Theoretical Computer Science 702 (2017) 48–59 Contents lists available at ScienceDirect Theoretical Computer Science www.elsevier.com/locate/tcs α -Concave hull, a generalization of convex hull Saeed Asaeedi, Farzad Didehvar ∗ , Ali Mohades Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran a r t i c l e i n f o Article history: Received May 2017 Received in revised form 14 August 2017 Accepted 19 August 2017 Available online 24 August 2017 Communicated by P.G Spirakis Keywords: Convex hull α -Shape α -Concave hull Minimum area polygon NP-complete Approximation algorithm a b s t r a c t Bounding hulls such as convex hull, α -shape, χ -hull, concave hull, crust, etc offer a wide variety of useful applications In this paper, we explore another bounding hull, namely α -concave hull, as a generalization of convex hull The parameter α determines the smoothness level of the constructed hull on a set of points We show that it is NPhard to compute α -concave hull on a set of points for any < α < π This leads us to a generalization of Fekete work (when α = π ) We also introduce α − M AC P as an NP-hard problem similar to the problem of computing α -concave hull and present an approximation algorithm for α − M AC P The paper ends by implementing the proposed algorithm and comparing the experimental results against those of convex hull and α -shape models © 2017 Elsevier B.V All rights reserved Introduction The convex hull of a set of points S in the plane, is the smallest convex polygon containing S Many algorithms have been introduced for computing the convex hull of points [1–8] Convex hull is used in many fields such as pattern recognition, image processing, GIS, sensor networks, path planning, etc [9–15] Computing the convex hull of S has an optimal O (n log n) algorithm [3] The Minimum Area Polygon (M A P ) of a set of points S in the plane, is the problem of computing the smallest simple polygon containing S The Maximum Area Polygon (M A X P ) of a set of points S in the plane, is the problem of computing the simple polygon with maximum area that passes through all points of S Fekete in [16] considered M A P for the grid points and denoted this problem by G A P He demonstrated that G A P and M A X P are NP-complete [16,17] He also presented a 12 -approximation algorithm for M X A P [18] and proved that there does not exist any ( 23 + ε )-approximation algorithm for this problem for < ε < 13 , unless P = N P Here, we consider convex and concave hulls on the set of points in the plane We introduce a generalization of convex hull and M A P called α -concave hull such that the parameter α limits the internal angles of the constructed hull We then show that it is NP-hard to compute α -concave hull on a set of points for any < α < π Bae et al [19] used convex hull to cover a set of points with convex sets of minimum total area or perimeter size In [20], Jin-Seo Park and Se-Jong Oh showed that for identifying the exact area occupied by a set of points, concave hull region was more useful than convex hull The concept of Concave Hull was first introduced as (non-convex footprint) by A Galton and M Duckham in [21] and then it was expanded in [22] In [20] an algorithm was presented to compute * Corresponding author E-mail address: didehvar@aut.ac.ir (F Didehvar) http://dx.doi.org/10.1016/j.tcs.2017.08.014 0304-3975/© 2017 Elsevier B.V All rights reserved S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 49 Concave Hull in d dimensions Concave Hull has an effective applicability in the fields of shape reconstruction [23,24], material computation [24], GIS [25], dataset classification [26], etc The α -shape is another generalization of convex hull which was introduced by Edelsbrunner in [27] The α -shape has expanded applicability in shape reconstruction [28], space decomposition [29], sensor networks [30], bioinformatics [31], feature detection [32], 3D visualization (of brain tumor) [33], etc χ -hull and Crust are another non-convex hulls to cover a set of points The Crust concept was developed as a graph on the set of points in R [34] and R [35] and χ -hull was introduced in [36] as a simple subpolygon of Delaunay triangulation of the set of points The α -concave hull on a set of points in the plane is a non-convex hull with angular constraints under the minimum area condition For α = 0, computing α -concave hull is equivalent to that of computing convex hull with O (n log n) optimal algorithm For