Given a set of point sites in the plane, the Voronoi diagram is the corresponding partition of the plane into cells, where each cell is the locus of points in the plane sharing the same closest site. This fundamental structure has a wide variety of applications. When coupled with a point location data structure, it can be used to quickly answer nearest neighbor queries. Other applications include robot motion planning, modeling natural processes in areas such as biology, chemistry, and physics, and moreover, the dual of the Voronoi diagram is the well known Delaunay triangulation. See the book  for an extensive coverage of the topic.
Many of the varied applications of Voronoi diagrams require generalizing its definition in one way or another, such as adding weights or otherwise altering the distance function, moving to higher dimensions, or considering alternative types of sites. While some of these generalizations retain the highly desirable linear worst-case complexity of the standard Voronoi diagram, many others unfortunately have quadratic worst-case complexity or more.
Within the field of Computational Geometry, particularly in recent years, there have been a number of works analyzing the expected complexity of various geometric structures when the input is assumed to have some form of randomness. Here we continue this line of work, by studying the expected complexity of two previously considered Voronoi diagram variants.
Directional and weighted Voronoi diagrams. A natural generalization to consider is when each site is only visible to some subset of the plane. Here we are interested in the so called visual restriction Voronoi diagram (VRVD) [5, 11], where a given site is only visible to the subset of points contained in some cone with base point and angle . These diagrams model scenarios where the site has a restricted field of view, such as may be the case with various optical sensors or human vision. For example, in a football game, each player has their own field of view at any given time, and the location of the ball in the VRVD tells us which player is the closest to the ball among those who can see it. When the visible region for each site is a half-plane whose boundary passes through the site (i.e., a VRVD where for each site ), such diagrams are called semi Voronoi diagrams . Just like general VRVD’s, semi Voronoi diagrams have worst-case complexity. Our expected analysis is shown for the semi Voronoi diagram case, however, we remark a similar analysis implies the same bounds hold more generally for VRVD’s.
The other generalization we consider is when sites have weights. There are many natural ways to incorporate weights into the distance function of each site. Three of the most common are additive, power, and multiplicative Voronoi diagrams (see ). (For brevity, throughout we use the prefix multiplicative, rather than the more common multiplicatively-weighted.) For additive Voronoi diagrams the distance from a point in the plane to a given site is , for some constant , which can vary for each site. For power diagrams the distance is given by . For multiplicative diagrams the distance is given by . The worst-case complexity for additive and power diagrams is only linear. Here our focus is on multiplicative diagrams, whose worst-case complexity is known to be . These diagrams are used to model, for example, crystal growth where crystals grow together from a set of sites at different rates.
Previous expected complexity bounds. There are many previous results on the expected complexity of various geometric structures under one type of randomness assumption or another. Here we focus on previous results relating to Voronoi diagrams. For point sites in , the worst-case complexity of the Voronoi diagram is known to be , however, if the sites are sampled uniformly at random from a -ball or hypercube, for constant , then the expected complexity is [6, 10]. For a set of point sites on a terrain consisting of triangles, Aronov et al.  showed that, if the terrain satisfies certain realistic input assumptions then the worst-case complexity of the geodesic Voronoi diagram is . Under a relaxed set of assumption, Driemel et al.  showed that the expected complexity is only , when the sites are sampled uniformly at random from the terrain domain.
Relevant to the current paper, Har-Peled and Raichel  showed that, for any set of point site locations in the plane and any set of weights, if the weights are assigned to the points according to a random permutation, then the expected complexity of the multiplicative Voronoi diagram is . The motivation for this work came from Agarwal et al., who showed that if one randomly fattens a set of segments in the plane, by taking the Minkowski sum of each segment with a ball of random radius, then the expected complexity of the union is near linear, despite having quadratic worst-case complexity. In a follow-up work to , Chang et al. defined the candidate diagram, a generalization of weighted diagrams to multi-criteria objective functions, and showed that under similar randomized input assumptions the expected complexity of such diagrams is .
Our results and significance. Our first result concerns semi Voronoi diagrams, where the visible region of each site is a half-plane whose bounding line passes through the site, and where the worst-case complexity is quadratic. For any set of site locations and bounding lines, we show that if the visible side of each site’s bounding line is sampled uniformly at random, then the expected complexity of the semi Voronoi diagram is linear, and
with high probability. To achieve this we argue that our randomness assumption implies that any point in the plane is likely to be visible by one of itsnearest neighbors, for large enough . Thus within each cell of the order- Voronoi diagram, one can argue the complexity of semi Voronoi diagram is , and so summing over all cells gives an bound. This is a variant of the candidate diagram approach introduced in . Unfortunately, in general this approach requires to be more than a constant, which will not yield the desired linear bound. Thus here, in order to get a linear bound, we give a new refined version of this approach, by carefully allowing to vary as needed, in the end producing a diagram which is the union of order- cells, for various values of .
