Matching random colored points with rectangles

Abstract

Given $n>0$, let $S\subset [0,1]^2$ be a set of n points, chosen uniformly at random. Let $R\cup B$ be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed axis-aligned rectangles that cover exactly two points of S of the same color, and is strong in the sense that all of its rectangles are pairwise disjoint. We prove that almost surely $M(n)\ge 0.83\,n$ for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm that runs over the random point set as a Markov chain.

Bichromatic matching

In this section, we present our results when matching red points with blue points. The Markov chain for this case consists of 20 states $\{e_1,e_2,\ldots ,e_{20}\}$, and it is described in Fig. 9. Its transition matrix is showed in Fig. 8.

We have that $f(e)=2$ for all $e\in \{e_3,e_7,e_9,e_{10},e_{12},e_{17},e_{18},e_{19},e_{20}\}$, $f(e_{11})=4$, and $f(e)=0$ for every other state e. Furthermore, the stationary distribution $s=(s_1,\ldots ,s_{20})$ satisfies

$$\begin{aligned}{} & {} s_3 ~=~ \frac{4694}{26397},~ s_7 ~=~ \frac{2560}{26397},~ s_9 ~=~ \frac{340}{26397},~ s_{10} ~=~ \frac{85}{2933},~ s_{11} ~=~ \frac{85}{8799}\\{} & {} s_{12} ~=~ \frac{340}{26397},~ s_{17} ~=~ \frac{340}{26397},~ s_{18} ~=~ \frac{68}{26397},~ s_{19} ~=~ \frac{68}{26397},~ s_{20} ~=~ \frac{68}{26397}. \end{aligned}$$

Then, we obtain

$$\begin{aligned} \alpha _3 ~{} & {} =~ 2(s_3+s_7+s_9+s_{10}+s_{17}+s_{18}+s_{19}+s_{20}) + 4s_{11} ~\nonumber \\{} & {} =~ \frac{19506}{26397} ~\approx ~ 0.738947608. \end{aligned}$$

By Theorem 1, taking $\varepsilon =\alpha _3-0.7389>0$, for n large enough we have almost surely that at least $0.7389\,n$ points can be matched (Fig. 9).

Fig. 8

The transition matrix of the Markov chain for $k=3$, when matching red points with blue points (Color figure online)

Fig. 9

The table shows the 20 states of the Markov chain for $k=3$, when matching red points with blue points. In the second column we show the first component of $e_i$, and in the third column we show $f(e_i)$. In the last column we show the neighborhood of $e_i$ as pairs $(e_j,P_{i,j})$, where $P_{i,j}$ is the transition probability from $e_i$ to $e_j$ (Color figure online)