α = π , this problem converts to M A P as it is proved to be NP-complete In this paper, we put forward an NP-hard problem called Maximum Area Clustering Problem (α − M AC P ) on a set of points which produces the same results as the α -concave hull We also present a 14 -approximation algorithm for α − M AC P The structure of the paper is as follows In section we unveil the concept of α -polygon and α -concave hull and prove that computing an α -polygon with a given area on a set of points is an NP-complete problem Likewise, we demonstrate that it is NP-hard to compute α -concave hull on a set of points for any < α < π In section we introduce α − M AC P as a problem equivalent to that of computing α -concave hull and an approximation algorithm is presented to deal with it This is then followed by a discussion on the factor of the approximation algorithms for α -concave hull Section is devoted to experimental work and implementing our proposed approximation algorithm We then compare the results achieved by this solution against known methods that employ convex hull and α -shape to verify its outperformance Section concludes the paper α -Concave hull As was mentioned in the previous section, many concepts have been studied as concave hull of a set of points Previous studies on concave hull did not consider any limitation on angles nor area of the constructed polygons In this section we disclose the concept of α -concave hull as a concave hull with angular constraint under the minimum area condition Let us first define the concept of α -polygon as follows: Definition 2.1 Let A be a simple polygon A is called α -polygon if all internal angles of A are equal or less than π + α Definition 2.2 α -Concave hull on a set of points S is an vertices of the polygon are a subset of S α -polygon containing S with minimum area such that all the Based on those definitions, when α = 0, α -polygon is equal to a convex polygon, hence, the α -concave hull of S for α = is equal to convex hull of S In the case of α = π , α -polygon is equal to a simple polygon, hence, computing α -concave hull of S for α = π converts to M A P for S Consequently, α -concave hull definition is a generalization of the convex hull concept and computing α -concave hull, is a generalization of M A P The computing convex hull problem has O (n log n) optimal algorithm while M A P is NP-complete Fig illustrates α -concave hulls on a set of points where we consider various values for the parameter α As stated above, when α = 0, α -concave hull is equal to convex hull of the points For α = 12◦ , concave angles would be constructed in α -concave hull When α > 120◦ , the boundary of α -concave hull would pass through all of the points When α = 180◦ , α -concave hull is a simple polygon with minimum area that passes through the points which is a solution of M A P for this set of points Remark α -Concave hull on a set of points is not unique for a fixed value of α Fig shows an example of two different α -concave hulls on a set of points Based on the Remark 1, α -concave hull on a set of points is not unique for a fixed value of α The following theorem, however, expresses that the set of boundary points on various α -concave hulls with a fixed value of α is unique on the set of points Theorem 2.1 For a set of points S and a fixed value of α , if A and A are two α -concave hulls on S, then boundary points of A and A will be equal Proof Let B , B and C H be the boundary points of A , A and convex hull of S, respectively By reductio ad absurdum, suppose B and B are not equal Without loss of generality, we have a point called z in which z ∈ / B and z ∈ B So, the polygonal chain C = ax1 x2 z xn b from a to b is on B such that a, b ∈ C H are two adjacent vertices in C H As a, b ∈ B , so the polygonal chain C = ay y ym b is also placed on B Fig shows the chains Since the beginning and ending vertices of the chains C and C are equal and z ∈ C but z ∈ / C , C and C cross each other at least once at a point c As 50 S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Fig Various values of Fig Images a and c display two α for α -concave hull on a set of points α -concave hulls, image b shows both of them having the same area Fig The edges ax1 x2 b and ay y b show the chain related to A and A , respectively far as A and A contain all of the points of S, the cross point c will be located on the first and last edges In other words, c ∈ xn b and c ∈ ay If the cross