Computational Complexity

Relevant source files

The following files were used as context for generating this wiki page:

• README.md
• package.json
• src/index.js
• src/distance.js
• src/perpendicular-distance.js

The geo-path-reducer library implements the Douglas-Peucker algorithm to simplify curves represented by geographical coordinates. The computational complexity of this implementation is determined by the recursive nature of the algorithm and the mathematical calculations required to determine spatial relationships between points in a 2D plane mapped to a sphere.

Sources: README.md:3-7, src/index.js:3-23

Algorithm Structure and Recursive Logic

The core complexity arises from the geoPathReducer function (referenced internally as douglasPeucker), which employs a divide-and-conquer strategy. The algorithm identifies the point with the maximum perpendicular distance from a line segment formed by the start and end points of a path.

Execution Flow

1. Iterative Search: The function iterates through all intermediate points ($1$ to $n-1$) to calculate the maximum distance (dmax).
2. Conditional Branching:
   • If dmax exceeds the user-defined epsilon, the path is split at the index of the furthest point.
   • The function recursively calls itself for the "left" and "right" sub-paths.
   • If dmax is less than or equal to epsilon, the intermediate points are discarded, returning only the endpoints.

Sources: src/index.js:5-22

The following diagram illustrates the recursive decision-making process:

This flow demonstrates how the path is subdivided until all points fall within the specified epsilon tolerance.
Sources: src/index.js:3-23

Mathematical Computation Complexity

Each iteration of the search loop invokes perpendicularDistance, which in turn relies on multiple calls to a Great-circle distance formula.

Distance Calculations

The perpendicularDistance function calculates three distinct distances for every intermediate point to determine the height of the triangle formed by the point and the segment bounds.

Function	Operations	File
distance	7 Math.sin/cos/atan2 calls, 1 sqrt	src/distance.js:4-14
perpendicularDistance	3 distance calls, Heron's formula (sqrt)	src/perpendicular-distance.js:4-11
geoPathReducer	Loop $n-2$ times calling perpendicularDistance	src/index.js:8-14

The dependency graph for these calculations is shown below:

The complexity of a single point evaluation is constant $O(1)$, but it involves significant floating-point arithmetic.
Sources: src/distance.js:3-16, src/perpendicular-distance.js:3-12

Data Management and Memory

The implementation uses array slicing and concatenation during the recursion process, which impacts memory overhead and processing time.

const leftPoints = points.slice(0, index + 1);
const rightPoints = points.slice(index);
const simplifiedLeft = douglasPeucker(leftPoints, epsilon);
const simplifiedRight = douglasPeucker(rightPoints, epsilon);
return simplifiedLeft.slice(0, simplifiedLeft.length - 1).concat(simplifiedRight);

Sources: src/index.js:18-22

Complexity Factors

• Best Case: $O(n)$ when the distance of all points is less than epsilon in the first pass, resulting in no recursion.
• Worst Case: $O(n^2)$ when the recursion tree is unbalanced (e.g., the furthest point is always adjacent to an endpoint).
• Space Complexity: Related to the depth of the recursion stack and the creation of sub-arrays via slice and concat.

Sources: src/index.js:16-23, package.json:1-12

Summary of Complexity Components

Component	Logic	Impact
Point Comparison	for (let i = 1; i < n - 1; i++)	Linear scan per recursion level
Distance Logic	Haversine/Great-circle distance	Constant factor overhead per point
Recursion	douglasPeucker(leftPoints...)	Logarithmic to Linear depth
Array Slicing	points.slice(0, index + 1)	Linear time/space relative to segment size

Sources: src/index.js:8-22, src/distance.js:4-15, src/perpendicular-distance.js:5-11

The computational efficiency of the geo-path-reducer is highly dependent on the topology of the input path and the chosen epsilon value. A larger epsilon generally leads to fewer recursive steps and faster execution, whereas a very small epsilon may result in deeper recursion and $O(n^2)$ performance in suboptimal path configurations.