Essential Geospatial Algorithms

Why This Matters

Geospatial algorithms are the computational backbone of everything you'll do in Geospatial Engineering—from routing delivery trucks to predicting flood zones to analyzing urban sprawl. You're being tested not just on what these algorithms do, but on when to apply them and why one approach outperforms another for a given problem. Understanding the underlying mechanics—spatial indexing, interpolation theory, graph traversal, coordinate mathematics—separates engineers who can solve novel problems from those who just push buttons in software.

These algorithms connect directly to core exam concepts: data structures and efficiency, uncertainty and error propagation, scale and resolution trade-offs, and spatial relationships. When you encounter a problem asking you to optimize a network, predict values at unmeasured locations, or combine datasets from different sources, you need to immediately recognize which algorithmic family applies. Don't just memorize algorithm names—know what problem each one solves and what assumptions it makes.

---

Data Organization & Retrieval

Efficient spatial queries depend on hierarchical data structures that partition space, reducing search complexity from linear to logarithmic time.

R-Trees

• Hierarchical bounding rectangles—group nearby objects into minimum bounding rectangles (MBRs) that nest into a tree structure for fast range queries
• Multi-dimensional indexing makes R-trees ideal for complex geometries like polygons and lines, not just points
• Dynamic insertion and deletion allows real-time updates without rebuilding the entire index—critical for live GIS applications

Quadtrees

• Recursive spatial subdivision divides 2D space into four equal quadrants, continuing until each cell contains a manageable number of points
• Point data optimization means quadtrees excel when your dataset consists primarily of discrete locations rather than complex shapes
• Variable resolution naturally adapts to data density—sparse regions get large cells, dense regions get fine subdivisions

---

Network Analysis & Routing

Graph-based algorithms model spatial networks as nodes and edges, finding optimal paths by systematically exploring connections while tracking cumulative costs.

Dijkstra's Algorithm

• Guaranteed shortest path in weighted graphs by expanding outward from the source, always selecting the unvisited node with the smallest cumulative distance
• Computational complexity of $O((V + E) \log V)$ with a priority queue, where $V$ is vertices and $E$ is edges
• No negative weights allowed—the algorithm assumes adding distance never makes a path shorter, which holds for real-world travel networks

A* Search Algorithm

• Heuristic-guided exploration uses an estimated distance to the goal (often Euclidean or Manhattan distance) to prioritize promising paths
• Admissible heuristics must never overestimate the true cost; this guarantees optimality while dramatically reducing nodes explored
• Real-time navigation systems rely on A* because it finds optimal routes faster than Dijkstra's when the destination is known

---

Surface Modeling & Interpolation

Interpolation algorithms estimate unknown values by weighting contributions from nearby known points, with different methods making different assumptions about spatial correlation.

Inverse Distance Weighting (IDW)

• Distance-decay function assigns weights inversely proportional to distance: $w_i = \frac{1}{d_i^p}$ where $p$ controls how quickly influence drops off
• Deterministic method produces the same result every time—no statistical uncertainty estimates, making it simple but limited
• Bull's-eye artifacts appear around sample points when $p$ is high; predicted values can never exceed the range of input data

Kriging

• Geostatistical foundation models spatial autocorrelation through a semivariogram, capturing how similarity decreases with distance
• Prediction uncertainty is quantified at every location—you get both an estimated value and a confidence interval
• Stationarity assumption requires that spatial patterns are consistent across the study area; violating this produces unreliable results

---

Coordinate Systems & Projections

All flat maps distort the curved Earth; projection algorithms control which properties (area, shape, distance, direction) are preserved and which are sacrificed.

Map Projection Algorithms

• Projection families include cylindrical (Mercator), conic (Lambert), and azimuthal (stereographic), each optimized for different regions and purposes
• Distortion trade-offs mean no projection preserves everything: conformal projections preserve angles, equal-area projections preserve size, but never both
• Tissot's indicatrix visualizes distortion patterns—exam questions often ask you to identify projection type from distortion characteristics

Coordinate Transformation Algorithms

• Datum shifts account for different Earth models (e.g., WGS84 vs. NAD27) using translation, rotation, and scale parameters
• Seven-parameter transformation (Helmert) handles 3D conversions: three translations, three rotations, and one scale factor
• Error propagation means small datum errors compound across large distances—critical for integrating datasets from different sources

---

Pattern Detection & Clustering

Clustering algorithms identify spatial structure in point data, grouping observations by proximity while handling noise and irregular distributions differently.

K-Means Clustering

• Centroid-based partitioning assigns each point to the nearest of $k$ cluster centers, then recalculates centers iteratively until convergence
• User-specified $k$ requires prior knowledge or trial-and-error; the elbow method helps identify optimal cluster count
• Spherical cluster assumption means K-means struggles with elongated or irregular cluster shapes common in geographic data

DBSCAN

• Density-based detection defines clusters as regions where points are densely packed, separated by areas of low density
• Noise handling explicitly labels outliers that don't belong to any cluster—critical for messy real-world spatial data
• Arbitrary cluster shapes emerge naturally because DBSCAN doesn't assume geometric structure; it finds whatever patterns exist

---

Spatial Statistics & Autocorrelation

Spatial autocorrelation measures whether nearby locations have similar values—violating the independence assumption that underlies standard statistics.

