Loading in 2 Seconds...

Distance Trisector curve and Voronoi diagram with neutral zone

Loading in 2 Seconds...

258 Views

Download Presentation
##### Distance Trisector curve and Voronoi diagram with neutral zone

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Distance Trisector curve and Voronoi diagram with neutral**zone Takeshi Tokuyama (Tohoku U) joint work with Tetsuo Asano (JAIST) and Jiri Matousek (Charles U)**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**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**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,….**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.**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.**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**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?**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**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**Questions**• Bisector ?? • Distance trisector curves: • Known or unknown? • As easy as angle trisection? Ruler, Compass, and red-ink (Archimedes)**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**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**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)**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)**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)**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**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)**y=gi(x)**P = (0,1) y=fi(x) y= -fi(x) (0, -1) y= -gi-1(x)**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)**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**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)**Backward computation to have (x, F(x))**P = (0,1) y=F(x) t2(x) t(x) x z y= -F(x) (0, -1)**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**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**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??**What happens if the third point is given**Insertion of a new point**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)**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.**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.**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……**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**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)**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.**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**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