Analysis of half-space range search using the k-d search skip list. Here we analyse the expected time for half-space

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1 Analysis of half-space range search using the k- search skip list Mario A. Lopez Brafor G. Nickerson y 1 Abstract We analyse the average cost of half-space range reporting for the k- search skip list. The k- search skip list is a ynamic ata structure requiring (kn) space for a set of n k-imensional points. Uner the assumption of uniform ranom istribution of points in k- space, an a ranom query hyperplane, our results show that for the - case, the average running time for half-space range reporting using the k- search skip list is O( p n + t), where t is the number of reporte points. Keywors: multiimensional ata, algorithms, half-space range reporting, ranom hyperplane Introuction Algorithms for orthogonal range search an half space range search are extensively covere in the literature (e.g. [1, 3, 5, 11, 13]). For a survey of extensive literature on the subject the reaer may want to consult [6] or []. As mentione in [1], algorithms for range search traitionally fall into two categories: best worst case search time achievable by a linear space ata structure, an minimum storage space to enable polylog (worst case) search time. The algorithm we analyse here falls into the former category. The istribution of multiimensional ata is unlikely to occur in patterns leaing to worst case range search cases. Expecte case analysis of range search algorithms is often a goo preictor of their performance (e.g. [, 7, 1]). The k- search skip list [9] is a linear space ata structure (base on the 1-3 eterministic skip list of [8]) supporting k- range search in O(kn) time, with ynamic upate time of O(k log n). Figure 1 illustrates the eight rectangles of the internal noes of a - search skip list containing 1 points. Figure shows a simpli- e version of the - search skip list arising from inserting the 1 points of Figure 1. Notice that each interior noe contains the bouning box of all noes in its own sublist. Department of Computer Science, University of Denver, Denver, Colorao, U.S.A., mlopez@cs.u.eu y Faculty of Computer Science, University of New Brunswick, Freericton, N.B. Canaa, bgn@unb.ca Research supporte by the Natural Sciences an Engineering Research Council (NSERC) of Canaa. Here we analyse the expecte time for half-space k- range reporting uner the assumption that both the input points an the query hyperplane are ranomly an uniformly istribute. In aition, we change the structure of the k- skip list in two ways, as follows: (1) we use a ranomize skip list instea of the eterministic one use in [9], an () points are inserte into the list, not by x-coorinate, but by their position along a space-lling curve. These changes result in a simpler analysis an a better average performance. Other than these two changes, the ata structure analyse here is ientical to that of [9]. 3 Ranom Hyperplane In < k, a hyperplane h is ene as the set of points satisfying : a x = c where a = (a 1 ; a ; : : : ; a k ) are the hyperplane coef- cients, x, an c < is a constant. We say x = (x 1 ; x ; : : : ; x k ) belongs to positive half-space h, i.e. x h, if a x c >. A ranom hyperplane is equally likely to be in any orientation or position. Assume, without loss of generality, that a ranom hyperplane intersects the unit hypercube [; 1] k. Vector a is a unit normal to the hyperplance, an a = (a 1 ; a ; : : : ; a k ) are irection cosines, each in [1; 1]. the irection cosines of the unit vector perpenicular to the hyperplane, an c [; 1] is a ranom variable that xes the hyperplane such that it intersects the unit hypercube, then we have ene a ranom hyperplane. Consiering k 1 of the angles ening the irection cosines as inepenent ranom variables, then along with c, we have k ranom variables ening a ranom hyperplane. Rectangle Intersection Internal noes of the k- search skip list store the k- bouning box of all points in the own sublist. In -, the bouning box is a rectangle, an we have the situation as shown in Figure 3. Coorinates of the lower left an upper right corners of a rectangle are (L, B) an (R, T ), respectively. We consier

2 8 6 a v f g h r j l m c p q k s o e i u n 6 b t Figure 1: Rectangles inuce by 1 points (labelle a through n) store in a - search skip list. v t u o p q r s a b c e f g 5 5 h i j k 1 1 l m n Figure : Simplie representation of a - search skip list for the points shown in Figure 1.