point locates on another point of C such as c ∈ xi x j , then all of the boundary points of C which are succeeded by x j i.e x j +1 , x j +2 , , xn will be located outside of the polygon A which contradicts the definition of α -concave hull Let αi denote the internal angle of xi in A and βi denote the internal angle of y i in A We construct polygon A by eliminating xn b and ay edges and connecting the two xn and y vertices to each other The area of polygon A is less than polygons A and A Since the points a and b are on the boundary of convex hull, the internal angle of xn in A is smaller than αn and the internal angle of y in A is smaller than β1 Hence, all internal angles of A will be smaller than π + α and A will be an α -polygon with smaller area than A and A which contradicts the definition of α -concave hull ✷ Computing an α -polygon on a set of points is a P problem (it is enough to construct a polygon that contains the points and its internal angles are less than π + α ) The convex hull of the set of points has these features for any value of α But, we show that computing an α -polygon with a given area on a set of points is an NP-complete problem Theorem 2.2 Computing an α -polygon with a given area on a set of points is NP-complete for any < α < π S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Fig Regular 6-gon with the area of A R = 6∗416 cot( π6 ) and α -concave hull with the area of x = A R − 51 14∗16 Proof We compute and analyze the internal angles and the area of polygon in polynomial time Checking the points which are inside or outside of a polygon is also executable in polynomial time Hence, computing an α -polygon with a given area on a set of points is in NP Consider S as a set of points, x as a real number and α as a given angle The goal is to find an α -polygon containing S with the area of x We reduce the subset sum problem to this problem The subset sum problem is a well-known set recognition problem which is NP-complete [37] Let M = {m1 , m2 , , mn } be a set of natural numbers and k be a given natural number The subset sum problem is to determine whether or not there exists a subset of M whose sum is equal to k The following steps reduce the subset sum problem to that of computing α -polygon with a given area for any < α < π : set r = max {mi }, 1≤i ≤n set t = 2r cot(α /2), let R be a regular n-gon such that the length of each edge is t, let E = {e , e , , en } be the set of all edges of R, insert n points Q = {q1 , q2 , , qn } within the R such that each point q i places on the perpendicular bisector of e i at the distance of mi Any point q i corresponds to mi , set x = A R − kt and P = Q V R such that A R = nt4 cot( πn ) refers to the area of R and V R refers to the all vertices of R, compute α -polygon on S with area of x, let M ⊆ M be the answer of subset sum problem Any q i on the boundary of computed α -polygon corresponds to mi ∈ M Fig illustrates further the reduction steps ✷ Fig shows a solution for the subset sum problem using α -polygon when α = π /2 in which M = {2, 3, 4, 6, 7, 8} and k = 14 α -polygon with area x passes through the points that correspond to the numbers 2, and So M = {2, 4, 8} is an answer to the given subset sum problem In [16], Fekete demonstrated that the grid avoiding polygon problem (G A P ) is NP-complete G A P is the problem of computing a simple polygon on a set of points on a grid that avoids any grid points inside or on the boundary He argued that such polygon is equivalent to a polygon with minimum area of n2 − on its set of points As far as Theorem 2.2 is concerned, computing an α -polygon with an area of n2 − on a set of points on a grid, is still an NP-complete problem Corollary 2.2.1 Let S be a set of points on a grid and < α < π , the problem of finding a simple polygon on S which avoids any grid points in the boundary or inside of it such that all of the internal angles are less than or equal to π + α is a generalization of G A P Based on Theorem 2.2, this extended version of problem is NP-complete Theorem 2.2 shows that computing α -polygon with a given area on a set of points is NP-complete An α -concave hull on a set of points is an α -polygon with minimum area on these points In sequel, we reduce the clustering problem to that of computing α -concave hull on a set of points The clustering problem is presented as follows: Definition 2.3 [37] Let d : S × S → Z 0+ be a distance function such that S is a finite set of