
| Grouping | The Box Packer Groups nearby objects into bounding boxes. | R-Tree | Maps (GIS) āFind all restaurants in this map view.ā |
The Three Laws of Space
- Hierarchy: To search space efficiently, you must group empty space together. If a huge box is empty, you donāt need to look inside it.
- Overlap: Unlike sorting numbers, spatial objects can overlap. A robust algorithm must handle the ambiguity of ābeing in two places at once.ā
- The Curse: As dimensions increase, distance becomes meaningless. In 1000-dimensional space, everything is āfarā from everything else. Standard spatial trees fail here (approximate methods are needed).
Summary
In this section, we will see how computer graphics, databases, and GPS systems solve the problem of āWhere.ā We will learn that organizing space is surprisingly similar to organizing a messy closet: putting things in boxes, and putting those boxes in bigger boxes.
Letās start with the simplest dimension: Time (1D Intervals).