3 (1, 1) (R,T) c =a 1 +a (L,B) c =a 1 R +a T (, ) c =a 1 L +a B c = Figure 3: A rectangle with ranom half-space in orientation I. four possible orientations of a query plane epening on the sign of a 1 an a. Orientation I refers to the unit normal irection cosines a i [; 1], as shown in Figure 3. Theorem 1 In -, the probability P of a ranom half-space intersecting a rectangle in the unit square [; 1] is 1 (T B)+(RL) +. (, 1) c =a c =a 1 L +a T c =a 1 R +a B c =a 1 (1, ) Orientation II (, ) (1, 1) c =a 1 +a c =a 1 R +a T c =a 1 L +a B c = Orientation III Proof Halfspace h, in orientation I, intersects the rectangle i (R; T ) is above line h. Since c is istribute uniformly in the range [; a 1 +a ], the probability that h intersects the rectangle is c =a c =a 1 L +a T c =a 1 R +a B Ra 1 + T a a 1 + a = R cos + T sin c =a 1 Accoringly, the probability of the positive half space h in orientation I intersecting the rectangle is P I = R cos + T sin (1) for all possible orientations ; of the unit normal to the ranom line ening h, as well as the fraction of the unit square where h intersects the rectangle. Evaluating equation 1 gives P I = R+T. The three other possible orientations of the line ening h are now consiere. They are Orientation II: a 1 [1; ], a [; 1], Orientation III: (a 1 ; a ) [1; ], an Orientation IV: a 1 [; 1], a [1; ]. These three orientations are shown in Figure. The Orientation IV Figure : A rectangle with ranom half-space in the three other orientations. probability of the positive half space h intersecting the rectangle in orientations II, III an IV are P II = P III = = 1 + T L = 1 L + B L cos + T sin cos sin cos () (1 L) cos + (1 B) sin sin + cos (3)

4 P IV = = 1 + R B R cos + B sin sin cos sin () Equations, 3 an account for all possible orien- tations ;, ;, an ;, respectively, of the unit normal to the ranom line ening h, as well as the fraction of the unit square where h intersects the rectangle. Aing all four components of the probability together, an iviing by four to account for the fact that each component contributes 1 P = P I + P II + P III + P IV of the probability gives = 1 + R L + T B : Thus, the probability of a ranom half-space intersecting a rectangle in the unit square is 1. Note, in particular, that this probability epens exclusively on the size of the rectangle an not on its location. For k-, we oer the following conjecture: Conjecture 1 For k-, P = k Pk i=1 x i In conjecture 1, x i is the sie length of the rectangle in imension i. 5 - Search Skip List For Q n = time complexity of half-space range search in a ata structure containing n points, we have EfQ n g = no:ofinteriornoes i=1 P fr i \ h 6= g; (5) where E is the expecte value, P is the probability, an R i refers to a rectangle representing the bouning box of one internal noe of the search skip list. The algorithm for half-space range search in a k- search skip list [9] prunes a noe an it's own sublist if all points in R i are not in h ; i.e. R i 's own sublist is prune if x = h, 8x R i. Symmetrically, the algorithm reports all points within a noe's own sublist as in range if all points in R i are in h ; i.e. report all points in noe R i 's own sublist if x h, 8x R i. Without counting the cost of noes traverse solely for reporting purposes (similar to Willar's [1] worst-case locate runtime), the cost of half-space range search arises from those noes whose bouning box intersects the hyperplane h. Lemma In -, the probability P of a ranom half-plane h intersecting a rectangle in the unit square [; 1] is (T B)+(RL).. Proof The proof follows the same technique use for proving Theorem 1, but accounting for the probability of a ranom half-plane rather than a halfspace intersecting a rectangle. Equations (1), (), (3) an () become P I = P II = P III = P IV = (R L) cos + (T B) sin (6) (L R) cos + (T B) sin (7) sin cos (L R) cos + (B T ) sin sin cos (8) (R L) cos + (B T ) sin (9) cos sin Evaluating these equations gives (T B) + (R L) P I = P II = P III = P IV = (1) Aing all four components of the probability together, an iviing by four to account for the fact that each component contributes 1 of the probability gives the result. With Lemma in place, we can now procee to the main result. Theorem 3 Consier a - search skip list S of n points rawn at ranom from a uniform istribution an inserte into S by Hilbert orer. The expecte time EfQ n g of half-space range searching with a ranom query plane is O( p n + t), where t is the number of reporte points. Proof Let w + 1 be the number of levels of S, labele ; : : : ; w, an S, the set of internal noes of S, i.e., the set of noes at levels : : : ; w 1. With each noe v of S we ene an inicator variable I v associate with the event h \ R 6= ;, where R is the bouning rectangle store at v. Thus, v is further explore by going into its own sublist i I v = 1. Let v be an arbitrary noe of S with bouning rectangle R of size w h. Let v i be the chilren of v with bouning rectangles R i. By Lemma, EfI v g = (w + h)=. Since the expecte length of a sie of R i is no more than 1= the length of the corresponing sie of R, we have EfI vi g (w + h)=. Couple with the fact that EfI r g 1 for the root r, we have that EfI v g 1 (1j ) = j, where j is the level of noe v (an j = at the root). Since Efwg = log ne an the expecte number of noes of S at level j is j then the expecte number of noes explore in S is at most vl EfI v g w1 j= j j = w1 j= j = w 1 = O( p n)