points For two integers k, B ∈ N, the decision whether or not a partition of S into k disjoint subsets C , C , , C k exists such that for all ≤ i ≤ k and all x, y ∈ C i , d(x, y ) ≤ B, is called a clustering problem A Clustering Problem remains NP-complete even for fixed k = [37] In [38] a global optimization algorithm is presented to solve the hard clustering problem We now introduce the triangle clustering problem as an example of a clustering problem, but first let us define α -path as below: 52 S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Definition 2.4 Let a1 a2 a3 be a triangle and S be a set of points inside the triangle For p , p , , p r ∈ S and i = j ∈ {1, 2, 3}, we call the chain L = , p , p , , pr , a j an α -path, if the polygon p p pr a j is a simple empty polygon such that all internal angles of pˆi in polygon is greater than α Definition 2.5 Let S be a set of points inside the triangle a1 a2 a3 Triangle clustering problem is a decision problem that asks whether or not there exists a clustering of S into partitions C , C and C such that for all ≤ i ≤ and all x, y ∈ C i , dα (x, y ) ≤ in which the distance function dα : S × S → {0, 1, 2} is defined as follows: ⎧ ⎨ if x = y dα (x, y ) = if ∃α − path = , , x, , y , , a j ⎩ (1) o.w Based on [39] and [37], a triangle clustering problem is NP-complete In sequel, by applying Lemma 2.3, we reduce this problem to that of computing α -concave hull Lemma 2.3 Let S be a set of points inside the triangle a1a2 a3 and A be an α -concave hull on S ∪ {a1 , a2 , a3 } A passes through all points of S if and only if the triangle clustering problem with distance function dβ : S × S → {0, 1, 2} (which β = π − α ) clustered S into partitions Proof To prove the “if” part, let us assume A = a1 C a2 C a3 C a1 passes through all points of S Based on the definition of α -concave hull, a1 C a2 , a2 C a3 and a3 C a1 are all β -paths Hence, C , C and C are partitions of S using distance function dβ In order to prove the “only if” part, without loss of generality, assume that points of S are on the grid Based on the Pick’s theorem [40], the area of a polygon on S is equal to b2 + m − as b is the number of grid points on the boundary of polygon and m is the number of grid points which are inside the polygon Now let us suppose that S is clustered into partitions C , C and C Since dβ (x, y ) = for any x, y ∈ C , C a j is a β -path Similarly, a j C ak and ak C are β -paths such that i = j = k ∈ {1, 2, 3} As α = π − β , the polygon A = C a j C ak C , is an α -polygon which passes through all points of S By reductio ad absurdum, suppose that there is an α -concave hull as A which does not pass through all points of S Based on the Pick’s theorem, and the fact that the number of internal points of A is more than that of A and the number of boundary points of A are less than that of A, the area of A is less than that of A This contradicts the definition of α -concave hull ✷ Theorem 2.4 Computing α -concave hull on a set of points is an NP-complete problem for any < α < π Proof The following steps reduce the triangle clustering problem to that of computing α -concave hull: set S = S ∪ {a1 , a2 , a3 } and β = π − α , compute β -concave hull on S and let A = a1 , C , a2 , C , a3 , C , a1 be the computed hull such that C , C and C are disjoint subsets of S, if A passes through all points of S, the partitions C , C and C cluster S as an answer to the triangle clustering problem Otherwise, S cannot be clustered into partitions using dα Based on Lemma 2.3, S clusters into C , C and C if and only if A passes through all points of S ✷ Fekete showed that M A P is an NP-complete problem As long as α = π , computing α -concave hull converts to M A P , and Theorem 2.4 showed that the generalization of this problem is NP-hard for any < α < π Approximation algorithm Let S be a set of points in the plane To construct a polygon containing S we employ two methods, called incremental and decremental methods In the decremental method, first we compute the convex hull of S and then by eliminating the subpolygons in a decremental manner, we obtain the desired polygon But in the incremental method, first by finding small polygons on the points of subsets of S and gathering them, we obtain the desired polygon in an incremental manner So, these methods enable us to construct a polygon with minimum area on a set of points: First by eliminating the subpolygons with maximum area from convex