Convex Polygon Containment: Improving Quadratic to Near Linear Time

We revisit a standard polygon containment problem: given a convex k-gon P and a convex n-gon Q in the plane, find a placement of P inside Q under translation and rotation (if it exists), or more generally, find the largest copy of P inside Q under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required (n²) time, even in the simplest k=3 case. We present a significantly faster new algorithm for k=3 achieving O (npolylog n) running time. Moreover, we extend the result for general k, achieving O (k^O (1/) n^1+) running time for any >0. Along the way, we also prove a new O (k^O (1) npolylog n) bound on the number of similar copies of P inside Q that have 4 vertices of P in contact with the boundary of Q (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).