J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inform. Process. Lett. 33 (1989), 169--174.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....also the bound for the case where Q = Q 2 found in Hamacher et al. 5] 4. Bicriteria case In the case where we only have two criteria, we may use the image of the network mapped into criterion space Z to solve the problem faster. This is done by calculating the lower envelope, see Hershberger [8]. This can be done in O(p log p) time, where p is the number of line segments. There are three di erent situations. Q 1 = denoted min min (1 G d(V; G) 2 ( Par ) jQ 1 j = jQ 2 j = 1 denoted max min (1 G d(V; G) obnox ; Par ) and Q 2 = denoted maxmax (1 G d(V; G) 2 ( ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Info Process Lett, 33:169-174, 1989.

....lower envelope E of the bottom edges of rectangles in G and the boundary of tAG2 is given by the upper envelope E2 of the top edges of rectangles in 62. Since both of these are envelopes of at most s parallel line segments they each have at most 2s vertices and can be computed in O(s log s) time [8]. E and E2 are both x monotone and the space above E is tAG and the space above E2 is UG2. Any vertex of U(G U G2) is either a vertex of G or G2 or an intersection point of E and E2. Since two monotone polygonal chains of length O(s) intersect at most O(s) times, the boundary of the 1 occupied ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters, 33(4):169-174, December 1989.

....envelope E 1 of the bottom edges of rectangles in G 1 and the boundary of [G 2 is given by the upper envelope E 2 of the top edges of rectangles in G 2 . Since both of these are envelopes of at most s parallel line segments they each have at most 2s vertices and can be computed in O(s log s) time [9]. E 1 and E 2 are both x monotone and the space above E 1 is [G 1 and the space above E 2 is [G 2 . Any vertex of [ G 1 [G 2 ) is either a vertex of G 1 or G 2 or an intersection point of E 1 and E 2 . Since two monotone polygonal chains of length O(s) intersect at most O(s) times, the boundary ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters, 33(4):169-174, December 1989.

.... For s 4; n; s) O(n 2 O( n) s 2) 2 ) if s is even; O(n 2 O( n) s 3) 2 log( n) if s is odd [AShSh89] In the following, we assume that k is a positive integer and that S is a set of functions, polynomials of degree at most k (or more generally, k intersecting [Hersh89], i.e. each pair of members of S has graphs that intersect in at most k points) As was done in [Atal85a, B M89a, B M89b, Hersh89] we also assume that for ff i ; f j g S, i 6= j, all solutions of the equation f i (x) f j (x) may be determined in (1) serial time. We note that somewhat ....

....integer and let S be a set of polynomial functions, each of degree at most k. If the domain of each member of S is an interval of R 1 (not necessarily the same interval for each member of S) then the lower envelope of S has at most (n; k 2) pieces generated by members of S. Theorem 3. 6 [Hersh89] Let k be a xed positive integer and let S be a set of polynomial functions, each of degree at most k. If the domain of each member of S is an interval of R 1 (not necessarily the same interval for each member of S) then a description of the lower envelope of S may be computed in O( n; k 1) ....

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J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Information Processing Letters 33 (1989), 169-174.

....once the Delaunay triangulation of the footholds has been computed, which can be done in O(n log n) time [5, 17] By Proposition 4. 9, the upper and lower envelopes can be computed in O(k i 0 log k i 0 ) time using O(k i 0 ff(k i 0 ) space where ff is the pseudo inverse of the Ackerman s function [6]. Also by Proposition 4.9, the union of Omega 1 and Omega 2 can be done in linear time in the size of the edge chains, that is O(k i 0 ff(k i 0 ) time. Thus, we can compute [ i Z i in O(k i 0 log k i 0 ) time using O(k i 0 ff(k i 0 ) space after O(n log n) preprocessing time. We can compute ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inform. Process. Lett., 33 (1989), pp. 169174.

