On Searching for Cables and Pipes : The Opaque Cover Problem

1. On Searching for Cables and Pipes : The Opaque Cover Problem Scott Provan Department of Statistics and Operations Research University of North Carolina Marcus Brazil, Doreen Thomas Department of Electrical and Electronic Engineering University of Melbourne Jia Weng National Institute of Information and Communications Technology of Australia TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

2. Two Search Problems Find a cable/pipe running through your property Find an ore vein lying beneath your property

3. Related Problems Searching on • Finite sets (Onaga) • Lines (Demaine, Fekete, Gal) • Multiple lines (Kao,Reif,Tate) • Graphs (Deng, Papadimitriou) • Plane regions (Baeza-Yates, Culberson, Rawlins) Searching for • Specific objects (Fiat, Rinaldi) • Probabilistically placed objects (Koopman, Richardson, Alpern, Gal) Search Games (Alpern,Gal)

4. Searching in the Plane: The Opaque Cover Problem (OCP) Given: polygonally bounded convex region S in the plane Find: the minimum length set Fof lines that will intersect any straight line passing through S

5. Opaque Covers Block Lines of Sight The idea: Find a setof lines thatblocks all light from going through S.

6. Related Papers Faber, Mycielski (1986), The shortest curve that meets all the lines that meet a convex body. Akman (1988), An algorithm for determining than opaque minimal forest for a convex polygon. Brakke (1992) The opaque cube problem Richardson, Shepp (2003). The “point” goalie problem. Kern, Wanka (1990). On a problem about covering lines by squares.

7. Some Variations • Does it have to be a single polygonal line? • Does it have to be connected? • Does it have to lie entirely inside S?

8. Covers for a regular pentagon with sides of unit length

9. Opaque Covers and Steiner Trees When the solution is required to be connected and to lie in S, then the solution is the Steiner tree on the corners of S. In any case, each of the components of the solution will be Steiner trees on the corners of their own convex hulls

10. Akman’s Heuristic for the OCP Triangulate S, put a Steiner tree on one of the triangles, and place altitudes on the remaining triangles so as to block all remaining lines through those triangles. Optimal triangulation/line placement for this type of solution can be done in O(n6) (improved to O(n3) by Dublish).

11. Critical Lines Let F be a solution to OCP on S A critical line with respect to F is any line that separates the components of F nontrivially into opposite half-planes.

12. Some Facts about Critical Lines

13. Critical Lines and Adjacent Points of F

14. Critical Lines with 3 Critical Points v1 a2 v3 L a1 a3 d1 d3 v2

15. Two Nasty Examples No single-vertex perturbation Multiple critical lines per vertex Steiner tree length 4.589 OCP length 4.373

16. Some Research Questions • Let F be a solution to the OCP on convex polygonal set S with c corners. • What is the largest number of components F can have, as a function of c ? • What can the components of F look like? • What is the largest number of critical lines can there be w.r.t. F, as a function of c ? • How many critical lines can a given point of F be adjacent to ?

17. A Special Version of the OCP The w-Cover Problem (w-OCP)

18. Examples p/2-cover (all horizontal and vertical lines covered) p-cover (all horizontal lines covered)

19. A solution to the p- and p/2-OCP

20. Examples p-cover p/2-cover

21. Fitting w-Covers Inside of S p-fat region p/2-fat region

22. Fitting w-Covers Inside S

23. Proof for p-Covers Place vertical lines from the bottom coordinate, working up diagonally to the top of the set

24. Proof for p/2-Covers Place a set of sufficiently small rectangles similar to B into S, covering all x-and y-coordinates. Find a set of non-overlapping squares covering all coordinates Place a diagonal in each of these squares

25. Fitting w-Covers Inside General S Problem: How do you fit vertical or diagonal lines into this figure to cover all coordinates?

26. p-Covers for General S

27. Proof Start from middle, continue upward and downward diagonally, possibly adding a final point at the corners.

28. p/2-Covers for General S

29. However . . . e/4 e/4 p/2-fat

30. p/3-Covers All lines with slopes of 0, 60, and 120 degrees must intersect a line of F.

31. p/3-Covers hexagonal coordinates: 90o Idea: Any p/3-cover F for S must contain points having every hexagonal coordinate found in the set of points in S. q 210o 330o

32. p/3-Covers The sum of the 3 coordinate ranges covered by a line segment Lis maximized when L has slope 30, 90, or 150 degrees. Therefore any set of lines that contains all hexagonal coordinates of Sexactly once (except possibly endpoints) and with all of its line segments having 30-, 90-, or 150-degree slopes will constitute an optimal solution to the p/3-cover problem for S.

33. p/3-Covers W1 W2 W3

34. Two Examples Having a p/3-Cover that Meets the Lower Bound Are these the only two?

35. Conjectured p/3-Covers for Equilateral Triangles with Side 1 Solution size: Lower bound: Solution size: Lower bound: Size of Steiner tree =

36. Open Questions • What is the solution to the p/3-cover problem ? • Are there efficient algorithms to solve the w-OCP for other values of w ? • For what values of w is the w-OCP solution guaran-teed to be a set of disjoint lines ? • Is there a sufficiently small value of w that guarantees that the w-OCP solution will be the OCP solution? (Answer: No, if the OCP solution for a triangle is in fact a Steiner tree.)