Efficient Algorithms for Intersecting Line Segments and Finding Cycles in Graphs

In this article, we explore the computational geometry problem of detecting polygons in a set of lines. We present an efficient algorithm that can handle large datasets with less than three hundred lines, making it suitable for online processing. The algorithm works by first finding the minimum cycle basis of the graph, which is then used to detect the polygons.
To understand this concept, imagine you have a bunch of strings tied together in different shapes and sizes. You want to find all the possible ways these strings can be connected without overlapping or touching each other. That’s similar to what our algorithm does – it finds all the possible polygons that can be formed by connecting the lines in the dataset.
The algorithm has two main steps. The first step is to find the minimum cycle basis of the graph, which is like finding the simplest way to connect all the strings together without any overlapping or tangling. Once we have this basis, we can use it to detect the polygons by connecting the lines in a specific way.
The second step is to find all the cycles in the graph, which is like counting how many times each string can be wrapped around without passing through itself. We then use these cycles to form the polygons.
We tested our algorithm on several datasets with different sizes and found that it works well and efficiently. In fact, for datasets with less than three hundred lines, our algorithm takes less than twenty minutes to detect the polygons, making it suitable for online processing.
Overall, this article provides a practical solution for detecting polygons in a set of lines, which is useful in various fields such as computer graphics, engineering, and architecture. By understanding the minimum cycle basis of a graph and the all-pairs shortest cycle basis problem, we can develop efficient algorithms to solve real-world problems.