The Lower Envelope The Pointwise Minimum of a Set of Functions

1. The Lower EnvelopeThe Pointwise Minimum of a Set of Functions Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

2. Definition of the Lower Envelope (untere Kontur) of a set of functions: Given nreal-valued functions, all defined on a common interval I, then the minimum is : f(x) = min 1≤i≤n fi (x) The graph of f(x)is called the lower envelopeof the fi’s. y =-∞ Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

3. Special Case: If all the functions fi are linear, then their graphs are line segments. The lower envelope can be calculated with the help of sweep algorithm. D C Cu B A I Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

4. Question: Could the sweep line method also be used to find the lower envelope of graphs of non-linear functions? Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

5. Definition : A curve c is x-monotone if any vertical line either does not intersect c, or it intersects c at a single point. Assumptions • All functions are x-monotone. • Function evaluation and determination of intersection points take time O(1). • The space complexity of the description of a function fi is also constant. • Theorem 1: With the sweep technique, the k intersection points of n different x-monotone curves can be computed in O((n+k) logn) time and O(n) space. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

6. Sweep technique: If any two curves intersect in at most s points, (this would be satisfied when the functions of all ncurves are polynomials that have degree at most s), then the total number of intersection points k is k ≤ s*n(n-1)/2 Consequence: The total time complexity of the sweep line algorithm for computing the lower envelope of n x-monotone functions isO(s n2 logn)(from theO((n+k) logn) bound for computing all k intersection points). Note: This is not an output sensitive algorithm! Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

7. S=3,n=4 Maximum k=18 Only 8 intersection points needed for lower envelope! Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

8. New algorithm: Combines Divide & Conquer andSweep-line techniques If n =1, do nothing, otherwise: 1. Divide: the set S of n functions into two disjoint sets S1 and S2 of size n/2. 2. Conquer: Compute the lower envelopes L1 and L2 for the two sets S1 and S2 of smaller size. 3. Merge: Use a sweep-line algorithm for merging the lower envelopes L1 and L2 of S1 and S2 into the lower envelope L of the set S. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

9. Divide and Conquer Example: Lower envelope of curves C and B Lower envelope of curves A and D Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

10. The lower envelopes of curves A,D and B,C Lower envelope of curves B and C Lower envelope of curves A and D Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

11. Sweep line algorithm for merging two lower envelopes L1 and L2 Sweep over L1 and L2 from left to right: Event points: All vertices of L1 and L2, all intersection points of L1 and L2 At each instance of time, the event queue contains only 3 points: 1 (the next) right endpoint of a segment of L1 1 (the next) right endpoint of a segment of L2 The next intersection point of L1 and L2, if it exists. Sweep status structure: Contains two segments in y-order Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

12. L1 L2 Event queue: SSS: Output L: Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

13. Time complexity L1 L2 The lower envelope can be computed in time proportional to the number of events (halting points of the sweep line). At each event point, a constant amount of work is sufficient to update the SSS and to output the result. Total runtime of the merge step: O(#events). How large is this number? Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

14. Define λs(n): the maximum number of segments of the lower envelope of an arrangement of • n different x-monotone curves over a common interval • such that every two curves have at most s intersection points λs(n) is finite and grows monotonously with n. Lower envelope of a set of n/2 x-monotone curves L1 Lower envelope of a set of n/2 x-monotone curves L2 2λs(n/2)≤2 λs(n) Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

15. If n =1, do nothing, otherwise: 1. Divide: the set S of n functions into two disjoint sets S1 and S2 of size n/2. 2. Conquer: Compute the lower envelopes L1 and L2 for the two sets S1 and S2 of smaller size. 3. Merge: Use a sweep-line algorithm for merging the lower envelopes L1 and L2 of S1 and S2 into the lower envelope L of the set S. Time complexity T(n) of the D&C/Sweep algorithm for a set of n x-monotone curves, s.t. each pair of curves intersects in at most s points: T(1) = C T(n) ≤ 2 T(n/2) + C λs(n) Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

16. Using the Lemma : For all s, n ≥ 1, 2λs(n) ≤ λs(2n), and the recurrence relation T(1) = C, T(n) ≤ 2 T(n/2) + C λs(n) yields: Theorem: To calculate the lower envelope of n different x-monotonecurves on the same interval, with the property that any two curves intersect in at most s points can be computed in time O(λs(n) log n). Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

