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.

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

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(n6) (improved to O(n3) 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

T heorem 1: Let F bea solution to theOCP onset S. Then

1. Let C bea component of F , and let ¹C beitsconvex hull. Then every corner of C that isnot a corner of S is incident to at least onecritical line.

2. Every critical line L has at least 3 intersec-tion points with F , occuring on alternatesides when traversing L.

Critical Lines and Adjacent Points of F

Let F bea solution to theOCP on set S, and letL be a critical line with respect to F . Call thepoints of F lying on L critical points. A freecritical point is onewhich is neither an corner ofS nor on any other critical line.

Lemma 1: Let v be a free critical point of L.Then

1. v has degree1 or 2

2. If v is of degree 1 then its adjacent edge isperpendicular to L.

3. If v is of degree 2 then its adjacent edgesformequal angles with L.

Critical Lines with 3 Critical Points

1

2

3L

d1 d3

v3

v2

v1

Lemma 2: Let L be a critical line with exactly 3 free criticalpoints. Then

(i) v1 and v3 havedegree2;

(i) If v2 has degree1 then sin®i = 12

³1¡ di

d1+d3

´; i = 1;3;

(ii) If v2 hasdegree2then sin®i = 12

³1¡ di

d1+d3

´sin®2; i = 1;3.

Two Nasty Examples

Steiner tree length 4.589

OCP length 4.373

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

Given Region S in the plane and slopeangle ! =¼=k for some integer k

F ind the minimum length set oflines that intersects all lines ofslope 0;! ;2! ; : : :; (k¡ 1)! passingthrough S.

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

T heorem 2:

A solution to the ¼-cover problem is anyvertical line going from the smallest to thelargest y-coorinateof S.

A solution to the ¼=2-cover problem is ei-ther diagonal of the smallest box B thatcontains S.

Examples

/2-cover-cover

Fitting Covers Inside of S

-fat region /2-fat region

De¯nition: A region S is ! -fat if noline L perpendicular to a valid slope j !intersects S in exactly onepoint.

Fitting Covers Inside S

T heorem 3: For ! = ¼or ¼=2, If S is! -fat then thereexists a solution to the ! -OCP with length as given by Theorem 1,but lying entirely insideS.

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

T heorem 4:

If S is line segment, then theonly solutionto the¼-cover problemthat lies entirely in-side S is either a single point of S (if S ishorizontal) or S itself.

Otherwisethereis a solution to the¼-coverproblem | with possibly in¯nite numberof lines segments | lying entirely in S andhaving length as given in Theorem1.

Proof

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

Covers for General S

Fact: For ageneral set S theremay beno¼-cover| evenwithan in¯nitenumber of linesegments|lying entirely insideS and having length as givenby Theorem1.

However . . .

-fat

T heorem 5: For any ² > 0, there is a ¼=2-cover with a¯nitenumber of linesegments lyingentirely insideS andwhose length is within ² of that given in Theorem1.

Covers

All lines with slopes of 0, 60, and 120 degrees must intersect a line of F.

Covers

hexagonal coordinates:

210o

330o

90o

r [sin(90±¡ µ);sin(210±¡ µ);sin(330±¡ µ)]

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

Corollary: A lower bound on the length of a¼=3-cover for set S with hexagonal dimensions(W1;W2;W3) is (W1+W2+W3)=2

W1

W3

W2

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

3.14/33

5.1

3.14/33

5.1

3.14/33

5.1

Solution size:

Lower bound:

Solution size:

Lower bound: 5.1

577.13/31

Size of Steiner tree = 732.13

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.)