CCCG 2018, Winnipeg, Canada, August 8{10, 2018 An efficient approximation for point-set diameter in higher dimensions Mahdi Imanparast∗ Seyed Naser Hashemi∗ Ali Mohades∗y Abstract ministic algorithms with running time O(n log3 n) and O(n log2 n) are found for this problem in three dimen- In this paper, we study the problem of computing the sions. Finally, Ramos [4] introduced an optimal deter- diameter of a set of n points in d-dimensional Euclidean ministic O(n log n)-time algorithm in R3. space for a fixed dimension d, and propose a new (1+")- In the absence of fast algorithms, many attempts approximation algorithm with O(n + 1="d−2) time and have been made to approximate the diameter in low O(n) space, where 0 < " 6 1. We also show that the and high dimensions. A 2-approximation algorithm in proposed algorithm can be modified to a (1 + O("))- O(dn) time can be found easily by selecting a point of 2d − 1 approximation algorithm with O(n + 1=" 3 2 ) running S and then finding the farthest point of it by brute- time. These results provide some improvements in com- force manner for the dimension d. The first non- parison with existing algorithms in terms of simplicity, trivial approximation algorithm for the diameter is pre- and data structure. sented by Egecioglu and Kalantarip [5] that approxi- mates the diameter with factor 3 and operations cost O(dn). They also present an iterative algorithm with 1 Introduction t ≤ n iterations and the cost O(dn) for each itera- p p Given a finite set S of n points, the diameter of S, de- tion that has approximate factor 5 − 2 3. Agar- noted by D(S) is the maximum distance between two wal et al. [6] present a (1 + ")-approximation algorithm points of S. Namely, we want to find a diametrical pair in Rd with O(n="(d−1)=2) running time by projection p and q such that D(S) = maxp;q2S (jjp − qjj). Comput- pto directions. Barequet and Har Peled [7] present a ing the diameter of a set of points has a large history, d-approximation diameter method with O(dn) time. and it may be required in various fields such as database, They also describe a (1 + ")-approximation approach data mining, and vision. A trivial brute-force algorithm with O(n + 1="2d) time. They show that the running for this problem takes O(dn2) time, but this is too slow time can be improved to O(n + 1="2(d−1)). Similarly, for large-scale data sets that occur in the fields. Hence, Har Peled [8] presents an approach which for the most we need a faster algorithm which may be exact or is an inputs is able to compute very fast the exact diame- approximation. ter, or an approximation with O((n + 1="2d) log 1=") By reducing from the set disjointness problem, it running time. Although, in the worst case, the algo- can be shown that computing the diameter of n points rithm running time is still quadratic, and it is sensitive d to the hardness of the input. Chan [9] observes that in R requires Ω(n log n) operations in the algebraic computation-tree model [1]. It is shown by Yao that a combination of two approaches in [6] and [7] yields 3(d−1)=2 it is possible to compute the diameter in sub-quadratic a (1 + ")-approximation with O(n + 1=" ) time d− 1 time in each dimension [2]. There are well-known so- and a (1 + O("))-approximation with O(n + 1=" 2 ) lutions in two and three dimensions. In the plane, this time. He also introduces a core-set theorem, and shows problem can be computed in optimal time O(n log n), that using this theorem, a (1 + O("))-approximation d− 3 but in three dimensions, it is more difficult. Clarkson in O(n + 1=" 2 ) time can be found [10]. Recently, and Shor [3] present an O(n log n)-time randomized al- Chan [11] has proposed an approximation algorithm p d 2 +1 1 O(1) gorithm. Their algorithm needs to compute the inter- with O((n= " + 1=" )(log " ) ) time by applying section of n balls (with the same radius) in R3. It may the Chebyshev polynomials in low constant dimensions, be slower than the brute-force algorithm for the most and Arya et al. [12] show that by applying an efficient practical data sets, and it is not an efficient method for decomposition of a convex body using a hierarchy of higher dimensions because the intersection of n balls Macbeath regions, it is possible to