Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces?

Abstract

Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming problems. For each problem, we study the intersection between a sphere and a Voronoi-like polyhedron. That way, we get information about a possible overlap between the frontier of the union and the intersection of balls. If the polyhedra are non-degenerate, the initial nonconvex geometric problem, which is NP-hard in general, is tractable in polynomial time by convex optimization tools and vertex enumeration. Under some mild conditions, the vertex enumeration can be skipped. Simulations highlight the accuracy and efficiency of our approach compared with competing algorithms in Python for nonconvex quadratically constrained quadratic programming. This work is motivated by an application in statistics to the problem of multidimensional changepoint detection using pruned dynamic programming algorithms.

Appendices

Appendices

Motivation

Our covering problem has a direct application for the implementation of the pruned penalized³ dynamic programming algorithm for changepoint detection in a multidimensional setting [25,26,27]. This problem consists in finding the optimal changepoint within the set ${\mathbb {S}}_m = \{\tau =(\tau _1,\ldots ,\tau _k) \in {\mathbb {N}}^k \,|\, 1 = \tau _0< \tau _1< \cdots< \tau _k < m+1 = \tau _{k+1}\}$ such that we minimize a quadratic (for Gaussian modelization) penalized cost (by $\beta >0$):

$$\begin{aligned} Q_m= & {} \min _{\tau \in {\mathbb {S}}_m}\left[ \sum _{i=0}^{k}\lbrace {\mathcal {C}}(y_{\tau _i:\tau _{i+1}-1}) + \beta \rbrace \right] \,,\\ \hbox {with}\quad {\mathcal {C}}(y_{\tau _i:\tau _{i+1}-1})= & {} \min _{x \in {\mathbb {R}}^n} \sum _{j=\tau _i}^{\tau _{i+1}-1} \sum _{k=1}^{n}(x_k- y_{jk})^2\,, \end{aligned}$$

and $(y_1,\ldots ,y_m)^T \in ({\mathbb {R}}^n)^m$ the data to segment. Using a dynamic programming procedure, we build the recursion $Q_{t+1}(x) = \min \lbrace Q_{t}(x),\, m_{t}+\beta \rbrace + \sum _{k=1}^{n}(x_k- y_{(t+1)k})^2$ with $m_t = \min _{x} Q_t(x)$ and the initialization $Q_0(x) = 0$. We solve this recursion iteratively from $t=0$ to $t=m-1$.

At each t, the recursive function $Q_t(\cdot )$ is a piecewise quadratic function defined on t non-overlapping regions $R^1_t,\ldots ,R^t_t \subset {\mathbb {R}}^n$, for which each quadratics is active on a set $R^i_t$ of type “${\mathcal {I}}\setminus {\mathcal {U}}$”. Precisely,

$$\begin{aligned} \begin{aligned}&R_t^1 = \cap _{i=1}^t B_i^1\,, \\&R_t^2 = \cap _{i=2}^t B_i^2 \setminus B_1^1\,, \\&R_t^3 = \cap _{i=3}^t B_i^3 \setminus (B_2^1 \cup B_2^2)\,, \\&\quad \quad \vdots \\&R_t^t = R_t^t \setminus (\cup _{j=1}^{t-1}B_{t-1}^j)\,.\\ \end{aligned} \end{aligned}$$

where all the “$B_l^k$” sets designate balls determined by the data $(y_k,\ldots ,y_l)$. At the next iteration $t+1$, each $R_t^i$ is intersected by a new ball (that is, we add a ball in each ${\mathcal {I}}$) and the set $R_{t+1}^{t+1}$ is created.

In order to get the global minimum $m_t$, we compare the minima of all present quadratics. To speed-up the procedure, it is worthwhile to search for vanishing sets, that is to detect efficiently the emptiness of sets $R_t^1,\ldots ,R_t^t$. In fact, once a set is proved to be empty, we do not need to consider its minimum anymore at any further iteration. This method is called pruning in the changepoint detection literature.

In a paper in preparation, we will compare this exact method to heuristic approaches in order to build a fast algorithm for an application to genomic data. Moreover, our methods can be extended to other distributions of the natural exponential family.

An Alternative Method

In some applications, as for example in “Appendix A,” we need to test sequentially the covering. This means that we add at each iteration a new ball in the intersection ${\mathcal {I}}$. For large q, it could be expensive to solve q QP problems at each iteration for minimum and maximum. Another approach consists in detecting an intersection between ${\mathcal {I}}\setminus {\mathcal {U}}$ and ${\mathbb {S}}$ (as soon as $\#{\mathcal {I}} >0$), where ${\mathbb {S}}$ is the frontier of the new ball ${\mathbb {B}}$ (with center c and radius R) to add in ${\mathcal {I}}$. The problem is now centered on the “newest” ball of set $\varLambda$ rather than the successive balls of V. Thus, a unique QP problem centered on the ball ${\mathbb {B}}$ is built and we get a result similar to Theorem 3.1:

Theorem B.1

If $V_n(\varPi )>0$, we have the following results.

• (a) If there exists $(x^-, x^+)\in \varPi \times \varPi$ such that $\Vert x^--c\Vert ^2< R^2 < \Vert x^+-c\Vert ^2$,

  then $({\mathbb {B}} \cap {\mathcal {I}}) \setminus {\mathcal {U}} \ne ~\emptyset$ and $V_n(({\mathbb {B}} \cap {\mathcal {I}}) \setminus {\mathcal {U}}) > 0$;

• (b) if $R^2 \le \Vert x^*_{min}-c\Vert ^2$ or $\Vert x^*_{max}-c\Vert ^2 \le R^2$, then ${\mathbb {S}}\cap ({\mathcal {I}}\setminus {\mathcal {U}}) = \emptyset$.

In case (b), we can conclude knowing a point ${\tilde{x}}_k$ for each connected component $C_k$ $(k=1,\ldots ,K)$ of ${\mathcal {I}}\setminus {\mathcal {U}}$ and testing ${\tilde{x}}_k \in {\mathbb {B}}$. If always false, we get $({\mathbb {B}} \cap {\mathcal {I}}) \setminus {\mathcal {U}} = \emptyset$. The drawback of this approach is the necessity to know the number of connected components in ${\mathcal {I}}\setminus {\mathcal {U}}$. However, with the conditions of Theorem 4.1 the set ${\mathcal {I}}\setminus {\mathcal {U}}$ is connected (see the proof) and a unique point in ${\mathcal {I}}\setminus {\mathcal {U}}$ is required $(K = 1)$.

Proposition 3.2 remains the same with ${\mathbb {B}}$ instead of ${\mathbf {B}}$. The detection of the inclusion ${\mathcal {I}} \subset {\mathbb {B}}$ is now made possible and is important for sequential problems, since in this case, the ball ${\mathbb {B}}$ is not added to the set $\varLambda$.