The main focus of the paper is the second part concerning multiplicative Voronoi diagrams, where the distance to each site is the Euclidean distance multiplied by some site-dependent weight, and where the worst-case complexity is quadratic. We first argue that if the weights are sampled from a set with constant size, then interestingly a similar refined candidate diagram approach yields a linear bound. Our main result, however, considers the more challenging case when no restrictions are made on the weights, but instead the site locations are sampled uniformly at random from the unit square. For this case we show that in any grid cell of side length in the unit square the expected complexity of the multiplicative diagram is . This implies an expected bound for the multiplicative diagram over the whole unit square. This improves over the bound of  for this case, by using an interesting new approach which “stretches” sites about a given cell based on their weights, thus approximately transforming the weighted diagram into an unweighted one with respect to a given cell. As this main result is technically very challenging, it is placed last.
It is important to note the significance of our bounds being linear. Given a randomness assumption, one wishes to prove an optimal expected complexity bound, and for the diagrams we consider linear bounds are immediately optimal. However, even when a linear bound seems natural, many standard approaches to bounding the expected complexity yield additional factors. That is, not only are linear bounds desirable, but also they are nontrivial to obtain. However, when they are obtained, they reveal the true expected complexity of the structure, free from artifacts of the analysis.
The standard Voronoi diagram: Let be a set of point sites in the plane. Let denote the Euclidean distance from to , and for two sites let denote their bisector, that is the set of points in plane such that . Any site induces a distance function defined for any point in the plane. For any subset , the Voronoi cell of with respect to , , is the locus of points in the plane having as their closest site from . We define the Voronoi diagram of , denoted , as the partition of the plane into Voronoi cells induced by the minimization diagram (see ) of the set of distance functions , that is the projection onto the plane of the lower envelope of the surfaces defined by these bivariate functions.
One can view the union, , of the boundaries of the cells in the Voronoi diagram as a planar graph. Specifically, define a Voronoi vertex as any point in which is equidistant to three sites in (happening at the intersection of bisectors). For simplicity, we make the general position assumption that no point is equidistant to four or more sites. Furthermore, define a Voronoi edge as any maximal connected subset of which does not contain a Voronoi vertex. (For each edge to have two endpoints we include the “point” at infinity, i.e., the graph is defined on the stereographic projection of the plane onto the sphere.) The complexity of the Voronoi diagram is then defined as the total number of Voronoi edges, vertices, and cells. As the cells are simply connected sets, which are faces of a straight-line planar graph, the overall complexity is .
Order- Voronoi diagram: Let be a set of point sites in the plane. The order- Voronoi diagram of is the partition of the plane into cells, where each cell is the locus of points having the same set of nearest sites of (the ordering of these sites by distance can vary within the cell). It is not hard to see that this again defines a straight-line partition of the plane into cells where the edges on the boundary of a cell are composed of bisector pieces. The worst-case complexity of this diagram is (see [4, Section 6.5]).
We also consider the diagram where every point in a cell not only has the same nearest sites, but also the same ordering of distances to these sites. We refer to this as the order- sequence Voronoi diagram, known to have worst-case complexity (see Appendix B).
Semi Voronoi diagram: Let be a set of point sites in the plane, where for each there is an associated closed half-plane , whose boundary passes through . We use to denote the bounding line of . For any point in the plane and any site , we say that and are visible to each other when . Given a point , let denote the set of sites which are visible to .
For any subset , define the semi Voronoi cell of with respect to as, . It is possible that there are points in the plane which are not visible by any site in . If desired, this technicality can be avoided by adding a pair of sites far away from which combined can see the entire plane. As before, semi Voronoi cells define a straight-line partition of the plane, where now the edges on the boundary of a cell are either portions of a bisector or of a half-plane boundary. In the worst case, the semi Voronoi diagram can have quadratic complexity .
Random semi Voronoi diagram: We consider semi Voronoi diagrams where the set of sites is allowed to be any fixed set of points in general position. For each site , the line bounding the half-plane of , , is allowed to be any fixed line in passing through . Such a line defines two possible visible closed half-spaces. We assume that independently for each site , one of these two spaces is sampled uniformly at random.
An alternative natural assumption is that the normal of the half-plane for each site is sampled uniformly at random from . Note our model is strictly stronger, that is any bound we prove will imply the same bound for this alternative formulation. This is because one can think of sampling normals from , as instead first sampling directions for the bounding lines from , and then sampling one of the two sides of each line for the normal.