hull and second by gathering the triangles with minimum area on the points In this section we investigate α -M AC P as an NP-hard problem to compute α -concave hull in a decremental manner We also present an approximation algorithm for α -M AC P and discuss the factor of incremental approximation algorithms (if it exists) for M A P on a set of points In order to compute an approximated polygon with minimum area, triangles with minimum area are gathered consecutively Considering the area of smallest triangle on a set of points leads us to a well-known problem in the field of discrete geometry and discrepancy theory [41], called the Heilbronn’s triangle problem S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 53 Table Works on Heilbronn’s triangle problem Year Author Comment Bound 1950 1951 1951 1972 1972 1972 1981 1982 1999 2001 2003 Heilbronn ˝ (Appendix of [45]) P Erdos K.F Roth [45] W.M Schmidt [46] K.F Roth [47] K.F Roth [43] Komlós, Pintz, Szemerédi [44] Komlós, Pintz, Szemerédi [42] T Jiang, M Li, P Vitányi [48] Barequet [49] Lefmann [50] Conjecture of the upper bound The Heilbronn’s conjecture is tight if it is true The first nontrivial upper bound Improving the upper bound Improving the upper bound Improving the upper bound Improving the upper bound Presenting a lower bound and disproving the Heilbronn’s conjecture The expected size of Heilbronn’s triangles Presenting a lower bound in R d Improving lower bound in R d = O (1/n2 ) = (1/n2 ) n = O (1/n log log n) n = O (1/n log n) 1.105−ε ) n = O (1/n 1.117−ε ) n = O (1/n 8/7−ε ) n = O (1/n (log n/n ) n = Exp ( n ) = (1/n3 ) d (1/nd ) n = d (log n/nd ) n = 2005 2004–2016 Brass [51] Presenting an upper bound in R d Online case and approximation in R and R d n n d n = O (1/n 2d+1 2d ) [52–55] Fig A – a simple polygon CH – the convex hull Let us consider S as a set of n points in a unit square and ( S ) as the area of the smallest triangle determined by points of S Let also Q n be the set of all point sets with cardinality n in a unit square Thus, n = Sup { ( S )| S ∈ Q n } Heilbronn’s triangle problem is the problem of analyzing the value of n This problem was first raised by Heilbronn in 1950, who expressed n = O ( n1 ) as a trivial upper bound and claimed that n = O ( n12 ) In 1982, Komlós, Pintz and Szemerédi disproved the Heilbronn’s conjecture and expressed ε > we could have n n = n ( log ) [42] Also in 1972, Roth showed that by any n2 = O ( n1.117−ε ) [43] In 1981, Komlós, Pintz and Szemerédi improved the upper bound and obtained = O ( n8/7−ε ) [44] Table displays some of the studies on this problem In what follows next, we explore α -M AC P as a decremental method and then use the bound Szemerédi, et al achieved in [44] to investigate the approximation factor for M A P by employing the incremental methods Consider S as a set of points, C H as the convex hull of S with m edges and E i as the ith edge of C H As depicted in the Fig 5, any polygon A on S, clusters the internal points of C H into m + partitions C , C , , C m+1 For ≤ i ≤ m, any partition C i corresponds to the edge E i and C m+1 corresponds to the internal points of A The corresponding region to E i , which is denoted by R i , is equal to the region between E i and edges of A which passes through C i Hence, the area of A is computed as: n Area( A ) = Area(C H ) − m i =1 Area( R i ) (2) m Area( R i ) i =1 According to equation (2), the area of polygon A is minimum if and only if the value of is maximum Hence, M A P for a set of points is equal to the clustering problem of points in a way that the sum of areas of corresponding regions of convex hull edges should be maximum In the following, we define α -M AC P Definition 3.1 Let S be a set of points, C H be the convex hull of S with m edges and E i be the ith edge of C H The α -M AC P on S is the problem of clustering the internal points of C H into m + partitions C , C , , C m+1 such that the sum of the area of all corresponding regions of E i is maximum and all internal angles of the constructed polygon are less than π + α Suppose S is a set of points on a grid and A is a simple polygon on S which clusters the points of S into m + partitions C , C , , C m+1 Based on the Pick’s theorem, the area of R i is equal to |b i |/2 + |mi | − such that b i is the m set of grid points on the boundary of R i and mi is the set of grid points contained in the interior of R i If R = i =1 R i , m m m ∗ M = i =1 |mi | and B = i =1 |b i |, then Area( R ) = i =1 Area( R i ) = B /2 + M − m Let polygon A be the optimal answer to 54 S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Algorithm 1 let F = S − { boundary points of C H } as available points, for i = to i = m (a) construct R i with maximum area using available points F , (b) let B = boundary points of R i , (c) set F = F − B, set R = im=1 R i m Area( R ∗i ) = B ∗ /2 + M ∗ − m will be maximum In the following, we present a greedy i =1 algorithm as an approximation algorithm for α -M AC P by maximizing the area of each region R i : M X A P for a set of points is the problem of finding a polygon on points with maximum area that passes through all points Fekete in [16] proved that M X A P is an NP-complete problem In the following lemma, we find the approximation factor of Algorithm based on the factor for M X A P α -M AC P on S Hence, Area( R ∗ ) = Lemma 3.1 If there exists any α -M AC P τ -approximation algorithm for M X A P , then Algorithm yields a τ2 -approximation algorithm for Proof Let S be a set of points on a grid and suppose that C H is the convex hull and A ∗ is the α -concave hull on S Suppose also that E X is an exact algorithm and A P X is a τ -approximation algorithm for M X A P on S The computed polygons which have used the E X and A P X in the second step of Algorithm are denoted by A and A p , respectively The union of constructed regions in the optimal solution for α -M AC P is denoted by R ∗ and the computed regions which have used the E X and A P X are denoted by R and R p , respectively Now we have Area( R ) = im=1 Area( R i ), Area( R ∗ ) = im=1 Area( R ∗i ) and |b∗ | p |b | |b | Area( R p ) = im=1 Area( R i ) such that Area( R ∗i ) = 2i + |m∗i | − 1, Area( R i ) = 2i + |mi | − and Area( R i ) = 2i + |mi | − (b denotes the set of boundary grid points, m denotes the set of internal grid points and | S | denotes the cardinality of the set S.) We denote the cardinality of the set of internal, external and boundary grid points of polygon A ( A ∗ ) by M (M ∗ ), O (O ∗ ) and B (B ∗ ), respectively In other words, B = im=1 |b i |, B ∗ = im=1 |b∗i |, M = im=1 |mi | and M ∗ = im=1 |m∗i | If mci is the set of common internal grid points of R i and R ∗i , and moi is the set of internal grid points of R ∗i − R i , then moi = m∗i − mci Similarly, boi = b∗i − bci Considering M o = im=1 |moi |, M c = im=1 |mci |, B o = im=1 |boi | and B c = im=1 |bci |, we have: p Area( R ∗ ) = B∗ ≤ B + M∗ − m = + Bo Bc + Bo p + Mc + Mo − m + M + M o − m = Area( R ) + Bo In inequality (3), we have Area( R ∗ ) ≤ Area( R ) + M o + o that M + B2 m i =1 Area( R i ) o |moj | + + Mo Bo (3) So, to prove Area( R ∗ ) ≤ Area( R ), it is enough to show ≤ Area( R ) By reductio ad absurdum, suppose that M o + and there exists a j such that: |boj | p Bo > Area( R ) Consequently, > Area( R j ) |bo | m |moi | + 2i i =1 > (4) Since the moj and boj belong to the available points F of Algorithm 1, inequality (4) contradicts step 2.a of Algorithm in which the region R j constructs the largest polygon using the available points F Hence, Area( R ∗ ) ≤ Area( R ) ⇒ Area( R ∗ ) ≤ Area( R ) ⇒ τ Area( R ∗ ) ≤ τ Area( R ) p Since R i and R i are the ith constructed regions by the exact and have: τ -approximation algorithms, respectively, then we p ∀1 ≤ i ≤ m τ Area( R i ) ≤ Area( R i ) ⇒ τ Area( R ) ≤ Area( R p ) By inequalities (5) and (6), we have: τ2 Area( R ∗ ) ≤ Area( R p ) ≤ Area( R ∗ ) Remark Fekete presented a ( 23 -approximation (5) (6) ✷ algorithm for M X A P [18] If < ε < 13 , he proved that there was + ε )-approximation algorithm for this problem unless P = N P Hence, Based on Lemma 3.1, Algorithm is a algorithm for α -M AC P and this problem has no ρ -approximation algorithm for ρ > 23 unless P = N P Since the time complexity of the presented algorithm in [18] is O (n log n), then the time complexity of Algorithm is O (nm log n) such that n is the number of points and m is the number of convex hull edges no -approximation S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 55 Remark If A is an optimal solution for α -M AC P on S and C H is the convex hull of S, then B = C H − A is an optimal solution for M A P Based on Lemma 3.1, a better approximation for M X A P leads us to a better approximation for α -M AC P Likewise, a better approximation for α -M AC P yields us a better approximation for M A P The presented algorithm in this section is a decremental approximation algorithm for α -M