Local Complexity of Polygons
LLocal Complexity of Polygons
Fabian Klute ∗ , Meghana M. Reddy †‡ , and Tillmann Miltzow § ETH Z¨urich, Department of Computer Science Utrecht University, Information and Computing ScienceDepartmentJanuary 20, 2021
Abstract
Many problems in Discrete and Computational Geometry deal withsimple polygons or polygonal regions. Many algorithms and data-structuresperform considerably faster, if the underlying polygonal region has lowlocal complexity. One obstacle to make this intuition rigorous, is thelack of a formal definition of local complexity. Here, we give two possibledefinitions and show how they are related in a combinatorial sense. We saythat a polygon P has point visibility width w = (cid:74) pvw (cid:75) , if there is no point q ∈ P that sees more than w reflex vertices. We say that a polygon P has chord visibility width w = (cid:74) cvw (cid:75) , if there is no chord c = seg( a, b ) ⊂ P thatsees more than w reflex vertices. We show that (cid:74) cvw (cid:75) ≤ (cid:74) pvw (cid:75) O ( (cid:74) pvw (cid:75) ) , for any simple polygon. Furthermore, we show that there exists a simplepolygon with (cid:74) cvw (cid:75) ≥ Ω( (cid:74) pvw (cid:75) ) . In Discrete and Computational Geometry we study many problems with respectto the input size n and other natural parameters. One famous example is thecomputation of the convex hull of a set of points in the plane. While Θ( n log n )time is worst case possible, this can be improved to Θ( n log h ), where h is thenumber of vertices on the convex hull [4]. Here, the number of vertices onthe convex hull is a natural parameter to study this problem. We also saysometimes that the algorithm is output-sensitive. Another famous example, isthe spread ∆ of a set of points in the plane. That is the ratio between the largest ∗ Supported by the Netherlands Organisation for Scientific Research (NWO) under projectno. 612.001.651. † Supported by the Swiss National Science Foundation within the collaborative DACHproject
Arrangements and Drawings as SNSF Project 200021E-171681. ‡ The second author’s full last name consists of two words and is
Mallik Reddy . However,she consistently refers to herself with the first word of her last name being abbreviated. § Supported by the NWO Veni project EAGER. a r X i v : . [ c s . C G ] J a n igure 1: The polygon on the left has intuitively lower local complexity than onthe right.and the smallest distance, between any two points. Efrat and Har-Peled werethe first to find an approximation algorithm for the art gallery problem underthe assumption that the underlying set of vertices has bounded spread [2]. Athird example is the number of reflex vertices of a polygon. This parameter gaveraise to some FPT algorithms for the art gallery problem [1].In this work, we introduce two new parameters that are meant to capturerigorously the idea of local complexity. Consider the polygons shown in Figure 1,most researchers would probably agree that the polygon on the left has lowerlocal complexity than the polygon on the right. Yet, it is not straightforwardhow to define this rigorously in a mathematical sense.Here, we give two possible definitions and show how they are related ina combinatorial sense. We say that a polygon P has point visibility width w = (cid:74) pvw (cid:75) , if w is the smallest number such that there is no point q ∈ P that sees more than w reflex vertices. We say that a polygon P has chordvisibility width w = (cid:74) cvw (cid:75) , if w is the smallest number such that there is nochord c = seg( a, b ) ⊂ P that sees more than w reflex vertices.We show the following theorem. Theorem 1.
