Well Separated Pair Decomposition 16 GigaYears and 3.3 Million Light Years Playing.
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Transcript of Well Separated Pair Decomposition 16 GigaYears and 3.3 Million Light Years Playing.
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Well Separated Pair Well Separated Pair DecompositionDecomposition
16 GigaYears and 3.3 Million Light Years Playing
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Applications Astronomy Molecular Dynamics Fluid Dynamics Plasma Physics And Guess What:
Surface Reconstruction
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N-Body Simulation
How do I do it Provably Fast??
Trees!! Trees!! Trees!!
A Simple, Provable and Powerful Algorithm
(Callahan-Kosaraju, FMM)
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Andromeda from Earth
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The Galactic Empire
Enough of Sci-Fi
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Well Separated Pair Decomposition
A Decomposition of the complete EucledianGraph into O(n) pairs.
The Curse of Dimensionality/Constants
Can be used to do Surface ReconstructionIn O(nlogn) theoretically!
Practically????
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Well Separated Pair
Dumbbells !!
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An Example of WSPD
100 pointsOn a Circle
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Another Picture
An Output of myimplementationthat computesD-dimensional
WSPD’s
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The Fair Split Tree
The tree on which the Well Separated PairsAre computed.
An adaptation of QuadTrees to give theoreticalGuarantees
Notion: R(P) minimum bounding hyper-rectangleThat encloses the point set P
The FST, breaks R(P) in the longest dimension
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FST Algorithm
CreateTree(vector<points> P){if( P.size() == 1) return Leaf(P);Cut = Plane that cuts the longest dim of R(P)P1 = Points on left of CutP2 = Points on right of CutNode-> left = CreateTree(P1)Node->right= CreateTree(P2)
}
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The Fair Split Tree
Guarantees that at least one point is splitO( ) by Trivial AlgorithmO( ) by pre-sorting in every dimension
For O(nlogn) uses Partial Fair Split TreeComputation.<Double Sided pointer chasing>
nn 2log
2n
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Partial Fair Split Tree
•Keep points d- list<points> sorted in every dimension•In selected dimension, walk from both sides•Remove cut elements and recurse till the largest size of the set is less than (2/3)n•Create subsets and make recursive call
)()()(1
nOnTnTk
ii
nniki 3
2)(max 1
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WSPD Algorithm
On Each Internal node of FST Call Find_Pairs(Node->Left,Node->Right)
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Find_Pairs(N1,N2)
If Well-Separate(N1,N2) return pair<>If Lmax(N1) > Lmax(N2)
Find_Pairs(N1->left,N2)Find_Pairs(N1->right,N2)
ElseFind_Pairs(N1,N2->left)Find_Pairs(N1,N2->right)
endif
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Time for a pic
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Existence
Does a WSPD for a point set always Exist??
Yes!!!
Just take all pairs of points in P, they are well separated by definition
What about Linear Size?? O(n) pairs?
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Another Picture
Not too many disjoint
“good/big” rectangles
can intersect a cube
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Packing cubes
Lemma: A hypercube C which overlaps with a set S of disjoint hypercubes of length l’ follows:
d
l
ClS )2
'
)((||
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Proof by picture!!!
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Canonical Realization
)()( BpApB
)()( ApBpA
If A,B are well separated
(Notation for Lmax(A) < Lmax(B))
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Lemma 4.1
If C be a d-cube, and S={Ai} be a set of disjoint nodes in T and Lmax(p(Ai))>=l(C)/c and R(Ai)overlaps C, then:
dcdcKS )23(),(||
If u recall here l(C)/l’ is replaced by 3c here
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Where did that 3 come from?
3
))(())(ˆ( max
min
AplARl
Hint for Imagination: Think in 1D , use Induction for d-dimension
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Bounding the size of Canonical Realization
Lemma: Let P={p1,p2,…,pn} with FST T , For any node A in T , there are O(1) canonicalpairs of the form (A,B)
Proof: We know that p(A) and B are not well separated , so
))(()2
())(()()),(( maxmax Apld
sApldBApdR
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Rewriting
))(()2
()),(( max Apld
sdBApdR
Intuitively, means a constant factor enlargement ofR(p(A)) intersects with all B well separated from A.
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Bounding the size of Canonical Realization
We know that Lmax(p(B)) > Lmax(p(A))We know that all B are disjointWe know that B overlaps an enlargement of R(p(A))
We know that |SB| is bounded in these conditionsby a constant K(c,d)
Implies there can only be O(1) B’s well separatedfrom A’s, which in turn implies there are onlyO(n) WS-Pairs!!!
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Closest Pair
Theorem: Suppose that there is a k-nearest neighbor pair (a,b) such that
kABbAa ||&
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Closest Pair
By the previous theorem, putting k = 1the Closest pair :
{{a},{b}}
is a well separated pair in the WSPD. HenceGiven the WSPD, we can find the closest Pair of a d-dimensional point set in O(n) time.
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K-nearest neighbors
Put Yao’s cones on every point, anduses inheritance on the FST in these cones. In one cone, only k-neighborsneeds to be maintained.
Then a selection algorithm picks thenearest k-neighbors in O(nk) time.
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General Idea of Applications
Any application where one object affectsevery other object in a set of objects, issusceptible to WSPD paradigm. N-Bodysimulation is a glaring example.
If there is a lower bound of O(n^2) forthe problem, it might be susceptible to an Approximation algorithm using WSPD paradigm.
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Applications II
Nearest Neighbour queriesAnswering what is close in a r-radius ballClustering ApplicationsDynamic Nearest Neighbor queriesApproximate Voronoi diagramsMesh GenerationCombinatoricsMST Approximations/SpannersN-Body Simulation…
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Voronoi Diagrams
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Approximate Voronoi May not be a perfect Voronoi
Diagram but each cell is some kind of Approximation
May not be a decomposition of the entire space.
Of Theoretical Interest Till date no Implementation exists!
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Another Nice Picture
Courtesy Sariel Har Pelad
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Open Problems
•Is there a way to sort the edges of a completegraph in O(n^2) time?
•Is there a way to compute LFS(p) on a manifoldin O(nlogn) using WSPD’s?
•How practical is it for k-nearest neighbour queries?•How about dimensions >= 5??•Compute Medial Axis in O(nlogn)
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Medial Axis Approximation
Lfs Approximation isnt so easy!!
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That’s All Folks!
--Piyush