AC P It is notable that this algorithm does not give us any approximation factor for M A P The next theorem discusses the factor of incremental approximation algorithms (if it exists) for M A P Definition 3.2 M A P n is the problem of finding a minimum area polygon on a set of n points on a grid in a unit square Theorem 3.2 Let S be a set of n points on a grid If M A P n on S has a ρn ≥ − n1/C7−ε such that C is a positive constant ρn -approximation algorithm, then for any ε > we have Proof Let u be the area of grid cells, O P T be the polygon with minimum area A ∗ on S and A P X be the constructed polygon by the ρn -approximation algorithm on S with area A Hence, we have: A ∗ ≤ A ≤ ρn A ∗ (7) Based on the Pick’s theorem, the minimum possible value for the area of polygon with n points is equal to ( n− )u Suppose that this optimal polygon G A P exists on S Hence, by inequality (7) we have: n−2 A ≤ ρn u ( (8) ) We define A as the area of smallest triangle in triangulation of A P X polygon The number of A P X triangles is equal to n − 2, so, we have: A ≥ (n − 2) (9) A Based on the inequality (8) and (9): ρn u ≤ A Clearly A (10) A ≤ ≤ n n So, by inequality (10): ρn u + ⇒ ρn ≥ u (2 A − Based on [42,44], there are positive constants C and c such that for any c log n n2 < n (11) n) ε > 0, we have: C < n (12) −ε inequalities (11) and (12) yield: ρn ≥ (2 A u Obviously, ρn ≥ − A − C n −ε ) (13) ≥ u /2 Hence, by inequality (13) we have: 2C (14) un −ε Let Q be the set of all grid points in the unit square Since S is the set of n points on Q , the value of n is maximum when S = Q Clearly, in this configuration we have u = 1/n The inequality (14) yields: ρn ≥ − 2C n −ε ✷ (15) Corollary 3.2.1 If M A P for a set of points has a ρ -approximation algorithm, we have ρ = supn∈N {ρn } By inequality (15), we conclude ρ ≥ Corollary 3.2.2 Let γ be the factor of approximation algorithm for the problem of computing an Since this problem is a generalization of M A P , we conclude from Theorem 3.2 that γ ≥ α -concave hull on a set of points 56 S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Fig Approximation errors (Er) using convex hull (CH), α -shape (AS) and α -concave hull (AH) Implementation, application and experimental results Often in path planning, approximated simple shapes are used instead of complex obstacles [56,57] In [56,57] convex hull is used to decompose a complex object to simpler pieces In spite of above, Ghosh and Amato showed that convex hull approximation might extremely wastes the free space [58] α -shape is a more suitable concept to reconstruct shapes and surfaces Edelsbrunner used α -shape for reconstruction of a shape and surface [28] Shape representations and description techniques were examined in [59] and found to be applicable for searching images from remote database In [60] polygonal shapes were reconstructed in the real plane using Fourier samples In this section, we will compare the results obtained from Algorithm 1, dealt with in the previous section, with those from reported methods of polygon approximation such as convex hull and α -shape 10 test samples were considered for this operation Each sample contained 100 random scaled polygons with fixed cardinality Since computing α -concave hull is NP-hard, we used Algorithm to construct approximated α -concave hull Definition 4.1 Let S be a set of points An α -shape on S which constructs a simple polygon is called α -shape approximated polygon of S and the constructed polygon on S by using Algorithm is called α -concave hull approximated polygon of S In the approximation of any random polygon using α -shape, the value of α has been computed in a way that the α -shape approximated polygon of vertices constructs the smallest polygon containing the random polygon Likewise, in the approximation of any random polygon using Algorithm 1, the value of α has been computed in a way that the α -concave hull approximated polygon of vertices constructs the smallest polygon containing the random polygon Fig compares α -shape and α -concave hull approximation methods based on the approximation error The x-axis indicates vertices cardinality for each of the samples and the y-axis indicates approximation errors of each of the methods Here, the approximation error of a method is equal to the area difference between constructed random polygon and the approximated polygon which is developed by the same method Fig compares accuracy of the two methods