The Lower Envelope: The Pointwise Minimum of a Set of Functions
Computational Geometry, WS 2007/08Lecture 4
Prof. Dr. Thomas Ottmann
Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg
Computational Geometry, WS 2007/08Prof. Dr. Thomas Ottmann 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.
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The Lower Envelope
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Definition of the Lower Envelope
Given n real-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 envelope of the fi’s.
y =-∞
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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.
A
B
C
D
Cu
I
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Non-Linear Functions
Question: Could the sweep line method also be used to find the lower envelope of graphs of non-linear functions?
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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.
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The Sweep Technique
• If any two curves intersect in at most s points, (this would be satisfied when the functions of all n curves 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 is O(s n2 log n) (from the O((n+k) log n) bound for computing all k intersection points).
Note:• This is NOT an output-sensitive algorithm.
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Example
S=3,n=4
Maximum k=18
Only 8 intersection points needed for lower envelope!
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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.
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Example: Divide & Conquer
Lower envelope of curves A and D
Lower envelope of curves C and B
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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
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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
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Example: Sweep-line
Event Q:
SSS:
Output L:
L1
L2
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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?
L1
L2
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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.
λs(n/2) ≤ λs(n)
Lower envelopeof a set of n/2x-monotone curves
Lower envelopeof a set of n/2x-monotone curves
L1
L2
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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 n x-monotone curves, s.t. each pair ofcurves intersects in at most s points:
T(1) = CT(n) ≤
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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) + C λs(n) yields:
Theorem: To calculate the lower envelope of n different x-monotone curves 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 ).
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Recursion Tree
Back-substitution
The root has cost of Cλs(n)
each subtree has cost of Cλs(n/2)
By induction….
each subtree has cost of Cλs(n/4)
Marking each node with the cost of the divide and conquer step
T(n)
T(n/2) T(n/2)
T(n/4) T(n/4) T(n/4) T(n/4)
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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?
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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 lowerenvelope.
Proof part 2: Show that for each DSS of length n and order sthere 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 alower envelope of n segments.
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DSS: Proof (Part 1)
A AC
DC B
B
DC
Lower envelope contains the segments ABACDCBCD in this order.
It obviously has the same length as the l.e. Is this also a DSS?
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Example: DSS
A
B
C
A
A
A
B
B
C
C
C
Example: Davenport-Schinzel-Sequence: ABACACBC
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DSS: Proof (Part 2)
Proof part 2: Given a DSS w of order s over 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
A
B
C
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Lemma
Lemma: For all s,n ≥ 1: 2 λs(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.
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Properties of s(n)
1. λ1(n) = n
2. λ2(n) = 2n -1
3. λs(n) ≤ s (n – 1) n / 2 + 1
4. 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
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Lower Envelope of n Line-Segments
A
B
C
D
Cu
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
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Line-Segments in General Position
A
BC D
A
AB
B
D
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).
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Reduction to X-Monotone Curves
A
BC
D
A
AB
B
D
Any two curves mayIntersect at most3 times!
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Reduction to X-Monotone Curves
Any two curves mayIntersect at most3 times!
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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 ≥ 1A(k, 1) = A(k – 1, 1) , if k ≥ 2A(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).
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References
1. R. Klein. Algorithmische Geometrie, Kap. 2.3.3. Addison Wesley, 1996.
2. M. Sharir and P. K. Agarwal. Davenport-Schinzel Sequences and their Geometric Applications, Cambridge University Press, 1995.