5 Other noes visite correspon to points insie h an a O(t) to the complexity, establishing the claim. The above proof assumes a ranomize skip list is use, where the expecte number of noes in each own sublist is four. 6 Conclusions We have shown that the expecte cost of half-space range search using the k- search skip list containing n points is O( p n+t) in the - case. This result is competitive with the best known algorithms for half-space range search requiring linear space. It remains to expan the analysis to 3- an to k-. If our conjecture is correct, we anticipate goo upper bouns for the expecte case of half-space range search in higher imensions. The approach we have chosen for ening a ranom hyperplane will be useful in other expecte case analyses involving halfspace range search. Can our approach be use to analyse the expecte orthogonal range search time in an R-tree an other ata structures whose internal noes contain bouning boxes encompassing their chilren's bouning boxes References [1] P.K. Agarwal an J. Matousek. Dynamic halfspace range reporting an its applications. Algorithmica, 13:35{35, [8] J. Ian Munro, T. Papaakis, an R. Segewick. Deterministic skip lists. In Proc. of the Thir Annual Symposium on Discrete Algorithms, pages 367{375, Orlano, Floria, January [9] B.G. Nickerson an Y. Pan. On ynamic k- halfspace range reporting. In Proc. of the 1th Canaian Conference on Computational Geometry, pages 187{19, Freericton, N.B., Canaa, Aug Faculty of Computer Science. [1] R. Seiel an C.R. Aragon. Ranomize search trees. Algorithmica, 16:6{97, [11] P.M. Vaiya. Space-time trae-os for orthogonal range queries. SIAM Journal of Computing, 18():78{758, [1] D.E. Willar. Polygon retrieval. SIAM Journal of Computing, 11(1):19{165, 198. [13] A.C. Yao. Space-time traeo for answering range queries. In Proc. of the 1th Annual ACM Symposium on the Theory of Computing, pages 18{136, San Francisco, CA, May [1] F.F. Yao. Hanbook of Theoretical Computer Science, volume A: Algorithms an Complexity, chapter 7, pages 37{37. Elsevier, Amsteram, 199. [] M. J. Atallah, eitor. Algorithms an Theory of Computation Hanbook, chapter. CRC Press, Boca Raton, [3] J.L. Bentley an H.A. Maurer. Ecient worstcase ata structures for range searching. Acta Informatica, 13():155{168, 198. [] P. Chanzy, L. Devroye, an C. amora-cura. Analysis of range search for ranom k- trees. Acta Informatica, 37(Fasc./5):355{383, 1. [5] B. Chazelle. Lower bouns for orthogonal range searching: I. The reporting case. Journal of the Association for Computing Machinery, 37():{1, 199. [6] J.E. Gooman an J. O'Rourke, eitors. Hanbook of Discrete an Computational Geometry, chapter 31. CRC Press, Boca Raton, [7] S.T. Leutenegger an M.A. Lopez. The eect of buering on the performance of r-trees. IEEE Trans. on Knowlege an Data Engineering, 1(1):33{, Jan/Feb.