On Searching for Cables and Pipes : The Opaque Cover Problem TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAAA A A AAAA 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
Two Search Problems Find a cable/pipe running through your property Find an ore vein lying beneath your property
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)
Searching in the Plane: The Opaque Cover Problem (OCP) Given: polygonally bounded convex region S in the plane Find: the minimum length set F of lines that will intersect any straight line passing through S
Opaque Covers Block Lines of Sight The idea: Find a set of lines that blocks all light from going through S.
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.
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?
Covers for a regular pentagon with sides of unit length
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
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(n 6 ) (improved to O(n 3 ) by Dublish).
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.
Some Facts about Critical Lines
Critical Lines and Adjacent Points of F
Critical Lines with 3 Critical Points 11 22 33 L d1d1 d3d3 v3v3 v2v2 v1v1
Two Nasty Examples Steiner tree length OCP length No single-vertex perturbation Multiple critical lines per vertex
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 ?
A Special Version of the OCP The -Cover Problem ( -OCP)
Examples /2-cover (all horizontal and vertical lines covered) -cover (all horizontal lines covered)
A solution to the - and /2-OCP
Examples /2-cover -cover
Fitting Covers Inside of S -fat region /2-fat region
Fitting Covers Inside S
Proof for Covers Place vertical lines from the bottom coordinate, working up diagonally to the top of the set
Proof for Covers 1.Place a set of sufficiently small rectangles similar to B into S, covering all x-and y-coordinates. 2.Find a set of non-overlapping squares covering all coordinates 3.Place a diagonal in each of these squares
Fitting Covers Inside General S Problem: How do you fit vertical or diagonal lines into this figure to cover all coordinates?
Covers for General S
Proof Start from middle, continue upward and downward diagonally, possibly adding a final point at the corners.
Covers for General S
However... -fat
Covers All lines with slopes of 0, 60, and 120 degrees must intersect a line of F.
Covers hexagonal coordinates: 210 o 330 o 90 o Idea: Any /3-cover F for S must contain points having every hexagonal coordinate found in the set of points in S.
Covers The sum of the 3 coordinate ranges covered by a line segment L is maximized when L has slope 30, 90, or 150 degrees. Therefore any set of lines that contains all hexagonal coordinates of S exactly 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 /3-cover problem for S.
Covers W1W1 W3W3 W2W2
Two Examples Having a /3-Cover that Meets the Lower Bound Are these the only two?
Conjectured /3-Covers for Equilateral Triangles with Side 1 Solution size: Lower bound: Solution size: Lower bound: Size of Steiner tree =
Open Questions What is the solution to the /3-cover problem ? Are there efficient algorithms to solve the OCP for other values of ? For what values of is the -OCP solution guaran- teed to be a set of disjoint lines ? Is there a sufficiently small value of that guarantees that the -OCP solution will be the OCP solution? (Answer: No, if the OCP solution for a triangle is in fact a Steiner tree.)