Dynamic Geometric Data Structures via Shallow Cuttings

Abstract

We present new results on a number of fundamental problems about dynamic geometric data structures: (1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the discrete 1-center of a 2D point set, (v) the number of maximal (i.e., skyline) points in a 3D point set. The update times are near $n^{11/12}$ for (i) and (ii), $n^{5/6}$ for (iii) and (iv), and $n^{2/3}$ for (v). Previously, sublinear bounds were known only for restricted “semi-online” settings (Chan in SIAM J. Comput. 32(3), 700–716 (2003)). (2) We slightly improve previous fully dynamic data structures for answering extreme point queries for the convex hull of a 3D point set and nearest neighbor search for a 2D point set. The query time is $O(\log ^2\!n)$, and the amortized update time is $O(\log ^4\!n)$ instead of $O(\log ^5\!n)$ (Chan in J. ACM 57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495–2504. SIAM, Philadelphia (2017)). (3) We also improve previous fully dynamic data structures for maintaining the bichromatic closest pair between two 2D point sets and the diameter of a 2D point set. The amortized update time is $O(\log ^4\!n)$ instead of $O(\log ^7\!n)$ (Eppstein in Discrete Comput. Geom. 13(1), 111–122 (1995); Chan in J. ACM 57(3), # 16 (2010); Kaplan et al. in 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2495–2504. SIAM, Philadelphia (2017)).

Appendix: Proof of Lemma 4.1

We describe how a proof by Chan and Tsakalidis [20, Thm. 5] can be adapted to show Lemma 4.1, as restated below:

Lemma 4.1

  There exist constants b, c, and $c'$ such that the following is true: For a set H of at most n planes in ${\mathbb {R}}^3$ and a parameter $k\in [1,n]$, given a convex $(-\infty ,cbk)$-shallow cutting $\varGamma _{\mathrm{in}}$ with at most $c'n/(bk)$ downward cells, together with their conflict lists, we can construct a convex $\varGamma _{\mathrm{in}}$-restricted (k, ck)-shallow cutting $\varGamma _{\mathrm{out}}$ with at most $c'n/k$ downward cells, together with their conflict lists, in $O(n + (n/k)\log (n/k))$ deterministic time.

Proof

As in the previous paper  [20], it is more convenient to work with shallow cuttings in vertex form: given a set H of n planes in ${\mathbb {R}}^3$, a (k, K)-shallow cutting in vertex form is a set V of points whose upper hull $\mathrm{UH}(V)$ covers all points in ${\mathbb {R}}^3$ of level at most k, such that every point in V has level at most K. The conflict list of a point refers to the list of all planes of H below the point.

Chan and Tsakalidis [20, Thm. 5] proved the following statement for some constants B, C, and $C'$:

For a set H of at most n planes in ${\mathbb {R}}^3$ and a parameter $k\in [1,n]$, given a (Bk, CBk)-shallow cutting $V_{\mathrm{in}}$ in vertex form with at most $C'n/(Bk)$ vertices, together with their conflict lists, we can construct a (k, Ck)-shallow cutting $V_{\mathrm{out}}$ in vertex form with at most $C'n/k$ vertices, together with their conflict lists, in $O(n + (n/k)\log (n/k))$ deterministic time.

By a close inspection of their proof, we actually get the following stronger statement for any choice of constants B, $C'$, and t, where $a_0'$, $c_0$, and $c_0'$ are absolute constants:

For a set H of at most n planes in ${\mathbb {R}}^3$ and a parameter $k\in [1,n]$, given a $(12c_0^2k, CBk)$-shallow cutting $V_{\mathrm{in}}$ in vertex form with at most $C'n/(Bk)$ vertices, together with their conflict lists, we can construct a (k, Ck)-shallow cutting $V_{\mathrm{out}}$ in vertex form with at most $C''n/k$ vertices, together with their conflict lists, in $O(n + (n/k)\log (n/k))$ deterministic time, where $C=12c_0^2+1$ and $C''= 2c_0' + {8a_0'c_0'CC'}/({3c_0\sqrt{t}})$.

Set $b=B/6$, $c=3C$, and $c'=C'/18$. To derive Lemma 4.1 from the above statement, we first convert $\varGamma _{\mathrm{in}}$ to vertex form, simply by letting $V_{\mathrm{in}}$ to be the vertex set of the convex polyhedron $\mathcal{P}_{\mathrm{in}}$ formed by unioning the cells in $\varGamma _{\mathrm{in}}$. The polyhedron $\mathcal{P}_{\mathrm{in}}$ has at most $c'n/(bk)$ faces, at most $3c'n/(3bk)=c'n/(bk)$ vertices, and at most $3c'n/(2bk)$ edges. It is helpful to assume that each vertex has degree 3 in $\mathcal{P}_{\mathrm{in}}$; this can be guaranteed by intersecting $\mathcal{P}_{\mathrm{in}}$ with extra planes infinitesimally close to each vertex (the number of new vertices is equal to twice the number of old edges). After this modification, $\mathcal{P}_{\mathrm{in}}$ has $|V_{\mathrm{in}}|\le 3c'n/(bk)=C'n/(Bk)$ vertices, and at most $2c'n/(bk) = 2C'n/(3Bk)$ faces.

We cannot apply the above result to H directly. Instead, we make $12 c_0^2k$ copies of the plane through each face of $\mathcal{P}_{\mathrm{in}}$, and add them to H. The new set ${\widehat{H}}$ of planes has size ${\widehat{n}}\le n +12 c_0^2k\cdot 2C'n/(3Bk)\le 2n$, assuming $B\ge 8c_0^2C'$. Then $\mathcal{P}_{\mathrm{in}}=\mathrm{UH}(V_{\mathrm{in}})$ covers all points of level at most $12 c_0^2k$ in ${\widehat{H}}$. Each point in $V_{\mathrm{in}}$ has level at most $cbk + 3\cdot 12 c_0^2k\le CBk$ in ${\widehat{H}}$, assuming $B\ge 72 c_0^2$.

We can now apply the above result to ${\widehat{H}}$, and obtain a (k, Ck)-shallow cutting $V_{\mathrm{out}}$ for ${\widehat{H}}$ in vertex form. We can set $\varGamma _{\mathrm{out}}$ to be the vertical decomposition of $\mathrm{UH}(V_{\mathrm{out}})$, which has at most $2|V_{\mathrm{out}}|$ cells. Each cell of $\varGamma _{\mathrm{out}}$ intersects at most $3Ck=ck$ planes of H. Every point covered by $\varGamma _{\mathrm{in}}$ with level at most k in H has level at most k in ${\widehat{H}}$ and is thus covered by $\varGamma _{\mathrm{out}}$. Furthermore, $|\varGamma _{\mathrm{out}}|\le 2C''{\widehat{n}}/k\le 4C''n/k=4 (2c_0'+{8a_0'c_0'CC'}/({3c_0\sqrt{t}})) n/k\le 12 c_0'n/k$ by setting $t=({8a_0'CC'}/({3c_0}))^2$. Thus, $|\varGamma _{\mathrm{out}}|\le C'n/k$ by setting $C'=36\cdot 12 c_0'$. $\square$