Overview
Orthogonal range search preprocesses points so an axis-aligned query can discard large regions and report only points whose coordinates fall within each query bound.
CoreSource checked
Orthogonal Range Search
Reports points inside an axis-aligned query rectangle or box.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Geometry
- Difficulty
- Intermediate
- Family
- Range Search
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 0
- Relationships
- 6
- Code example
- Ready
- Visualization
- Ready
Core Concepts
axis-aligned range searchrange searchgeometric primitivesgeometry
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(n)
Average
O(n log n)
Worst
O(n^2)
Space
O(n)
When to Use
- Use when queries are axis-aligned and repeated over mostly static points.
- Use when preprocessing can reduce repeated scanning.
When Not to Use
- Avoid when query shapes are arbitrary polygons or rotated boxes.
- Avoid when updates dominate and the chosen index is static.
What It Does
- Computes range search within Range Search workloads (O(n log n) time, O(n) space).
- Works by Reports points inside an axis-aligned query rectangle or box.
- Maintains key invariants across axis-aligned range search, range search, geometric primitives.
Use Cases
- Production and research workloads requiring range search via Reports points inside an axis-aligned query rectangle or box.
- Range Search pipelines where O(n log n) time and O(n) space fit system constraints.
- Engineering comparisons against k-d Tree and Interval Tree when tuning range search.
Inputs
- Points, segments, polygons, or geometric primitives
- Query or construction parameters
Outputs
- Geometric decision, measurement, or constructed structure
Pseudocode
- 1.Preprocess points into an axis-aware search structure
- 2.Normalize query lower and upper bounds
- 3.Visit only regions whose coordinate ranges intersect the query
- 4.Report points that satisfy every axis bound
- 5.Return the collected points
Canonical Example
- Input
- points (1,1), (3,2), (5,5), query x in [0,3], y in [0,2]
- Output
- points (1,1) and (3,2)
- Explanation
- Both returned points satisfy the x and y bounds; (5,5) is outside.
Implementation Notes
- Choose a structure such as a range tree, k-d tree, or sorted lists based on dimension and update needs.
- Normalize query bounds so min <= max on every axis.
- Test points exactly on rectangle boundaries.
Common Pitfalls
- Inclusive versus exclusive bounds must be defined consistently.
- High-dimensional range searching can lose pruning effectiveness.
- Duplicate coordinates and boundary points need explicit handling.
Interactive Mechanism Walkthrough
Spatial Primitive & Sweep-Line CanvasShow filtering or indexing points by coordinate intervals on a small axis-aligned range search example.
Step 1 of 5: Preprocess points into an axis-aware search structure
geometry-mechanismNotes
Orthogonal range search assumes axis-aligned bounds; rotated or curved queries need different geometry.
Sources (2)
What each reference supports for this entry.
Computational Geometry: Algorithms and Applications
BookComputational geometry textbook support for Orthogonal Range Search's geometric model, data structures, and robustness concerns.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Orthogonal Range Search's problem context, known alternatives, and practical selection tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
UsesInterval Tree
Data Structures · Core
Interval trees support one-dimensional range filtering used inside orthogonal range-query structures.
Compare with thisUsesk-d Tree
Data Structures · Core
Orthogonal range search can use kd-trees to prune points outside axis-aligned query boxes.
Compare with thisUsesRange Tree
Data Structures · Core
Range trees are a standard data structure used to answer orthogonal range-search queries.
Compare with thisConceptual neighborsCircle-Line Intersection
Geometry · Core
Both use geometric intersection ideas for related circle and line intersection problems.
Compare with thisRelationships
View full relationship pageUsesk-d Tree
Orthogonal range search can use kd-trees to prune points outside axis-aligned query boxes.
UsesInterval Tree
Interval trees support one-dimensional range filtering used inside orthogonal range-query structures.
Conceptual neighborSegment Tree
Segment trees and orthogonal range structures both answer interval or range queries through hierarchical decomposition.
Related Algorithms
Tags
GeometryComputational Geometry