Approximating the Maximum Overlap of Polygons under Translation

Abstract

Let $P$ and $Q$ be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present a $(1-\varepsilon )$-approximation algorithm, for finding the translation of $Q$, which maximizes its area of overlap with $P$. Our algorithm runs in $O\left( {c n}\right)$ time, where c is a constant that depends only on k and $\varepsilon$. This suggests that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.

Appendices

Appendix 1: Proof of Lemma 4

The result of Lemma 4 follows by the work of Ahn et al. [3]. For the sake of completeness we include the details here (our construction is somewhat different).

Proof of Lemma 4

The width of $P$ is realized by two parallel lines $\ell _0$ and $\ell _{m+1}$ at a distance of at most $\omega \left( {P}\right)$ from each other. These two lines can be computed in linear time using rotating caliper [23] (and one of these two lines must pass through an edge of $P$). Similarly, it is easy to compute the two extreme points of $P$ in the direction of $\ell _0$. Adding the supporting lines to $P$ through these points, which are orthogonal to $\ell _0$, results in a rectangle, that has five vertices of $P$ on its boundary, and we mark these vertices. Next, we slice this rectangle by m parallel lines $\ell _1, \ldots , \ell _m$ that are parallel to $\ell _0$, into $m+1$ slices of the same width. We mark all the intersections of these lines with $P$. Next, let $P^{\prime }$ be the polygon formed by the convex-hull of all the marked points. It is easy to verify that $P^{\prime }$ can be computed in linear time.

Clearly, $P^{\prime } \subseteq P$. For any point $p\in P$, consider the two lines $\ell _i$ and $\ell _{i+1}$ that $p$ is contained between them. Consider the projection of $p$ into $\ell _i$ and $\ell _{i+1}$, denoted by $p_i$ and $p_{i+1}$ respectively. If any of these two projections are inside $P$, then we are done, as this portion of $P$ is inside $P^{\prime }$, and the distance of projection is at most $\omega \left( {P}\right) {/}m$. Thus, the only remaining possibility is that both projections are outside $P$.

But this implies that $P$’s extreme point in the parallel direction to $\ell$ (or its reverse) must be in $p$’s slice. In particular, the segment $p_i p_{i+1}$ (that includes $p$) must intersect $P$, and furthermore, since the aforementioned extreme point is a vertex of $P^{\prime }$, it follows that this segments must intersect $P'$, implying the claim. $\square$

Appendix 2: Counter Example for Some Natural Approaches

Here, we present a counterexample to the natural algorithm for our problem: Decompose the two polygons into their convex parts, approximate the parts, and then find the maximum overlap for the two reconstituted polygons.

Fig. 5

Counterexample to naive approach for approximating overlap. a, b Input polygons, c optimal placement, d approximate polygons fails to preserve optimal solution

So, consider the polygon P, depicted in Fig. 5a, and its arch-nemesis Q in Fig. 5b. The optimal overlap placement is depicted in Fig. 5c.

But any naive piecewise approximation of P by pieces will fail, because it creates cavities that might contain the whole overlap. For example, one possible solution for the approximate polygon $P'$ and the maximum overlap, is depicted in Fig. 5d.

In particular, any placement that makes the approximate polygon $P'$ cover two of the disks of Q, would fail to cover the third disk of Q. so, in this case, the best approximation suggested by the this algorithm is (roughly) 2/3. This can be made to be arbitrarily bad by enlarging k.

Another Approach That Does Not Work Another natural idea is to pick from each pair of polygons $P_i$ and $Q_j$ a set of directions such that if you approximate each pair with these directions than the pair overlap is approximated correctly. Note, however, that if the polygons are of completely different sizes then, inherently, you need unbounded number of directions if the maximum overlap of the original (non-convex polygons) align the pieces far from their piecewise overlap maximum.