Lecture Note for Computational Geometry-Geometric Sampling and Center Points Lectured by Pr

sets in C have a common point. For example, in 1-D, assume 3 intervals I1, I2, I3 and I i ∩ I j ≠ φ (i≠j, i,j = 1,2,3), then
I1 ∩ I 2 ∩ I 3 ≠ φ Proof: Assume
So Pr[s is a δ-median] = 1-2( −3δ 3 + 2δ 2 ) . When δ = ¼, Pr[s is a δ-median] = 1-2x5/32 = 11/16 >1/2. It shows this method improves the estimation of δ-median.
and
i i P_
P − = {Pi | ai < 0} .
j j
From the equations P•a = 0, or
Theorem 3: If P⊂ Rd has n points, then its set of δ-center is the intersection of a family of closed halfspaces whose member each contains at least n•(1-δ) points of P and has at least d points of P in its boundary hyperplanes.
Theorem 1: (Helly) Suppose C={C1,C2,…Cn,..C∞} is a family of convex sets in Rd and +1 for every d+1 of them (Ci1,…Cid+1), if there exist a common point ( ∩idj = i1 C j ≠ φ ), then all
Center point is the extension of median in 1-D to high dimension Rd.
Definition 3: c is a δ-center point of P if taking any projecting to 1D, c’s projection is still a δ-median in that projection direction.
let’s define Γ = {H + | H + ∩ P >
d n} , now we only need to prove: for all H1+, …, d +1 d +1 1 n and ∩ H i+ = R d − ∪ H i− , it infers Hd+1+ ⊂ Γ , ∩ H i+ ≠ φ . Recall | H i + ∩ P |> d +1 i =1 d +1 1 ( P ∩ H i− ) < (d + 1) ⋅ ⋅n = n, that P ∩ H i + = P ∩ ( R d − ∪ H i− ) . Now we have: ∪ d +1 i =1
Theorem 4: (Radon’s theorem) Let Let P = {p1,…pn} ⊂ Rd such that n ≥ d+2, then P can be partitioned into two sets P1 and P2 such that P1 ∪ P 2 = P and the Convex Hulls CH ( P 1 ) ∩ CH ( P 2) ≠ φ P1 ∩ P 2 = φ
I1 ∩ I 2 = S3 ≠ φ I1 ∩ I 3 = S2 ≠ φ
I 2 ∩ I 3 = S1 ≠ φ Without of generality, assume S1 is in the middle of S2 and S3. Recall that S 2 ⊂ I1 and S3 ⊂ I1 , so obviously, we have S1 ⊂ I1 . Also S1 ⊂ I 2 ∩ I 3 , now we have I1 ∩ I 2 ∩ I 3 = S1 ≠ φ
Definition 2: A center point of a set P of n points in Rd is a point c of Rd such that every hyperplance passing through c partitions P into two subsets each of size at most n•d/(d+1).
Theorem 2: Let P = {P1,…Pn} ⊂ Rd, there will exist a point c ∈ Rd such that c is a
1 d +1
center point.
Proof: For any half plane H, let | H + ∩ P |≥| H − ∩ P | . If | H + ∩ P |> d n , then c ∈ H + . d +1
Lecture Note for Computational Geometry -Geometred by Prof. Shang-Hua Teng on Nov. 29 Scribed by Zhuangli Liang (sliang@) Boston University
Here when we talk about halfspace, we mean significant halfspace defined by the hyperplane containing at least d points of P. Now the linear programming algorithm is to n find a point in the common intersection of a family of no more than < n d halfspaces. d In other words, the linear programming problem in d dimension is subject to no more than nd inequality constrains. From Helly’s theorem, it can be solved in O(nd) time.
n
11×n P a = 0( d +1)×1 Since n >d+1, there exists a non-trivial solution a ≠ 0 . From 11×n ⋅ a = 0 , we have
∑a
i
= 0 . Now we define the partitions P + = {Pi | ai ≥ 0}
which means there must exist at least a point c out of
∪ ( P ∩ H i− ) and c ∈ ∩ H i+ ≠ φ .
i =1 i =1
d +1
d +1
We can apply this theorem in the linear programming problem.
Definition 1: Given a set P of n points in Rd and c ∈ P, if | P ≥ c |> δ and | P < c |> δ , then c is a δ-median of P.
If we randomly choose an element s from the set P, then the probability Pr[s is a δ-median] = 1-2δ. This is a not bad approximation when δ is small. To improve this probability, we can test it in a t elements sample T⊂P. This procedure is described as below (1) After s is picked, t element T⊂P are randomly picked min(| T < s |,| T ≥ s |) (2) φ = is computed |T | (3) if φ ≤ δ − ε , then s is a good estimation of δ-median, with error ε . Otherwise, s is a bad estimation and another s need to be picked and test again. This is a useful tool to estimate a δ-median with small error. Given a sample number of t~800, the probability Pr[testing is wrong] < 4%. Also, this testing procedure doesn’t depend on n, which can be a very large number. Now, we give another way to estimate the δ-median. In k randomly picked elements S⊂P and let s be the median of S, what will be Pr[s is a δ-median of P] ? First, we examine a simple example. when k=3 and S={s1,s2,s3}, s=median{ s1,s2,s3}, since we only care about the relative rank in P, without generality, we assume 1 2 n −1 P= {0, , ,..., } . Obviously, Pr[s is a δ-median] = 1-Pr[ s<δ, or s>δ ] = 1-2Pr[ s<δ ]. n n n 3 Recall Pr[ s<δ ] = Pr[ at least two of { s1,s2,s3 } ≤ δ ] = δ 2 (1 − δ ) + δ 3 = −3δ 3 + 2δ 2 2