For every polygon with chord visibility width (cid:74) cvw (cid:75) and pointvisibility width (cid:74) pvw (cid:75) , it holds that (cid:74) pvw (cid:75) ≤ (cid:74) cvw (cid:75) ≤ (cid:74) pvw (cid:75) O ( (cid:74) pvw (cid:75) ) . Moreover, there are polygons such that (cid:74) cvw (cid:75) ≥ Ω( (cid:74) pvw (cid:75) ) . Note that Hengeveld and Miltzow already defined the notion of chord visibilitywidth in a very similar way [3]. Specifically, they showed that the art galleryproblem admits an FPT algorithm with respect to chord visibility width. For aparameter to be interesting to study, we usually have three criteria.naturalness: Although there is no definition of what it means to be mathemat-ically natural, many researchers seem to have a common under-standing of this notion. 2elevance: The parameter is at least for some fraction of instances reasonablylow.profitable: Using the parameter, we should be able to design better algorithmsand prove useful run time upper bounds.We believe that both parameters are mathematically natural. Theorem 1indicates that the chord visibility width can be exponentially larger than thepoint visibility width. Thus we would expect that chord visibility width ispotentially more profitable. We would expect that both parameters are equallyrelevant as the example that we give is fairly contrived. The remainder of thispaper is dedicated to proving Theorem 1.
We prove Theorem 1 in two parts. First, we show the second half of the theoremin Section 2.1 by constructing a polygon for which it holds that (cid:74) cvw (cid:75) ≥ Ω( (cid:74) pvw (cid:75) ) .Second, we show the first half of Theorem 1 in Section 2.2 by analysing how thereflex vertices visible from a chord in a simple polygon P restrict each othersvision and relating this to the point visibility width of the polygon. k − k − c Figure 2: Construction of the Iterated Comb.We construct a polygon P , called the Iterated Comb , see Figure 2. In thefollowing, let k ∈ N . The Iterated Comb consists of k layers , each layer consistsof two spikes and each spike further splits into two more spikes in the subsequentlayer. Observe, that the entire polygon is visible from the chord connecting thetwo left-most points of the polygon. The distance between consecutive spikesin a layer, referred to as the bridge , is adjusted such that if at least one vertexin the interior of a spike is visible from a point p on c , then p cannot see anyinterior vertex of any other spike. This property is achieved by stretching thebridges vertically. More specifically, for 1 < i ≤ k , the length of the bridge ofthe i th layer is increased such that the property holds for layer i and then thebridge of the previous layer is adjusted accordingly. By iteratively stretching the3ridges from the last layer to the first layer, it can be ensured that the propertyholds for every layer. This property is illustrated in Figure 3 for k = 2. In thefirst layer, the point p sees an interior vertex of the first spike and no interiorvertex of the second spike. Similarly, in the second layer point p sees an interiorvertex of the second spike and no interior vertex of the first spike. p Figure 3: Point p sees interior points of at most one spike of any layer Chord visibility width of the Iterated Comb
Clearly, the chord whichsees the highest number of reflex vertices is the chord defined by the two left-mostvertices. Let this chord be c . The number of reflex vertices of the first layervisible from c is two. Similarly, the number of reflex vertices of the i th layervisible from c is 2 i . Summing up over all k layers, the number of reflex verticesvisible from c is Θ(2 k +1 ), and hence (cid:74) cvw (cid:75) = Θ(2 k +1 ). Point visibility width of the Iterated CombClaim 1.
Chord c contains at least one of the points in P which see the highestnumber of reflex vertices of P .Proof. Let q be a point in polygon P which sees the highest number of reflexvertices of P . Let p be a point on chord c which has the same y -coordinate as q .Assume p (cid:54) = q . Let r be a reflex vertex visible from q . Since P is monotone withrespect to y-axis, the triangle pqr must be empty. This implies that r is visiblefrom p as well. Hence, the point p also sees the highest number of reflex verticesin P since p sees at least as much as q . Refer to Figure 4 for an illustration.Without loss of generality, assume the point with highest visibility is thetopmost point on c , denoted by p . Both the reflex vertices in layer one are visiblefrom p . In each subsequent layer, p can see the reflex vertices that are in theinterior of the first spike, which is two reflex vertices, p cannot see any of theother reflex vertices in the other spikes by construction. Summing it up, we canconclude that 2 k reflex vertices are visible from p , and thus (cid:74) pvw (cid:75) = 2 k . Hencethe Iterated Comb has (cid:74) cvw (cid:75) ≥ Ω( (cid:74) pvw (cid:75) ) .4 p r Figure 4: Point p sees all the reflex vertices visible from q u vI ( v ) a b Figure 5: The vertex v sees a subinterval I ( v ) ⊆ s which is restricted by a and u . Next, we show that we can upper bound the chord visibility width in terms ofthe point visibility width.To this end, we prove the following lemma.
