Lower Bounds for the Complexity of the Voronoi Diagram of Polygonal Curves under the Discrete Frechet Distance

Consider the following scenario: A set S of n polygonal curves in R d , each with at most k vertices, is given. The task is to find for several query curves the most similar curve in S under the (discrete) Fréchet distance. In this setting, Bereg, Gavrilova, and Zhu [2] propose to compute the Voronoi diagram of the given set of curves under the (discrete) Fréchet distance and then to locate samples of the transformed query curve in this. The Voronoi diagram of polygonal curves can be represented using the correspondence polygonal curve with k vertices in R d ↔ point in R dk (x 11 , . . . , x 1d ), . . . , (x k1 , . . . , x kd ) ↔ (x 11 , . . . , x 1d , . . . , x k1 , . . . , x kd ).

However, very little is known about the Voronoi diagram of polygonal curves under the (discrete) Fréchet distance. Recently, Bereg et al. [2] have shown for the discrete Fréchet distance an upper bound of O(n kd+ε ) and a lower bound of Ω(n k+1 2

) for the complexity of the Voronoi diagram of n polygonal curves in R d , for d = 2, 3, with at most k vertices each.

We prove the following lower bounds:

Theorem 1. For any d, k, n, there is a set of n polygonal curves in R d with k vertices each whose Voronoi diagram under the discrete Fréchet distance has combinatorial complexity Ω(n dk ) for d = 1, 2 and k ∈ N and complexity Ω(n d(k-1)+2 ) for d > 2 and k ∈ N.

Our lower bounds significantly improve the lower bounds of Bereg et al. [2]. For dimension 2 the bound matches (up to ε) their upper bound.

Although Bereg et al. [2] formulate the upper bound only for dimensions 2 and 3, their proof generalizes to other dimensions yielding an upper bound of O(n d•k+ε ) for d, k ∈ N. Thus, for dimensions d = 1, 2 the upper and lower bounds match (up to ε), while for d > 2 a gap of n d-2 between the lower and upper bound remains. For d > 2 our lower bound construction is a generalization of our two-dimensional construction. One part of the generalized construction is still inherently two-dimensional. We assume that by finding a suitable generalization of this part or by avoiding it, the gap of n d-2 can be closed. Therefore we conjecture that the upper bound is tight (up to ε). In the following, we always use the parameters n, d, and k to denote n input curves in R d , each with at most k vertices. We give lower bounds Ω(f d,k (n)) on the combinatorial complexity of the Voronoi diagram by showing that it contains at least f d,k (n) Voronoi regions. By a Voronoi region we mean a set of curves with a common set of nearest neighbors under the discrete Fréchet distance in the given set of input curves.

We show the lower bounds in Theorem 1 first for dimension d = 1 (Lemma 1) and then for dimensions d ≥ 2 (Lemma 2). For both lower bounds we construct a set S of n curves. Then we construct f (n) query curves which all lie in different Voronoi regions of the Voronoi diagram of S. This implies that the Voronoi diagram has complexity Ω(f (n)). Proof. We construct a set S of n curves with k vertices each for n = m • k with m ∈ N. S will be a union of k sets S 1 , . . . , S k of m curves each. We show that the Voronoi diagram of S contains m k Voronoi regions.

The construction for k = 3 is shown in Figure 1. We place k points p 1 , . . . , p k with distance 2m between consecutive points on the real line. A curve in S has the form (p 1 , . . . , p i-1 , p i , p i+1 , p k ) for some i ∈ {1, . . . , k} and point p i close to p i . Our construction uses the following points, curves, and sets of curves. See

We claim that for all 1 ≤ j 1 , . . . , j k ≤ m a query curve Q exists whose set of nearest neighbors in S under the discrete Fréchet distance, denoted by N S (Q), is

Since these are m k different sets, this implies that there are at least m k Voronoi regions.

The query curve Q will have k vertices q 1 , . . . , q k with q i close to p i for i = 1, . . . , k. The discrete Fréchet distance of Q to any curve in S will be realized by a bijection mapping each p i or p i to q i . Because the p i are placed at large pairwise distances, this is the best possible matching of the vertices for the discrete Fréchet distance.

Let r = (a 2j2a 1j1 )/2 denote half the distance between a 1j1 and a 2j2 . We choose the first vertex of Q as q 1 = -r. The second vertex q 2 we choose as midpoint between a 1j1 and a 2j2 , i.e., q 2 = (a 1j1 + a 2j2 )/2. Since p 1 = 0, the distance between p 1 and q 1 is r. Because all curves in S start at p 1 , this is the smallest possible discrete Fréchet distance between Q and any curve in S. We now construct the remaining points of Q, such that the curves in N S (Q) are exactly those given in equation 1 and these have discrete Fréchet distance r to Q.