based on the number of desired results in each case As an example, considering 100 random examples with 30 vertices, in 64 cases α -concave hull approximated polygon is more accurate than its α -shape and in the other hand, in 23 cases α -shape approximated polygon is more accurate than its α -concave hull counterpart and in 12 cases the same results are obtained Theorem 4.2 expresses that for any polygon, by adding another vertex to the polygon at a far enough distance from the other vertices, the approximation error of α -concave hull would be less than that of α -shape By employing Theorem 4.2, we have theoretical justification for the results Lemma 4.1 Let S be a set of points, x, y ∈ S and R be the largest radius of the empty circle which passes through the points x, y R is infinite if and only if x and y are neighboring boundary points of the convex hull of S Proof Based on the features of Voronoi diagram, both x and y points are neighboring boundary points of the convex hull, if and only if their Voronoi cells are infinite and the shared Voronoi edge between cells is infinite segment This infinite segment is the locus of the center of empty circles which passes through x and y Hence, if we consider these circles radius as R, it is infinite if and only if x and y are neighboring boundary points of the convex hull of S ✷ S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 57 Fig Number of superiority in approximation methods Fig On the left an α -concave hull and an α -shape have constructed the same polygon The upper right shows the the lower right shows the α -concave hull after adding x Fig Right side is a convex hull and the left side is an α -shape polygon after adding x and α -concave hull of points Theorem 4.2 Let S be a set of points and C H be the boundary points of the convex hull of S and I S = S − C H be the internal points of the convex hull of S There is a point x such that α -shape approximated polygon of P ∪ {x} passes through no points of I S Proof Let R S be the largest radius of the empty circle which passes through the points of I S Let the distance between the point x and the set S be more than d = 2R S No α -shape approximated polygon of S can pass through any point of I S to cover point x By Lemma 4.1 we conclude that R S is finite ✷ Based on Theorem 4.2, by adding a point x to a set of points S, if x is located at a far enough distance from S, it causes global changes (changes which are not only in adjacent edges of x) in the α -shape approximated polygon But in the approximated polygon constructed by α -concave hull, these changes would be local (changing just the adjacent edges of x) Fig shows these changes in an α -shape approximated polygon and α -concave hull approximated polygon Theorem 4.2 and Fig justify that α -concave hull is more desirable for polygon approximation in comparison to α -shape Theorem 4.2 and the similarity of α -M AC P to α -concave hull computation, confirm the results of Fig and Igrashi et al in [61] used the convex hull on obtained mesh from 3D scanning of an object to design a cover Lucieer et al used the α -shape to decompose a shape into visually meaningful parts in [62] In [63] a mesh reconstruction Delaunaybased method is presented and the quality of mesh reconstruction is considered Fig sketches fitness of α -concave hull in comparison with that of convex hull in shape detection There are many suitable algorithms such as [34,64,65] for shape detection and curve reconstruction but in this paper our goal is not to compete with these algorithms in general, but rather, to make a comparison among the three concepts α -concave hull, convex hull and alpha shape in the field setting of shape detection 58 S Asaeedi et al / Theoretical Computer Science 702 (2017) 48–59 Conclusion We introduced a generalization of the concept of convex hull named α -concave hull Here, the parameter α shows 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the number of boundary points of A are less than that of A, the area of A is less than that of A This contradicts... into partitions Proof To prove the “if” part, let us assume A = a1 C a2 C a3 C a1 passes through all points of S Based on the definition of α -concave hull, a1 C a2 , a2 C a3 and a3 C a1 are all... each other The area of polygon A is less than polygons A and A Since the points a and b are on the boundary of convex hull, the internal angle of xn in A is smaller than αn and the internal angle