.... lines in the plane (or at least the part of it that is after the initial time t 0 ) This upper envelope computation can be trivially done in O(n log n) time with a divide and conquer algorithm (this bound holds even if points can appear and disappear at arbitrary times, but then it is not trivial [32]) In the worst case, the number of times during the motion that the topmost point changes is Theta(n) Thus we have a method for computing the configuration function of interest in time that is is only a logarithmic factor higher than the maximum number of changes in the configuration function ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

....envelope E 1 of the bottom edges of rectangles in G 1 and the boundary of [G 2 is given by the upper envelope E 2 of the top edges of rectangles in G 2 . Since both of these are envelopes of at most s parallel line segments they each have at most 2s vertices and can be computed in O(s log s) time [8]. E 1 and E 2 are both x monotone and the space above E 1 is [G 1 and the 9 space above E 2 is [G 2 . Any vertex of [ G 1 [G 2 ) is either a vertex of G 1 or G 2 or an intersection point of E 1 and E 2 . Since two monotone polygonal chains of length O(s) intersect at most O(s) times, the ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters, 33(4):169-174, December 1989.

....i k ) and (P i k ) in the objective space (i; i 2 f1; 2g) The bold curves represent the sets of globally nondominated solutions, respectively. These observations will be incorporated into the reduction procedure as follows: In the rst part of the procedure, the Hershberger algorithm [16], that nds the lower envelope of a collection of line segments in linear time, is used to determine the lower envelope of the segments A i k B i k of all subproblems in List(P i k ) Since our goal is to nd a superset of the nondominated sets of the subproblems, we add an auxiliary ....

....procedure of Nickel and Wiecek ( 24] specially designed for bi objective piece wise linear programs can be used. Given the nondominated sets of all the problems (P i k ) 2 List(P i k ) we can determine the globally nondominated points, as proposed in [10] by means of the Hershberger algorithm [16]. As this algorithm nds a lower envelope of a collection of line segments, we again add an auxiliary horizontal line at point B i k and a vertical line at point A i k of every triangle T i k to eliminate points coming from other subproblems but dominated by the points of the subproblem (P i k ....

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J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters, 33:169-174, 1989. 24

....between the diameter of S and the length of the shortest segment of S. Step 5 can be done in time linear in the complexity of W 2 , that is, in time O(minfnae 2 log aen; nae log 2 n log log n) The calculation of the upper and lower envelopes 5 in Step 6 can be done in time O(n log n) [2, 12]. Note that there are only O(n) vertices on both envelopes. Finally, the line sweeping algorithm of Step 7 takes time O( N k) log N ) where N is the total number of segments forming the boundaries of W 1 and of f W 2 , and k is the number of events that the algorithm processes. The number ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Information Proc. Letters 33 (1989), 169--174.

....Hence, only the points of P with the maximum y value for each x value are important. The union of these points is called the upper envelope of the part. 13 The upper envelope can be computed in O(n log n) time, using an algorithm for computing the upper envelope of line segments by Hershberger [23]. The second observation is that supported points of P for # l are also supported for # # l with # # l # l . Hence, the supported area to the left of c only increases as # l decreases, and consequently, # l is monotonic. From the two observations, it follows that a geometric representation of ....

....the border of a single orientation, and the intersecting borders of multiple orientations is depicted. The shape that follows the uppermost points of the graph is called the graph s upper envelope, as the reader might recollect. We compute the upper envelope by means of an algorithm of Hershberger [23] which computes the upper envelope of a set of segments that intersect pairwise at most k times. The combinatorial complexity of the upper envelope is #(# k 2 (n 2 ) where # s (n 2 ) is the maximum length of a Davenport Schinzel sequence of order s on n 2 symbols (see e.g. 35] The ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters, 33:169--174, 1989.

....whenever some CW p i ,p j or CCW p i ,p j (y) is not defined. Lemma 5.1 The graph of U p i , L p i have O(# 4 (n) breakpoints. See the full version for a proof of the above lemma. In view of the above lemma, U p i , L p i can be computed by a sequential algorithm in in O(n# 3 (n) log n) time [24], or by a parallel algorithm in time O(log n) using O(# 4 (n) processors. It has been shown in [23, 5] that # 3 (n) #(n#(n) and # 4 (n) #(n2 #(n) After having computed U p i , L p i , a good point can also be determined within the same time bound. Hence, the fixed size problem can be ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inf. Proc. Lett. 33 (1989), 169--174.

