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On the Power of Discrete and of Lexicographic Helly Theorems

On the Power of Discrete and of Lexicographic Helly Theorems. Nir Halman, Technion, Israel. This work is part of my Ph.D. thesis, held in Tel Aviv University, under the supervision of Professor Arie Tamir, and appeared in FOCS 2004. Helly theorems.

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On the Power of Discrete and of Lexicographic Helly Theorems

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  1. On the Power of Discrete and of Lexicographic Helly Theorems Nir Halman, Technion, Israel This work is part of my Ph.D. thesis, held in Tel Aviv University, under the supervision of Professor Arie Tamir, and appeared in FOCS 2004

  2. Helly theorems Helly’s theorem (1911):Given a finite set H of convex objects in Rd, H has a non empty intersection if every k=d+1 of its elements have a common point Radius theorem: Given a finite set H of points in Rd, H is contained in a unit ball if every k=d+1 of its elements are contained in a unit ball Observation: radius theorem follows from Helly’s 

  3. Our results •Discrete/lex Helly theorems • Useful for solving discrete optimization problems in linear time (extending LP-type) • Characterization of Helly theorems that yield linear time algorithms

  4. Discrete Helly theorems Doignon (1973): Given a finite set D of convex objects in Rd, the objects in D have a common point in S=the integer latticeif every k=2d of its elements do Theorem 1: S arbitrary set of points  unbounded Helly number

  5. Example: Intersecting axis parallel boxes in Rd Theorem 2: Given a finite set D of axis parallel boxes in Rd, and a finite setS of points, the objects in D have a common point in Sif every k=2d of its elements do Proof sketch:

  6. Lexicographic Helly theorems Lex Helly’s theorem:Given a finite set H of convex objects in Rd and a point x Rd, H has a non empty intersection in a point not lex greater than x if every k=d+1 of its elements do

  7. Our results •Discrete/lex Helly theorems • Useful for solving discrete optimization problems in linear time (extending LP-type) • Characterization of Helly theorems that yield linear time algorithms 

  8. = radius of square = 2 = 1 Helly theorems and Optimization Example: discrete smallest enclosing cube Input: n green points and m redcube centers No linear time algorithm known = 0

  9. LP-type problems [SW92] Def: a pair (H, ), H:constraints :objective function satisfying: •monotonicity (FGH,(F) (G)) •locality Goal: calculate (H) Interpretation:(G)=minimumvalue s.t. constraints in G Dual LP-type problems : (H, ) with inequality signs reversed

  10. r An example Smallest Enclosing Ball H: points(H): radius of the smallest enclosing ball of H Wanted: • The optimal value r =(H) •A basis of H combinatorial dimension:d+1 Linear time (randomized) algorithms [Cl88], [Ka92],[SW92]

  11. Usage of LP-type framework Mostly in computational geometry / location theory: • distance between polytopes • smallest enclosing ball/ellipsoid • largest ball/ellipsoid in polytope • angle-optimal placement of point in polygon • line transversal of translates •p-center on the line/in the plane with rectilinear norm • convex Hausdorff distance• etc. •p-recovery points on the line and on directed trees •simple stochastic game

  12. Discretization of Center of ball (cube) must be an input point We losemonotonicity ! Lower bound (n log n) even for circles [LW86] Lower bound for cubes is (n) Can we solve discrete smallest enclosing (d-dimensional) cube in linear time ?

  13. Discretization of smallest enclosing cube problem Input: a set Dof demand points and a set S of cube center locations (supply) Output: the center and radius of a smallest enclosing cube Observation: problem obeys “double monotonicity”: • Adding a demand point cannot decrease the value • Adding a supply point cannot increase the value

  14. Discrete LP-type problems (DLP) Triple (D,S,). Ddemand set, Ssupply set,  objective function s.t. for any D’ D and S’ S : •(D’):=(D’,S)  (D, ) is LP-type • (S’):=(D,S’)  (S, ) is dual LP-type Theorem 5: fixed-dimensional DLP problems are solvable in linear time

  15. Discrete Helly theorems  DLP parameterized Helly system(PHS) unique minimum condition (UMC) Theorem 6:discreteopt. problems with discrete PHS s.t. UMC, are fixed-dimensional DLP Extends [A94] Corollary:discreteopt. problems with discrete PHS s.t. UMC, are solvable in linear time

  16.  Our results •Discrete/lex Helly theorems • Useful for solving discrete optimization problems in linear time (extending LP-type) • Characterization of Helly theorems that yield linear time algorithms

  17. Lex Helly theorems  linear time alg Theorem 7:lex PHS are fixed-dimensional LP-type problems Corollary:existence of a finite lex Helly number solvability of the corresponding optimization problem by a linear LP-type algorithm

  18. Future research Find more discrete/lex Helly theorems Develop more algorithms for DLP model Find more applications for DLP model

  19. Thank you !

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