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  1. Distance Trisector curve and Voronoi diagram with neutral zone Takeshi Tokuyama (Tohoku U) joint work with Tetsuo Asano (JAIST) and Jiri Matousek (Charles U)

  2. Grand Challenge of NHC project • Throw right on Computational Barriers • Identifying barriers (something looking impossible) • Breaking/proving/avoiding barriers • Making properties of barriers clearer • Well-known barriers in TCS • Computability • P vs NP, NP vs PSPACE etc • Approximation hardness, approximation ratio • Randomness (P vs PP, primarity checking, Yao’s minmax principle) • Lower bounds for online algorithms • Cryptographic barriers (Discrete log etc) • Hopefully, find some more new barriers, because they will help progress of TCS

  3. Classical barriers in math history Let no one ignorant of Mathematics enter here. (plato) • Incomputable geometric/arithmetic problems • Compute diagonal length of a unit square • Find a cube of volume 2 • Dilemma of Pythagoreans • Draw a circle with the unit area • It is easier to square the circle than to get round a mathematician.August de Morgan • Draw regular n-gon • Trisect a given angle

  4. Progress in math history God created the integers, all else is the work of man.( Leopold Kronecker ) • Necessity is the mother of invention. • Diagonal length computation • Cube with volume 2 • Unit area circle  Transcendental number π • Draw regular n-gon • Trisect a given angle • Irrational numbers • Radicals “Solvability”of algebraic equations Complex numbers, Groups, Elliptic functions Inventing tools to measure what we cannot directly measure. Same in CS: NP-complete theory, Proof checking,….

  5. Tool to resolve a small but curious barrier. Story of this talk • In a computational geometric problem , we find a simple and natural geometric tool named distance trisector curve. • A new transcendental curve?? • Some initial results have been obtained • Existence and uniqueness of the trisector curves • An algorithm to compute it. • Use of the trisector curves. • A new possibility in computational geometry • Relaxed “computability” of geometric problems.

  6. Voronoi diagram • Subdivision of plane (space) into cells • S = {p1,p2,…pn} points in the plane • V(pi) = { x : d(x, pi) < d(x, pj) for all j ≠ i} • Voronoi cell: dominating region of pi • Great geometric structure with many applications • Mesh generation , Graphics, • Simulating economic/political equilibrium • Simulating biological cells / crystallization • Efficient algorithm : O( n log n) time • Many variants: VD of lines/discs/regions, higher dimension, non-Euclidean metric, power diagrams.

  7. Voronoi diagram is easy • Drawn by using “perpendicular bisectors” • Anciently known how to draw the bisector • Tool to bisect an angle • Can be drawn by using rational arithmetic • It is trivial that a Voronoi diagram is in the arrangement of bisectors (size is polynomial) • Naïve polynomial time algorithm • Only need to speed up the computation • Known variants are also mathematically conventional

  8. Motivation from nature • We often see beautiful patterns with “objects” and “neutral zone” in natural structures • Can we have a variation of Voronoi diagram to draw such a picture?

  9. Voronoi diagram with neutral zone • Classical Voronoi diagram: Partition the plane (space) into domination regions of input points. • Idea: Partition the space (plane) into domination regions and a neutral zone • There may be many mathematical formulations • We adopt a natural formulation

  10. Definition of N-Voronoi diagram • S = {p1,p2,…pn} points in the plane • NV(pi) = { x : d(x, pi) < d(x, y) for         every y ∈NV(pj) for j ≠i} y x

  11. N-Voronoi diagram on 7 points

  12. Questions • Bisector ?? • Distance trisector curves: • Known or unknown? • As easy as angle trisection? Ruler, Compass, and red-ink (Archimedes)

  13. Distance trisector • Different from Apollonius’s circle • Natural and simple definition (see next slide) • Surprisingly, seems to be a new curve 2 1 Apollonius’s circle

  14. Distance Trisector Curves Equally-spaced curves CP and CQ: for any point p on CP, dist(p, P) = dist(p, CQ) and for any point q on CQ, dist(q, Q) = dist(q, CP). where dist(p, C) is the distance from p to a point on C that is closest to p. Q P Q P

  15. Note: Drawing curves was not easy. Only possible after revealing some theoretical results y=x+1 y=-x+1 Q(-1,0) P(1,0)

  16. Observation: Each point X on CP is the intersection of the normal line of CQ at its partner point X’ and the perpendicular bisector of X’P Corollary. If we exactly determine CP at a small neighbor of P, we can determine whole CP normal line to the curve CQ X bisector of segment X’P We have a system of differential equation X’ Q(-1,0) P(1,0)

  17. Implication • No point beyond the tangent point A can be a partner point of a point of NV(P) A Indeed, it suffices to compute CP for x<5.65 tangent line from P to the curve CQ Q(-1,0) P(1,0)

  18. Nice nature of the curve • Convex and smooth (revealed to be analytic) • Satisfies a system of differential equations • Specialists did not know how to solve it • It suffices to compute in the range [0, 5.65] • Important curves satisfy this • sin (x)  [0, 2π], log x  [1,10] • Problems: Existence, uniqueness, computation