compute an approx- (d−1) with the same radius has a large size. Some deter- 1 2 +α imation in O(n log " + 1=" ) time, where α is an arbitrarily small positive constant. ∗Department of Mathematics and Computer Sci- ence, Amirkabir University of Technology, Tehran, Iran, 1.1 Our results [email protected], [email protected] yLaboratory of Algorithms and Computational Geom- etry, Amirkabir University of Technology, Tehran, Iran, In this paper, we propose a new (1 + ")-approximation [email protected] algorithm for computing the diameter of a set S of 30th Canadian Conference on Computational Geometry, 2018 ` ` Table 1: A summary on the complexity of some non- constant approximation algorithm for the diameter of a point set. Our results are denoted by +. Ref. Approx. Factor Running Time n ξ [6] 1 + "O( ) ξ (a) (b) "(d−1)=2 2(d−1) [7] 1 + "O(n + 1=" ) 2 2d 1 Figure 1: (a) A set of points in R and an ξ-grid. Initial [8] 1 + "O((n + 1=" ) log " ) 3(d−1) points are shown by blue points and their correspond- [9] 1 + "O(n + 1=" 2 ) d−2 ing central cell-points are shown by circle points. (b) + 1 + "O(n + 1=" ) Rounded point set S^. d− 1 [9] 1 + O(") O(n + 1=" 2 ) d− 3 [10] 1 + O(") O(n + 1=" 2 ) n d +1 1 O(1) results in a point set S^ . Let, B (p) be a hypercube with [11] 1 + O(") O((p + 1=" 2 )(log ) ) 1 δ " " side length δ and central-point p. We restrict our search (d−1) 1 2 +α [12] 1 + O(") O(n log " + 1=" ) domain for finding diametrical pairs of the first rounded 2d − 1 ^ + 1 + O(") O(n + 1=" 3 2 ) point set S into two hypercubes B2ξ1 (^p1) and B2ξ1 (^q1) corresponding to two diametrical pair pointsp ^1 andq ^1 in the point set S^1. Let us use two point sets B1 and B2 for maintaining points of the rounded point set S^, which are inside two hypercubes B2ξ1 (^p1) and B2ξ1 (^q1), n points in Rd with O(n + 1="d−2) time and O(n) respectively (see Figure 2). Then, it is sufficient to find space, where 0 < " 6 1. Moreover, we show that the a diameter between points of S^, which are inside two proposed algorithm can be modified to a (1 + O("))- point sets B1 and B2. We use notation Diam(B1; B2) 2d − 1 approximation algorithm with O(n+1=" 3 2 ) time and for the process of computing the diameter of the point O(n) space. As stated above, two new results have been set B1 [B2. Altogether, we can present the following recently presented for this problem in [11] and [12]. It algorithm. should be noted that our algorithms are completely dif- ferent in terms of computational technique. The poly- nomial technique provided by Chan [11] is based on us- Algorithm 1: APPROXIMATE DIAMETER (S;") d ing Chebyshev polynomials and discrete upper envelope Input: a set S of n points in R and an error parameter ". subroutine [10], and the method presented by Arya et Output: Approximate diameter D~. al. [12] requires the use of complex data structures to 1: Compute the axis-parallel bounding box B(S) for the point set S. approximately answer queries for polytope membership, 2: ` Find the length of the largest side in B(S). p p p directional width, and nearest-neighbor. While our al- 3: Set ξ "`=2 d and ξ1 "`=2 d. gorithms in comparison with these algorithms are sim- 4: S^ Round each point of S to its central-cell point pler in terms of understanding and data structure. We in a ξ-grid. ^ ^ have provided a summary on the non-constant approx- 5: S1 Round each point of S to its nearest grid-point in a ξ -grid. imation algorithms for the diameter in Table 1. 1 6: D^1 Compute the diameter of the point set S^1 by brute- force manner, and simultaneously, a list of the diam- ^ 2 The proposed algorithm etrical pairs (^p1; q^1), such that D1 = jjp^1 − q^1jj. ^ 7: Find points of S which are in two hypercubes B1 = B2ξ1 (^p1) and B = B (^q ), for each diametrical pair (^p ; q^ ). In this section, we describe our new approximation al- 2 2ξ1 1 1 1 ^ gorithm to compute the diameter of a point set. In our 8: D Compute Diam(B1; B2), corresponding to each diamet- rical pair (^p1; q^1) by brute-force manner and return the algorithm, we first find the extreme points in each co- maximum value between them. ordinate and compute the axis-parallel bounding box of 9: D~ D^ + "`=2. S, which is denoted by B(S). We use the largest length 10: Output D~.