Multiplicative Voronoi diagram: Let be a set of point sites in the plane, where for each site there is an associated weight . Any site induces a distance function defined for any point in the plane. For any subset , the Voronoi cell of with respect to , , is the locus of points in the plane having as their closest site from . The multiplicative Voronoi diagram of , denoted , is the partition of the plane into Voronoi cells induced by the minimization diagram of the distance functions .
Note that unlike the standard Voronoi diagram, the bisector of two sites is in general an Apollonius disk, potentially leading to disconnected cells. Ultimately, the diagram still defines a planar arrangement and so its complexity, denoted by , can still be defined as the number of edges, faces, and vertices of this arrangement. Note the edges are circular arcs and straight line segments and thus are still constant complexity curves. In the worst case, the multiplicative Voronoi diagram has complexity .
3 The Expected Complexity of Random Semi Voronoi Diagrams
3.1 The probability of covering the plane
As it is used in our later calculations, we first bound the probability that for a given subset of sites of that there exists a point in the plane not visible to any site in .
For any set of sites .
Consider the arrangement of the bounding lines . Let denote the set of faces in this arrangement (i.e., the connected components of the complement of the union of lines), and note that . Observe that for any face and any fixed site , either every point in is visible by or no point in is visible by . Moreover, the probability that face is not visible by site is . Hence the probability that a face is not visible by any of the sites is
Hence the probability that at least one face in is is not visible by any sites in is
3.2 A simple near linear bound
Ultimately we can show that the expected complexity of a random semi Voronoi diagram is linear, however, here we first show that Lemma 3.1 implies a simple near linear bound which also holds with high probability. Specifically, we say that a quantity is bounded by with high probability, if for any constant there exists a constant , depending on , such that the quantity is at most with probability at least .
Let be a set of sites, where each site has a corresponding line passing through . For each , sample a half-plane uniformly at random from the two half-planes whose boundary is . Then the expected complexity of the semi Voronoi diagram on is , and moreover this bound holds with high probability.
Let , for some constant . Consider the order- Voronoi diagram of . First triangulate this diagram so the boundary of each cell has constant complexity. (Note triangulating does not asymptotically change the number of cells.) Fix any cell in this triangulation, which in turn fixes some (unordered) set of -nearest sites. By Lemma 3.1,
for any arbitrarily small value . Thus with polynomially high probability for every point in its closest visible site will be in . Now let be the set of all triangles in the triangulation of the order- diagram. Observe that the above high probability bound applies to any triangle . Thus taking the union bound we have that with probability at least (where is an arbitrarily small value), simultaneously for every triangle , every point in will be visible by one of its closest sites. Let denote this event happening (and denote it not happening). Conditioning on happening, there are only relevant sites which contribute to the semi Voronoi diagram of any cell . Thus the total complexity of the semi Voronoi diagram restricted to any cell is at most , as the semi Voronoi diagram has worst-case quadratic complexity . (Note that as is a triangle, we can ignore the added complexity due to clipping the semi Voronoi diagram of these sites to .) On the other hand, if happens, then the worst-case complexity of the entire semi Voronoi diagram is still .
Now the complexity of the semi Voronoi diagram is bounded by the sum over the cells in the triangulation of the complexity of the diagram restricted to each cell. Thus the above already implies that with high probability the complexity of the semi Voronoi diagram is . As for the expected value, by choosing sufficiently large,
3.3 An optimal linear bound
The previous subsection partitioned the plane based on the order- Voronoi diagram, for , and then argued that simultaneously for all cells one of the nearest sites will be visible. This argument is rather coarse, but instead of using a fixed large value for , if one is more careful and allows to vary, then one can argue the expected complexity is linear. Note that in the following, rather than using the standard unordered order- Voronoi diagram, we use the more refined order- sequence Voronoi diagram as defined above.
Let be a set of sites, where each site has a corresponding line passing through . For each , sample a half-plane uniformly at random from the set of two half-planes whose boundary is (i.e., each has probability). Then the expected complexity of the semi Voronoi diagram on is .
Consider the partition of the plane by the first order Voronoi diagram of , i.e., the standard Voronoi diagram. We iteratively refine the cells of this partition into higher order sequence Voronoi diagram cells. At each iteration we have a collection of order- sequence Voronoi diagram cells, and for each cell we either mark it final and stop processing, or further refine the cell into its constituent order- sequence cells. Specifically, a cell is marked final in the th iteration if every point in is visible by one of its nearest sites, or when and the cell cannot be refined further. The process stops when all cells are marked final. Below we use to denote the set of cells which were marked final in the th iteration.
