Section 5.5 The Intermediate Value Theorem Rolle’s Theorem The Mean Value Theorem 3.6.
The Centervertex Theorem (CCCG)
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Transcript of The Centervertex Theorem (CCCG)
![Page 1: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/1.jpg)
The Centervertex Theorem for Wedge Depth
Gary Miller, Todd Phillips, and Don Sheehy
Carnegie Mellon University
![Page 2: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/2.jpg)
![Page 3: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/3.jpg)
Medians
![Page 4: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/4.jpg)
![Page 5: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/5.jpg)
Warm up
![Page 6: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/6.jpg)
Warm up in 1-D
![Page 7: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/7.jpg)
Warm up in 1-D
![Page 8: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/8.jpg)
Warm up in 1-D
Rank
1 2 3
![Page 9: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/9.jpg)
Warm up in 1-D
RankMean
1 2 3
![Page 10: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/10.jpg)
Warm up in 1-D
RankMean
Median
![Page 11: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/11.jpg)
Warm up in 1-D
RankMean
MedianIn the set
![Page 12: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/12.jpg)
Warm up in 1-D
RankMean
MedianIn the set
Robust
![Page 13: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/13.jpg)
Warm up in 1-D
RankMean
Median
!
In the set
Robust
![Page 14: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/14.jpg)
Warm up in 1-D
RankMean
Median
!
In the set
Robust
!
![Page 15: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/15.jpg)
Warm up in 1-D
RankMean
MedianIn the set
Robust
![Page 16: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/16.jpg)
Warm up in 1-D
RankMean
MedianIn the set
RobustCombinatorial
![Page 17: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/17.jpg)
Warm up in 1-D
RankMean
MedianIn the set
RobustCombinatorial
![Page 18: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/18.jpg)
Warm up in 1-D
RankMean
MedianIn the set
RobustCombinatorial
![Page 19: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/19.jpg)
Warm up in 1-D
RankMean
Median
Depth
In the set
RobustCombinatorial
![Page 20: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/20.jpg)
Warm up in 1-D
RankMean
Median
Depth
In the set
RobustCombinatorial
![Page 21: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/21.jpg)
Warm up in 1-D
RankMean
Median
Depth
We want toget this back.
In the set
RobustCombinatorial
![Page 22: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/22.jpg)
Warm up in 1-D
wedge depth candistinguish between
these two points.
![Page 23: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/23.jpg)
![Page 24: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/24.jpg)
Tukey Depth
![Page 25: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/25.jpg)
![Page 26: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/26.jpg)
S ! Rd, |S| = n
Input (vertices)
![Page 27: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/27.jpg)
Depth MeasuresD : R
d! Z
S ! Rd, |S| = n
Input (vertices)
![Page 28: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/28.jpg)
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
Depth MeasuresD : R
d! Z
S ! Rd, |S| = n
Input (vertices)
![Page 29: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/29.jpg)
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![Page 39: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/39.jpg)
D!(x) !n
d + 1
Centerpoints
![Page 40: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/40.jpg)
D!(x) !n
d + 1
Centerpoints
Centerpoints always exist! [Rado ’47, Danzer et al ’63]
![Page 41: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/41.jpg)
![Page 42: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/42.jpg)
Wedge Depth
![Page 43: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/43.jpg)
tr
α-Wedges
![Page 44: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/44.jpg)
p
tr
α-Wedges
!(ptr) !!
2
![Page 45: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/45.jpg)
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
![Page 46: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/46.jpg)
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
![Page 47: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/47.jpg)
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
![Page 48: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/48.jpg)
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
![Page 49: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/49.jpg)
D!(x) = min {|W ! S| : W is an !-wedge with apex x}
D!(x) = min {|H ! S| : H is a closed halfspace, and x " H}Tukey Depth
α-wedge Depth
![Page 50: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/50.jpg)
![Page 51: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/51.jpg)
![Page 52: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/52.jpg)
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![Page 56: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/56.jpg)
The Centervertex Theorem
![Page 57: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/57.jpg)
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
![Page 58: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/58.jpg)
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
![Page 59: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/59.jpg)
Given a set S ! Rd, there exists a vertexv " S such that D3!/2(v) # n
d+1.
The Centervertex Theorem
![Page 60: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/60.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
![Page 61: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/61.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
![Page 62: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/62.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
![Page 63: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/63.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
![Page 64: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/64.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
![Page 65: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/65.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
![Page 66: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/66.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
![Page 67: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/67.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
![Page 68: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/68.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
![Page 69: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/69.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
![Page 70: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/70.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
![Page 71: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/71.jpg)
For all 3!/2-wedges W with apex v, |W ! S| > n
d+1
v
Equivalently, |S ! W | " nd
d+1.
W ! H " T
|H ! S| "nd
d + 1
|T ! S| = 0
![Page 72: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/72.jpg)
![Page 73: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/73.jpg)
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![Page 77: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/77.jpg)
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![Page 79: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/79.jpg)
Expected Depth
![Page 80: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/80.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 81: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/81.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 82: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/82.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 83: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/83.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 84: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/84.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 85: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/85.jpg)
Given a set S ! Rd, a point x " Rd anda vertex s " S, if s is the kth nearest vertexto x then D3!/2(s) # D!(x) $ k + 1.
Lemma:
![Page 86: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/86.jpg)
![Page 87: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/87.jpg)
Theorem:The expected 3!/2-wedge depth of a vertex of Sis at least n
2(d+1)2 .
![Page 88: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/88.jpg)
Theorem:
Proof:Order the vertices (v1, . . . , vn) by increasing distanceto a centerpoint.
By the previous Lemma, D3!/2(vi) !n
d+1" i + 1.
The expected 3!/2-wedge depth of a vertex of Sis at least n
2(d+1)2 .
1
n
n!
i=1
D3!/2(vi) !1
n
n
d+1!
i=1
D3!/2(vi)
Ei[D3!/2(vi)] !1
n
n
d+1!
i=1
i !n
2(d + 1)2.
![Page 89: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/89.jpg)
![Page 90: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/90.jpg)
Algorithms
![Page 91: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/91.jpg)
Algorithm 1: Pick a point at random.
With constant probability, the depth will be linear.( )
![Page 92: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/92.jpg)
Algorithm 2: Find a centerpoint and then find the nearest vertex.
![Page 93: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/93.jpg)
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(n) time in R2
![Page 94: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/94.jpg)
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(n) time in R2
![Page 95: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/95.jpg)
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(nlog d) time for approximate centerpoints
O(n) time in R2
![Page 96: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/96.jpg)
Algorithm 2: Find a centerpoint and then find the nearest vertex.
O(nd!1) time for computing centerpoints (tukey medians)
O(nlog d) time for approximate centerpoints
Monte Carlo algorithms: Sublinear timeapproximate centerpoints w.h.p.
O(n) time in R2
![Page 97: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/97.jpg)
![Page 98: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/98.jpg)
Thank you.Questions?
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![Page 101: The Centervertex Theorem (CCCG)](https://reader034.fdocuments.in/reader034/viewer/2022042714/5550975cb4c90590208b4684/html5/thumbnails/101.jpg)
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OpenQuestions
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How hard is it to compute a centervertex?
How hard is it to compute wedge depth?
How hard is it to test centervertices?
Is 270o tight?
Open Questions