DouglasPeucker

The Douglas-Peucker algorithm, also known as the Ramer-Douglas-Peucker algorithm, is a line simplification method used to reduce the number of points in a polyline or polygonal chain while preserving its overall shape within a user-specified tolerance. It is widely used in geographic information systems (GIS), cartography, and vector map rendering.

The algorithm operates on an ordered sequence of points. It begins by preserving the endpoints of the sequence and finding the point with the maximum perpendicular distance to the line segment joining those endpoints. If this maximum distance exceeds the tolerance, the sequence is split at that point and the process recurses on both resulting subsequences. If the distance is within the tolerance, all intermediate points are discarded, and only the endpoints are kept. The result is a simplified polyline that stays within the specified deviation from the original.

The canonical formulation is attributed to Ramer and independently to Douglas and Peucker; several variants and optimizations exist. The worst-case time complexity is O(n^2), with space complexity O(n). In practice, implementations are optimized and can handle large datasets efficiently, though the performance depends on the data distribution and chosen tolerance.

Applications and limitations: The method is widely used to reduce data for rendering, storage, and analysis while maintaining visual fidelity. It does not guarantee preservation of topology, exact distances, or feature boundaries, and the choice of tolerance critically affects the balance between simplification and accuracy.