Let be the set of all cells of the order- sequence Voronoi diagram of .
Note that the order- sequence Voronoi diagram can be constructed by iteratively refining lower order cells, and hence any cell seen at any point in the above process is a cell of the order- sequence diagram for some value of .
So consider any cell of the order- sequence diagram of . Let be the number of order- sequence diagram cells inside ,
and let be the indicator variable for the event that is refined into its constituent order- sequence cells in the th round of the above iterative process.
Note that is upper bounded by the probability that has not been marked final by the end of the th round, which in turn is bounded by Lemma 3.1. Thus letting be the random variable denoting the total number of order-
be the random variable denoting the total number of order-sequence cells created in the above process, we have
as is at most the total number of cells in the order- sequence Voronoi diagram, which is bounded by (see Appendix B).
Using the same argument as in Lemma 3.2, the complexity of the random semi Voronoi diagram is bounded by . (If then a cell may not be fully visible, though the bound still applies as the worst-case complexity is .) Thus by the above, the expected complexity is
For simplicity the results above were presented for semi Voronoi diagrams, though they extend to the more general VRVD case, where the visibility region of is determined by a cone with base point and angle , where the orientation of the cone is sampled uniformly at random. Specifically, the plane can be covered by a set of cones around each with angle . Any one of these smaller cones is completely contained in the randomly selected cone with probability . If the are lower bounded by a constant , one can then prove a variant of Lemma 3.1 (and hence Lemma 3.2 and Theorem 3.3), as the arrangement of all these smaller cones still has complexity, and the probability a face is not visible to any site is still exponential in but with base proportional to .
4 The Expected Complexity of Multiplicative Voronoi Diagrams
In this section we consider the expected complexity of multiplicative Voronoi diagrams under different randomness assumptions. First, by using the approach from the previous section, we show that for any set of site locations, if each site samples its weight from a set of constant size , then the expected complexity is . Next, we consider the case when the sites can have arbitrary weights, but the site locations are sampled uniformly at random from the unit square. As making no assumptions on the weights makes the problem considerably more challenging, as a warm-up, we first assume the weights are in an interval . In this case, we consider a side length grid, and argue that locally in each grid cell the expected complexity of the multiplicative diagram is (inspired by the approach in ), and thus over the entire unit square the complexity is . Finally, we remove the bounded weight assumption, and argue that for any set of weights, the expected complexity is linear when site locations are sampled uniformly at random from the unit square, by introducing the notion of “stretched” sites.
4.1 Sampling from a small set of weights
Let be a set of non-negative real weights and let be a set of point sites in the plane, where each site in is assigned a weight independently and uniformly at random from . Then the expected complexity of the multiplicative Voronoi diagram of is .
Consider the partition of the plane determined by the unweighted first order Voronoi diagram of , i.e., the standard Voronoi diagram. Following the strategy of the proof of Theorem 3.3, we iteratively refine the cells of this partition into higher order sequence Voronoi diagram cells, except now a cell is marked final in the th iteration if at least one of its -nearest sites has weight . Note that the probability that sites are all assigned weight larger than is .
Using the same notation as in the proof of Theorem 3.3, we have
As the worst-case complexity of the multiplicative Voronoi diagram of sites is , overall the complexity of the multiplicative Voronoi diagram is bounded by . Thus by the above, the expected complexity is
where the last step is obtained by viewing the sum as a power series in ,
4.2 Sampling site locations with bounded weights
In this section we argue that the expected complexity of the multiplicative Voronoi diagram is linear when the site locations are uniformly sampled and the weights are in a constant spread interval. In the next section we remove the bounded weight assumption. Thus the current section can be viewed as a warm-up, and serves to illustrate the extent to which assuming bounded weights simplifies the problem. However, as the results of the next section subsume those here, this section can be safely skipped if desired.
The following fact is used both in the proof of the lemma below and the next section.
Consider doing independent experiments, where the probability of success for each experiment is . Let be the total number of times the experiments succeed. Then .
Note that the above fact holds since is a binomial random variable, , and so .
Let be a set of point sites in the plane, where for some value , each site in is assigned a weight . Suppose that the location of each site in is sampled uniformly at random from the unit square . Then the expected complexity of the multiplicative Voronoi diagram of within is .
Place a regular grid over , where grid cells have side length . Fix any grid cell , where . We now argue that the expected complexity of the multiplicative Voronoi diagram within is , and thus by linearity of expectation, the expected complexity over all grid cells in is .