Moran's I

• Global autocorrelation measure produces a single value from $-1$ (perfect dispersion) through $0$ (random) to $+1$ (perfect clustering)
• Spatial weights matrix defines neighbor relationships—contiguity, distance threshold, or k-nearest neighbors all produce different results
• Statistical significance requires comparing observed $I$ to expected value under randomness; z-scores indicate whether patterns are real

Geary's C

• Local sensitivity emphasizes differences between neighboring values rather than deviations from the global mean
• Value range runs from $0$ (strong positive autocorrelation) through $1$ (no autocorrelation) to $2$ (negative autocorrelation)—inverse of Moran's I interpretation
• Outlier detection makes Geary's C better at identifying local anomalies that Moran's I might miss in the global average

---

Geometric Operations & Simplification

Geometric algorithms transform, combine, and simplify spatial features—essential for visualization, analysis, and managing computational complexity.

Douglas-Peucker Algorithm

• Recursive point elimination keeps endpoints fixed, finds the farthest intermediate point from the line connecting them, and removes points within tolerance
• Tolerance parameter ($\epsilon$) controls simplification aggressiveness—larger values produce simpler lines with more deviation from original
• Topology preservation is not guaranteed; self-intersections can occur at high tolerance levels, requiring post-processing checks

Convex Hull Algorithms

• Minimum enclosing polygon wraps the tightest convex boundary around a point set—like stretching a rubber band around pins
• Graham scan achieves $O(n \log n)$ complexity by sorting points angularly around a pivot, then processing in order
• Boundary analysis applications include defining species ranges, identifying facility coverage areas, and simplifying complex shapes

Buffer Generation Algorithms

• Proximity zones create polygons at specified distances from input features—points become circles, lines become rounded rectangles
• Buffer operations support impact assessment, setback analysis, and service area definition in planning applications
• Dissolve option merges overlapping buffers into single polygons—critical for calculating total affected area without double-counting

---

Terrain & Visibility Analysis

Terrain algorithms extract meaningful information from elevation data, revealing landform characteristics that drive hydrological, ecological, and planning decisions.

Slope and Aspect Calculation

• Slope measures steepness as the maximum rate of elevation change, typically expressed in degrees or percent grade
• Aspect indicates the compass direction a slope faces—critical for solar radiation modeling, vegetation patterns, and microclimate analysis
• DEM dependency means results are only as accurate as the input Digital Elevation Model; resolution affects detected feature size

Viewshed Analysis Algorithms

• Line-of-sight computation traces rays from an observer point across terrain, marking cells as visible or hidden based on intervening elevations
• Observer parameters include height above ground, viewing radius, and vertical angle limits—all affect results significantly
• Cumulative viewshed aggregates visibility from multiple points, revealing locations visible from many observers (useful for tower placement)

---

Data Integration & Overlay

Overlay operations combine spatial datasets to reveal relationships invisible in individual layers—the analytical heart of GIS.

Spatial Join Algorithms

• Attribute transfer attaches data from one layer to another based on spatial relationships (contains, intersects, nearest)
• Relationship types include one-to-one (point in polygon), one-to-many (polygon contains multiple points), and many-to-many (overlapping polygons)
• Computational scaling makes spatial joins expensive for large datasets; spatial indexing is essential for acceptable performance

Spatial Overlay Operations

• Intersection extracts only areas where both input layers overlap, combining attributes from each—useful for identifying coincident features
• Union merges all areas from both inputs, creating a comprehensive layer with combined attributes where features overlap
• Difference subtracts one layer from another, revealing areas unique to the first input—critical for change detection

Raster-Vector Conversion Algorithms

• Vectorization traces raster cell boundaries to create polygons, often requiring smoothing to reduce stair-step artifacts
• Rasterization assigns cell values based on vector features that fall within each cell—resolution determines detail preservation
• Information loss occurs in both directions; conversions should be minimized in analytical workflows

---

Quick Reference Table

---

Self-Check Questions

1. You need to find all fire stations within 10 minutes of a reported emergency, but you don't know which station is closest. Would you use Dijkstra's algorithm or A* search, and why?

2. Compare IDW and Kriging: which method would you choose if your client requires uncertainty estimates for predicted groundwater contamination levels?

3. A dataset shows Moran's I = 0.85 with p < 0.01. What does this tell you about the spatial pattern, and how would you describe it in a report?

4. You're combining a land use layer with a flood zone layer to identify residential areas at risk. Should you use intersection or union, and what would each operation produce?

5. Your map of Alaska appears stretched horizontally compared to a globe. What type of projection distortion is occurring, and what projection family would minimize this problem for an equal-area analysis?