  19. Existence and uniqueness • P=(0,1), Q = (0,-1) • y=f1(x) is x-axis (bisector of PQ) • y = g1(x) :bisector parabola of P and x-axis • y=fj(x): bisector curve of P and y=-gj-1(x) • y=gj(x):bisector curve of P and y= -fj(x) Lemma. The trisecting curve CP must be above f j (x) and below gj(x)

  20. y=gi(x) P = (0,1) y=fi(x) y= -fi(x) (0, -1) y= -gi-1(x)

  21. Lemma: The series fj(x) and gj(x) converges to continuous functions F(x) and G(x),respectively Theorem (Existence and Uniqueness) . G(x) ≡ F(x), and y= F(x) gives the trisector curve CP y=G(x) y=F(x)

  22. Computation of F(x) • Given a value of x >0 , compute F(x) to any precision (error is smaller than given ε) Theorem. We can compute F(x) to the accuracy εin O( log ((1+x) /ε)) time

  23. t(x): x-value of the nearest point from (x, F(x)) P = (0,1) y=F(x) t2(x) t(x) x z y= -F(x) (0, -1)

  24. Backward computation to have (x, F(x)) P = (0,1) y=F(x) t2(x) t(x) x z y= -F(x) (0, -1)

  25. Idea for computing F(x) • If -t(x) is the x-value of the partner point, t(x) < βxfor a constant • We can compute the Taylor expansion of F(x) and t(x) around zero to any precision. • F(x) and t(x) are computable if x < h for a small h. • F(x) and x are computable from t(x), t(t(x)), F(t(x)), and F(t(t(x))). • Guess z = tk(x) < h, compute F(z) and F(t(z)), and recursively compute ti(x) for i=k, k-1, k-2, …, 0 • Determine correct value of z via binary search

  26. Formulas to compute F(x) • x = Φ(t(x), t(t(x)),F(t(x)), F(t(t(x))) • F(x) = ψ(t(x), t(t(x)),F(t(x)), F(t(t(x))) • Φ(x,y,u,w) = x + y(x2+(1+u)2)/2Q(x,y,u,w) • ψ(x,y,u,w) = (2xyu +(1+u)(1+x2-u2)) /2Q(x,y,u,w) • Q(x,y,u,w) = (1+u)(1+v)-xy The above mentioned: STOC06,to appear

  27. N-Voronoi diagram on many points Now we have trisector curves. Can we draw N-Voronoi diagram of more than two points by using them??

  28. What happens if the third point is given Insertion of a new point

  29. New enemy may contribute to you 漁父の利

  30. Basic properties • Boundary curve of NV(pi) is the bisector between pi and (the union of) boundary curves of other regions • Each region is convex and nonempty • It may happen that there is no unbounded region (different from an ordinary Voronoi diagram)

  31. Voronoi edges • Definition: If p is a boundary point on NV(pi) and q is its nearest point among other region boundaries, we call q the partner point of p. • Definition: The boundary of NV(pi) is decomposed into curve segments each of which consists of points whose partner points are in a same region. The connected components are called Voronoi edges.

  32. Combinatorial complexity • Theorem. Number of Voronoi edges is O(n) in an N-Voronoi diagram on n points. Proof: Rays towards nearest enemy’s-boundary do not cross each other (cheating a little) reduced to the linearity of number of edges in a planar graph +Davenport-Schinzel sequence argument.

  33. Questions • N-Voronoi diagram always exists? • Yes • Unique for a given point set? • Yes • Efficient algorithm exist? • Yes, if we are given some oracles • Really efficient in practice? • No……

  34. Existence Given a set of regions R1,R2,…Rn such that Rj contains pj, consider an operator F F(R1, R2, … Rn) = (Q1, Q2, … Qn ) where Qj = { x: d(x, pj) < d(x, y) for ∀y ∈ ∪ i ≠j Ri } Theorem. F has at least one fixed point From Schauder-Kakutani’s fixed point theorem Corollary. N-Voronoi diagram is given as a fixed point of F

  35. Note: Fixed point theorem Brower(1910),Schauder(1930),Kakutani(1941) Z: Banach space, K: Compact convex subset of Z (nonempty) F: K  K continuous map Then, F has a fixed point • Z: space of n-tupple of convex regions (R1,R2,..Rn) • We need to introduce norm, and define convexity etc… • Continuous dimensional space (Shauder-Kakutani’s version)

  36. Uniquness • Fixed point theorem does not assure uniqueness. • Uniqueness is given in a constructive fashion • Crystallization algorithm: • Growing radius of disks • Analogous to the space-sweep algorithm for computing a (classical) Voronoi diagram as a lower envelope of parabolic cylinders.

  37. Curves in a N-Vonoroi diagram Property 1: N-Voronoi diagram is a subset of an arrangement of curves in a curve family F • F consists of : • Distance trisectors of pairs of input points are in F. • Bisector curves of an input point and a curve (or point) C in F. • Intersection points of curves in F. Property 2: The number of applications of bisector operations is finite to obtain F

  38. Analysis of crystallization algorithm • For planar case, the algorithm shows uniqueness of N-Voronoi diagram • Number of “structural changes” is O(n) • Terminates in finite steps if we can draw • bisector curve of a point and given point • distance trisector curve • Terminates in polynomial time if we assume that we can draw an “generalized” trisector curve