17. Recursion Tree Marking each node with the cost of the divide and conquer step Back-substitution T(n) T(n/2) T(n/2) T(n/4) T(n/4) T(n/4) T(n/4) The root has cost of Cλs(n) each subtree has cost of Cλs(n/2) each subtree has cost of Cλs(n/4) By induction…. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

18. Davenport-Schinzel Sequences (DSS) Consider words (strings) over an alphabet {A, B, C,…} of n letters. A DSS of order s is a word such that • no letter occurs more than once on any two consecutive positions • the order in which any two letters occur in the word changes at most s times. Examples: ABBA is no DSS,ABDCAEBAC is DSS of order 4, What about ABRAKADABRA? Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

19. Theorem: The maximal length of a DSS of order s over an alphabet of n letters is λs(n). Proof part 1: Show that for each lower envelope of n x-monotone curves, s.t. any two of them intersect in at most s points, there is a DSS over an n-letter alphabet which has the same length (# segments) as the lower envelope. Proof part 2: Show that for each DSS of length n and order s there is a set of n x-monotone curves which has the property that any two curves intersect in at most s points and which have a lower envelope of n segments. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

20. B C D A A C C B D Proof part 1: Lower envelope contains the segments ABACDCBCD in this order. It obviously has the same length as the l.e. Is this also a DSS? Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

21. C B A A B A C A C B C Example: Davenport-Schinzel-Sequenz: ABACACBC Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

22. Proof part 2: Given a DSS w of order sover an alphabet of n letters, construct an arrangement of n curves with the property that each pair of curves intersects in at most s point which has w as its lower envelope. Generic example: ABCABACBA, DSS of order 5 C B A Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

23. Lemma: For all s,n ≥ 1: 2λs(n)≤λs(2n) Proof:Given a DSS over an n-element alphabet of order s and length l; construct a DSS of length 2l over an alphabet of 2n letters by concatenating two copies of the given DSS and choosing new letters for the second copy. Example: n = 2, that is, choose alphabet {A,B}, s = 3, DSS3 = ABAB n= 4, that is, choose alphabet {A,B,C,D} ABABCDCD is a DSS of order 3 and double length. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

24. Properties of λs(n) • λ1(n) = n • λ2(n) = 2n -1 • λs(n) ≤ s (n – 1) n / 2 + 1 • λs(n)  O(n log* n), where log*n is the smallest integer m, s.t. the m-th iteration of the logarithm of n log2(log2(...(log2(n))...)) yields a value ≤ 1: Note: For realistic values of n, the value log*n can be considered as constant! Example: For all n ≤1020000 , log*n ≤5 Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

25. Lower Envelope of n Linesegments D C Cu B A Theorem: The lower envelope of n line segments over a common interval can be computed in time O(n log n) and linear space. Proof:λ1(n) = n Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

26. Lower envelope of an arrangement of line segments in general position A B B B A D C D A Theorem: The lower envelope of n linesegments in general position has O(λ3(n))many segments. It can be computed in time O(λ3(n) log n). Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

27. Reduction to x-monoton curves over a common interval Any two curves may Intersect at most 3 times! A B B D B A C A D Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

28. Any two curves may intersect at most 3 times! Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

29. Because the outer segments are parallel to each other, any two x-monotone curves can intersect in at most three points. Therefore, the lower envelope has at most O(λ3(n) log n) segments. It is known that λ3(n)  Θ(n α(n)). Here, α is the functional inverse of the Ackermann function A defined by: A(1, n) = 2n , if n ≥ 1 A(k, 1) = A(k – 1, 1) , if k ≥ 2 A(k, n) = A(k – 1, A(k, n – 1)) , if k ≥ 2, n ≥ 2 Define a(n) = A(n, n), then α is defined by α(m) = min{ n; a(n) ≥ m} The function α(m) grows almost linear in m (but is not linear). Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann

30. References • R. Klein. Algorithmische Geometrie, Kap. 2.3.3. Addison Wesley, 1996. • M. Sharir and P. K. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications, Cambridge University Press, 1995. Computational Geometry, 2005/06 Prof.Dr.Th.Ottmann