Lemma 2.
For every simple polygon, it holds that (cid:74) cvw (cid:75) ≤ (cid:74) pvw (cid:75) O ( (cid:74) pvw (cid:75) ) . The rest of this paragraph is dedicated to the proof of Lemma 2.For that purpose assume, we are given a simple polygon P together witha chord s ⊂ P . Furthermore, we assume that no point in P sees more than k = (cid:74) pvw (cid:75) reflex vertices of P . Let us denote by R the set of all reflex verticesthat see at least one point of s = seg( a, b ). Furthermore, we also include thetwo endpoints of s in the set R . As P is a simple polygon it holds that everyreflex vertex r ∈ R \ { a, b } sees a subsegment I ( r ) ⊆ s . For convenience, we alsocall I ( r ) an interval .Note that every interval is restricted by exactly two points in R , see Figure 5.In case of ambiguity, due to collinearities, we say the point in R closer to s is5he restricting point. Those vertices can be either the endpoints of s ( a and b )or a different reflex vertex in R . We show the following claim. Claim 2. If u is a reflex vertex that restricts the reflex vertex v then it holdsthat I ( v ) ⊆ I ( u ) . Proof.
The triangle T formed by I ( v ) and v is trivially convex and fully containedinside P . The reflex vertex u is on the boundary of the triangle and thus seesevery point of T . In particular also I ( v ).Given the previous claim, we construct the visibility restriction graph G asfollows. The vertices are formed by the points in R . We say that uv forms adirected edge, if u is restricted by v . We summarize a few useful properties of G in the following claim. Claim 3.
The visibility restriction graph of a polygon with point visibility widthat most k has the following properties.1. The segment endpoints a, b are the only two sinks.2. The out-degree is two for every vertex v ∈ R \ { a, b } .3. The in-degree is at most k − for every vertex v ∈ R .4. The longest path has at most k + 1 vertices.Proof. By definition, every reflex vertex is restricted by exactly two vertices in R . This implies Item 1 and 2.Any reflex vertex v can see itself and all its neighbors. Its in-degree neighborsare also reflex vertices. As no point can see more than k reflex vertices v has atmost k − p = u u . . . u l be a directed path. Then it holdsthat there is a point q ∈ I ( u l ) ⊆ . . . ⊆ I ( u ) ⊆ I ( u ) = s. The point q sees all reflex vertices of the path p . As no point sees more than k reflex vertices, it holds that p has at most k reflex vertices. As all but potentiallythe first vertex is a reflex vertex, we have l ≤ k + 1.The properties of the last claim enable us to give an upper bound on the sizeof G and thus also on the size of R . Claim 4.
The visibility restriction graph G of a polygon with point visibilitywidth (cid:74) pvw (cid:75) = k has at most k O ( k ) vertices.Proof. We organize G into layers depending on the distance from a and b . Notethat if layer i has t vertices then layer ( i + 1) has at most t · k vertices. As thereare at most k + 1 layers and the first layer has size two we get that G has atmost 2 + 2 k + 2 k + 2 k + . . . + 2 k k = k O ( k ) vertices. 6 Conclusion
We believe that local complexity has the potential to be a useful parameter. Wegave two ways to define local complexity in a rigorous way and showed how thosetwo ways relate to one another. We want to end with a few open questions.1. Can we find algorithms and data structures that can make use of low localcomplexity?2. Can we compute or approximate the point visibility width and chordvisibility width in an efficient manner? Note that this is more a theoreticalquestion. We do not necessarily need to know the chord visibility width ofa polygon to use the concept in the design and analysis of an algorithm.3. Are there other ways to formalize the idea of low local complexity withina polygonal region?
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