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The Lower Envelope: The Pointwise Minimum of a Set of Functions

The Lower Envelope: The Pointwise Minimum of a Set of Functions. Computational Geometry, WS 2007/08 Lecture 4 Prof. Dr. Thomas Ottmann. Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg. Overview.

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The Lower Envelope: The Pointwise Minimum of a Set of Functions

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  1. The Lower Envelope: The Pointwise Minimum of a Set of Functions Computational Geometry, WS 2007/08 Lecture 4 Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

  2. Overview • Definition of the Lower Envelope. • Functions: Non-linear, x-monotone. • Techniques: Divide & conquer, Sweep-line. • Definition: s(n). • Davenport-Schinzel Sequences (DSS). • Lower Envelope of n line segments. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  3. The Lower Envelope Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  4. Definition of the Lower Envelope 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  5. 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  6. Non-Linear Functions Question: Could the sweep line method also be used to find the lower envelope of graphs of non-linear functions? Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  7. X-Monotone Functions • 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) log n) time and O(n) space. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  8. The 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 nx-monotone functions isO(s n2log n) (from theO((n+k) log n) bound for computing all k intersection points). Note: • This is NOT an output-sensitive algorithm. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  9. Example S=3,n=4 Maximum k=18 Only 8 intersection points needed for lower envelope! Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  10. New: Divide & Conquer, Sweep-line 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  11. Example: Divide & Conquer Lower envelope of curves C and B Lower envelope of curves A and D Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  12. Sweep-line: Merging 2 Lower Envelopes 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  13. Handling the Events @Initialization: • Retrieve segments (a0,a1) and (b0,b1)  • Test for intersection(s)  • Initialize the event Q with @Segment endpoint: • Retrieve next right endpoint  • Test for intersection(s)  • Insert into the event Q @Intersection point: • Simply Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  14. L1 L2 Example: Sweep-line Event Q: SSS: Output L: Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  15. L1 L2 Time Complexity 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  16. Definition: s(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 L2 Lower envelope of a set of n/2 x-monotone curves λs(n/2) ≤ λs(n) Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  17. Analysis 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 nx-monotone curves, s.t. each pair of curves intersects in at most s points: T(1) = C T(n) ≤ Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  18. Analysis 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) + Cλs(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, WS 2007/08 Prof. Dr. Thomas Ottmann

  19. Recursion Tree Marking each node with the cost of the divide and conquer step The root has cost of Cλs(n) T(n) T(n/2) T(n/2) T(n/4) T(n/4) T(n/4) T(n/4) Back-substitution each subtree has cost of Cλs(n/2) each subtree has cost of Cλs(n/4) By induction…. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  20. 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  21. Davenport-Schinzel Sequences (DSS) 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  22. B C D A A C C B D DSS: 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  23. C B A A B A C A C B C Example: DSS Example: Davenport-Schinzel-Sequence: ABACACBC Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  24. DSS: Proof (Part 2) 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  25. Lemma 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  26. Properties of s(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, WS 2007/08 Prof. Dr. Thomas Ottmann

  27. Lower Envelope of n Line-Segments 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  28. 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  29. Reduction to X-Monotone Curves Any two curves may Intersect at most 3 times! A B B D B A C A D Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  30. Reduction to X-Monotone Curves Any two curves may Intersect at most 3 times! Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

  31. Analysis 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, WS 2007/08 Prof. Dr. Thomas Ottmann

  32. 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, WS 2007/08 Prof. Dr. Thomas Ottmann

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