Output-Sensitive Construction of the Union of Triangles Esther Ezra and Micha Sharir.

• Slide 1

Output-Sensitive Construction of the Union of Triangles Esther Ezra and Micha Sharir Slide 2 Definition: Union T ={ 1,, n } - collection of n triangles in the plane. The union U= i I is defined as all regions in the plane that are covered by the triangles of T. Slide 3 Constructing the Union: Motivation Robot motion planning Construct the forbidden portion of the configuration space. Ray shooting amid semi- algebraic sets in R 3 Construct the union of 4- dimensional regions. Slide 4 Computing the union Constructing the arrangement of the triangles too slow! O(n 2 ) Output-sensitive algorithm (in terms of the number of edges on the boundary)? unlikely to exist! 3SUM === HOLE-IN-UNION The best known solutions to problems from the 3SUM-hard family require (n 2 ) time in the worst case. Slide 5 Union of triangles in R 2 Known results Special cases Union size Ref Fat triangles O(n loglog n) MPSSW-94 Pseudodiscs O(n) KLPS-86 General triangles: Algorithm Running time Ref RIC O(n log n + 1 ) AH-01 DC O(n 2 ) EHS-02 Performs well in practice i |V i | / i i |V i | / i V i the set of vertices in depth i Slide 6 Output-sensitive union construction Given a collection T ={ 1,, n } of n triangles in the plane, such that there exists a subset S T (unknown to us), of