....edges to the center of mass of P . Firstly, only the upper envelope of the part is relevant for the supported area, since we consider a half plane below c. The upper envelope can be computed in O(n log n) time, using an algorithm for computing the upper envelope of line segments by Hershberger [22]. Secondly, we sweep the left edge of the gap across the upper envelope and determine the angle of the line tangent to S, and the center of mass as a function of the position of the gap edge. Clearly, this function is monotonically increasing, since the supported area of the part only augments as ....

....is to get a parameter setting which will preserve only one orientation of the part. The overlay of connected curves of all orientations form an arrangement. Acceptable parametersettings are these between the upper envelope and the 2 level of this arrangement. We use the algorithm of Hershberger [22] to compute the upper envelope of a set of curves that intersect pairwise at most k times. Therefore, the upper envelope of our curves has worst case complexity Theta( k 2 (n 2 ) where s(n 2 ) is the maximum length of a Davenport Schinzel sequence of order s on n 2 symbols (see e.g. ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

.... can be done in O(n log n) time [DD90, Tur91] Since two curves ae i and ae j intersect each other at most once (Propositions 12, 13, 15) the upper and lower envelopes can be computed in O(k i 0 log k i 0 ) time and O(k i 0 ff(k i 0 ) space where ff is the pseudo inverse of the Ackerman s function [Her89]. The union of Omega 1 and Omega 2 can be done in O(k i 0 ff(k i 0 ) time because each envelope is the graph of a function of u and two arcs of the two envelopes intersect each other at most once. At this time, we have computed [ i6=i 0 Z i . The computation of [ i6=i 0 Z i [Z i 0 can be done ....

....32, 33 and 34, during the computation of the union of the regions Z e i , any two curves involved in the upper and lower envelopes of the curves ae i and ae Gamma i , intersect at most six time. Thus, the computation can be done in O( 7 (k i 0 ) log k i 0 ) time and O( 8 (k i 0 ) space [Her89]. Thanks to Section A.2, the computation of [ i int(H e i ) C s i 0 can be done within the same time and space bounds. Since A e is the arrangement of the C e i for i 2 f1; ng, and A is the set of circles of radius R centered at the endpoints of each straight line segment e i , the ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169174, 1989.

....is equivalent to checking whether D 6= We observe that the stabbing region D = T i D i of D is precisely the set of points that lie above the upper envelope U B and below the lower envelope L T . We construct L T (actually LH ) and U B using the efficient algorithm of Hershberger [15] in O(nff(n) log n) time. We determine whether there is any point above U B and below L T as follows: We merge the vertices of the minimization diagram M T and of the maximization diagram M 0 B . Let b 1 ; b 2 ; b k be the merged list of vertices. The bounds on the combinatorial ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Info. Proc. Letters 33 (1989), 169--174.

.... running time of the algorithm is O( s (n) log n) If the functions in F are partially defined, an easy modification of the above algorithm constructs MF in time O( s 2 (n) log n) In this case, however, MF can be computed in time O( s 1 (n) log n) using a more clever algorithm due to Hershberger [75]. Theorem 2.6 ( 19, 75] The lower envelope of a set F of n continuous, totally defined, univariate functions, each pair of whose graphs intersect in at most s points, can be constructed, in an appropriate model of computation, in O( s (n) log n) time. If the functions in F are partially defined, ....

....algorithm is O( s (n) log n) If the functions in F are partially defined, an easy modification of the above algorithm constructs MF in time O( s 2 (n) log n) In this case, however, MF can be computed in time O( s 1 (n) log n) using a more clever algorithm due to Hershberger [75] Theorem 2. 6 ([19, 75]) The lower envelope of a set F of n continuous, totally defined, univariate functions, each pair of whose graphs intersect in at most s points, can be constructed, in an appropriate model of computation, in O( s (n) log n) time. If the functions in F are partially defined, then EF can be computed ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inform. Process. Lett., 33 (1989), 169--174.

....segments can have O(mnff(mn) HS86] segments where ff( is the extremely slowly growing inverse of the Ackermann s function. For practical purposes, it can be considered a constant. The straightforward implementation of the angular sweepline algorithm runs in O(mnff(mn) log mn) time. Hershberger [Her89] presented an algorithm for calculating the outer envelope of n line segments in O(n log n) time. Therefore, we have Theorem 3.15 The Minkowski sum of two starshaped polygons can be computed in O(mn log mn) time. Currently we are using a numerically robust implementation of angular sweepline ....