Let be the random variable denoting the unweighted distance of the closest site in to the center of the grid cell , and let be the random variable denoting the number of points which contribute to the multiplicative Voronoi diagram in conditioned on the value . (For now ignore the contribution of the point at distance exactly .) Observe that any point in which contributes to the multiplicative Voronoi diagram within must lie within the annulus centered at the center of , with inner radius and outer radius . Conditioned on the value , let be the probability for a point to fall in this annulus, and let be the binomial random variable representing the number of points which fall into this annulus. For now assume , in which case the region outside the disk centered at the center of and with radius has area at least . We then have that
Let be the random variable denoting the complexity of the multiplicative Voronoi diagram within when conditioned on the value . As the worst-case complexity of the multiplicative Voronoi diagram is quadratic, we have , where the plus 1 counts the point at distance exactly . Thus using Fact 4.6, and again assuming ,
Now consider the event that for some integer . For this to happen, the open disk with radius centered at the center of must be empty, and one of the points must lie in the annulus with inner radius and outer radius . Therefore,
Furthermore, by the above, when and , we have
and for we have the trivial bound .
Finally, let be the random variable denoting the complexity of the multiplicative Voronoi diagram within . By the law of total expectation we have,
4.3 Sampling sites locations in general
In this section we argue that the expected complexity of the multiplicative Voronoi diagram is linear when the site locations are uniformly sampled. Here the weights can be any arbitrary set of positive values. Without loss of generality we can assume the smallest weight is exactly , as dividing all weights by the same positive constant does not change the diagram. Throughout, denotes the maximum site weight, hence all weights are in the interval .
Let be an arbitrary point in the unit square. The high level idea is that we want to apply a transformation to the sites such that the weighted Voronoi diagram around can be interpreted as an unweighted Voronoi diagram. Specifically, for a site with weight and distance , let the stretched site of with respect to , denoted by , be the point at Euclidean distance from which lies on the ray from through . That is, the weighted distance from to is the same as the unweighted distance from to .
So let be an arbitrary fixed point in the unit square, be a set of sites with weights whose positions have been uniformly sampled from the unit square, and let be the corresponding set of stretched sites.
Let . Our goal is to argue that for any arbitrary choice of in the expected complexity of the multiplicative Voronoi diagram in the ball is constant, i.e., . Then by the grid argument in the previous section this immediately implies a linear bound on the expected complexity of the overall diagram in . Namely, place a uniform grid over , where the grid cell side length is . Then as each cell is contained in a ball for some , and there are cells overall, by the linearity of expectation the expected complexity of the overall diagram is .
At a high level, the idea is simple. We wish to argue that sites whose stretched location is far from will be blocked from contributing by sites whose stretched location is closer to . However, putting this basic plan into action is tricky and requires handling various edge cases. In particular, later it will become clear why we need to consider the following cases for where sites lie relative to .
For a fixed point in , let be the subset of which falls in , i.e., . Let be the complement set, i.e., .
In the following, we typically condition on the set of sites which fall in as being fixed, but not the actual precise locations of those sites. We refer to this as “Fixing ”. Ultimately the statements below hold regardless of what sites actually fall in .
Fix . Let be the radius such that the expected number of stretched sites from that are contained in is . Observe that a site can only be moved further from after stretching, thus . (Potentially is significantly larger.) Also, note that as is fixed, the value of is fixed.
For any integer , let be the disk with radius , centered at . Moreover, define the rings , for any .
Consider two sites , and such that either or with . For any , if and , then cannot contribute to the multiplicative Voronoi diagram in .
For site , let and . Similarly define and for site . Note that the furthest weighted distance of a point in to is , and the closest weighted distance of a point in to is . Thus it suffices to argue , or equivalently . To that end, observe . Thus we only need to argue .
Case 1, : In this case .
Case 2, : In this case .
Note that a site with weight does not move after stretching. Thus as always has a site with weight , there must be at least one stretched site in . Therefore, if we define to be the smallest value of such that , then must contain a stretched site. Thus the above lemma implies any site which contributes to the multiplicative Voronoi diagram within must lie within after being stretched, and so going forward we only consider stretched sites in .
Fix , and consider any . For , let denote the probability that is located in . For , let denote the probability that is located in and is located in , conditioned on the event that no stretched sites from are located in .111Note for , , thus the condition that no stretched sites are in has non-zero probability.
Fix . For any and we have .
Note in the following we always take as given that . Observe that
Then as the location of the sites in are independent, for we have,
If , then the above implies . On the other hand, if , then .
Fix For any , consider the event, , that is the largest index such that there are no stretched sites from located in . We have .
Fix Let be the number of sites that fall outside and contribute to the multiplicative Voronoi diagram within . Then we have