J. Hershberger. Finding the upper envelope of n line segments in o(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

....Facets are shown in bold. RR n Sigma2575 28 F. Nielsen , M. Yvinec Thus, for the case of line segments we obtain an O(nff(h) log h) time algorithm. We show in the following section how we can reach the optimal bound Omega Gamma n log h) by adapting the technique due to J. Hershberger [Her89] The main idea is to group the the line segments eOEciently. A family of functions is said to be k intersecting if the functions are partially dened (this means that their graph have two endpoints) and if they intersect pairwise in at most k points. A set of line segments is 1 intersecting. 7.1 ....

....of their endpoints is the abscissa a i , cross the vertical line x = a i . Their upper envelope is linear in the number of line segments. We say that these segments are classied. INRIA An Output Sensitive Convex Hull Algorithm for Planar Objects 29 Following the communication of J. Hershberger [Her89] we notice that the upper envelope of the line segments allocated into a same internal level of IT is linear in the number of line segments. Indeed, the upper envelope of the segments allocated to a given internal node is linear (see [Her89] because all these segments cross a vertical line and ....

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J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169174, 1989.

.... Edelsbrunner, Guibas and Sharir showed how to construct a representation of the line stabbers of convex polygons with a total of n vertices in O(nff(n) log n) time [7] Recently Hershberger 1 announced an O(n log n) algorithm for finding the lower envelope of a set of line segments in the plane [13]. His result gives an O(n log n) algorithm for line stabbing n simple objects in the plane and an O(n log n) algorithm for line stabbing convex polygons with a total of n vertices in the plane. In their paper [8] Edelsbrunner et al. showed an Omega Gamma n log n) lower bound for constructing a ....

Hershberger, J., Finding the upper envelope of n line segments in O(nlogn) time, manuscript.

....in general position in the plane. Our main result, given below, improves a previous bound (n) O(n log n) of [1] Theorem 1 (n) O(n log log n) Proof. We partition S into t = O(log n) subcollections S 1 ; S t in the following rather standard interval tree manner (see, e.g. [2]) We find a vertical line , that may meet some of the segments, so that the number of segments fully to its left is roughly equal to the number of segments fully to its right. Let S 1 be the set of segments stabbed by . We now consider the subset SL of segments lying fully to the left of , ....

J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inform. Process. Lett. 33 (1989), 169--174.

..... Associated with each interval is the pair of functions u( and l( which return the grasp points. 6. If 6= create an arc in the transition graph between s and f . The complexity of the algorithm is dominated by the construction of the envelopes which takes O(n log n) time per iteration [2]; the rest of the steps have linear complexity. Since there are O(m 2 ) pairs of stable configurations, the complexity of constructing the entire transition graph is O(m 2 n log n) For star shaped (wrt the center of gravity) polyhedra, this reduces to O(m 2 n) because the intersections ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

....segments can have O(mnff(mn) 10] segments where ff( is the extremely slowly growing inverse of the Ackermann s function. For practical purposes, it can be considered a constant. The straightforward implementation of the angular sweepline algorithm runs in O(mnff(mn) log mn) time. Hershberger [11] presented an algorithm for calculating the outer envelope of n line segments in O(n log n) time. Therefore, we have Theorem 4.7 The Minkowski sum of two starshaped polygons can be computed in O(mn log mn) time. Currently we are using a numerically robust implementation of angular sweepline ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

....of S s2S s where s denotes the unbounded trapezoid conv(s [ f(0; 1)g) for a given segment s. Convex hulls correspond to lower envelopes of lines in the dual. Let h be the output size, i.e. the number of edges in the envelope; it is known that h is at most O(nff(n) 16] Hershberger [17] has given a worst case optimal algorithm that computes lower envelopes in O(n log n) time. We now describe how his algorithm can be made output sensitive with our technique. First, observe that we can trace the h edges in L(S) from left to right by performing h ray shooting operations, where a ....

J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett., 33:169--174, 1989.

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J. Hershberger, Finding the upper envelope of n line segments in O(n log n) time, Inform. Process. Lett. 33 (1989), 169--174.

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J. Hershberger. Finding the upper envelope of n line segments in O(n log n) time. Information Processing Letters 33, pp. 169174, 1989.

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