Disc Covering Problem with Application
to Digital Halftoning
Dagstuhl Workshop, 2004
Tetsuo Asano
School of Information Science,
JAIST
Japan Advanced Institute of Science and Technology
Joint Work with
Shinji Sasahara, Fuji Xerox Co.Ltd.
Peter Brass, CUNY, U.S.A.
Problem Specification
Problem: R = {r11, r12, ... , rnn} : a matrix of n2 positive real
numbers. Each rij is a radius of a disc at (i,j). Choose disks
so as to maximize the total singly-covered area.
R=
2.4
3.3
3.6
4.1
2.5
2.5
3.5
3.8
3.3
1.9
2.6
3.7
3.2
2.5
1.5
2.2
3.3
2.2
1.2
1.0
1.9
1.7
3.5
3.6
4.2
circle of
radius 1.2
singly-covered area
Example:
a set of input discs
given by a matrix
a set of discs that
maximizes the total
singly-covered area
a set of discs
singly-covered area
Algorithmic questions and Motivation:
How hard is it? NP-hard?
Approximation algorithm with guaranteed ratio
One-dimensional problem
still hard? or an efficient algorithm?
Motivation
application to digital halftoning:
conversion from continuous-tone images to
binary-tone images
How hard is it? NP-hard?
NP-hard or polynomial-time algorithm?
open
One-dimensional problem
still hard? or an efficient algorithm?
polynomial-time algorithms
1. graph-based approach
2. plane-sweep and dynamic programming
Approximation algorithm with guaranteed performance
Cu: a disc with center at u
r(Cu): radius of the disc Cu
Cu
u
Algorithm 1:
・Sort all the discs in the decreasing order of their radii.
・for each disc Cu in the order do
・ if Cu does not intersect any previously accepted disc
・
then accept it else reject it
・Output all the accepted discs.
input set of discs
accepted discs by
Algorithm 1
Lemma 5: Algorithm 1 finds a 9-approximate solution.
Proof:
S: a set of all input discs
S’: a solution (set of discs) obtained by Algorithm 1.
all the discs in S’ are disjoint.
Cu: a disc which has not been accepted by Algorithm 1
there must be some disc Cv such that
(1) Cv has been examined before Cu, and
(2) Cv intersects Cu.
Cu
Cv
(1) r(Cv)≧r(Cu) : larger discs are examined first
If we enlarge Cv by a factor 3, the all the discs rejected
by Cv are completely contained in the enlarged disc.
Approximation algorithm with better performance
some terminologies and notations:
Cu: a disc with center at u
r(Cu): radius of the disc Cu
Rr(Cu): a disc contracted by r, 0<r<1,
which is called the core of the disc Cu.
Rr(Cv)
Rr(Cu)
Cu
Cv
In this case
Cv violates Cu
but Cu does not
violates Cv
Cv violates Cu
if Cv intersects the core Rr(Cu) of Cu.
Otherwise, two discs safely intersect.
Algorithm 2:
・Sort all the discs in the decreasing order of their radii.
・for each disc Cu in the order do
・ if Cu is not violated by any previously accepted disc
(i.e., Cu does not intersect the core of any previously accepted disc)
・
then accept it else reject it
・Output all the accepted discs.
Rr(Cv)
Rr(Cu)
Cu
Cv
in this case
Cu can be accepted
input set of discs
accepted discs
Lemma 6: Algorithm 2 finds a 5.83-approximate solution.
Proof:
Cu: a disc which has not been accepted by Algorithm 2
there must be some disc Cv such that
(1) Cv has been examined before Cu, and
(2) Cv violates Cu i.e., Cv intersects the core of Cu.
Rr(Cv)
Rr(Cu)
Cu
Cv
Enlarge each accepted disc by a factor 2 + r
Enlarge each accepted disc by a factor 2 + r
Rr(Cv)
Cv
Rr(Cu)
Cu
r  r ( Rr (Cv ))  2r (Cv )
 rr (Cv )  2r (Cv )
 (2  r )r (Cv )
Every disc rejected by Cv is completely contained in
the enlarged disc.
How much area is covered exactly once by accepted discs?
How much area is covered exactly once by accepted discs?
a set of accepted discs
additively weighted Voronoi diagram of the discs
truncate each Voronoi cell of an accepted disc
by its blown-up copy
evaluate how much area in each cell is covered by the disc
ratio > 1 / 5.83.
accepted discs
blow up a disc by 2+r
additively weighted
Voronoi diagram of the discs
truncate each cell by the copy
partition into angular sectors
Voronoi edge outside disc
In this angular sector,
at least 1/(2+r)2 is occupied
by the disc.
2+r
1
Voronoi edge inside disc
The core is not overlapped
by any other disc. So, the ratio
is at least r2.
Voronoi edge outside disc
In this angular sector,
at least 1/(2+r)2 is occupied
by the disc.
Voronoi edge inside disc
The core is not overlapped
by any other disc. So, the ratio
is at least r2.
We choose r so that r2 ＝ 1 / (2+r)2.
Then, the ratio of singly-covered area is at least
1/(2+r)2.
approximation ratio is (2+r)2=5.83.
NP-hardness proof (uncompleted)
reduction from planar 3SAT
variable part
we have to choose
either all of red circles
or all of blue circles.
NP-hardness proof(continued)
path part
NP-hardness proof(continued)
clause part
Open Problems
• Improve the approximation ratio
approximation ratio 2 or (1+e)?
O(n log n) time and O(n) space for
practical applications
• Complete NP-hardness proof
Concluding Remarks
considered a geometric optimization problem
to choose discs among n given discs
to maximize the singly-covered area.
Application: adaptive digital halftoning.
Proposed some heuristic algorithms
and verified their effectiveness by experiments.
Heuristic Algorithm 1
(1) Fix a contraction factor r appropriately.
(2) D = a set of all discs in the raster order
(3) for each disc C in D (in raster order)
accept C if core(C) does not intersect the core of any
previously accepted disc
Rr(Cv)
and reject it otherwise.
Rr(Cu)
Cu
Cv
In this algorithm, the first disc in the raster order is always
accepted, which may be a bad choice for future selection.
Heuristic Algorithm 2
(1) Fix a contraction factor r appropriately.
(2) D = a set of all discs in the decreasing order of their radii
(3) for each disc C in D (large discs first)
accept C if core(C) does not intersect the core of any
previously accepted disc
Rr(Cv)
and reject it otherwise.
Rr(Cu)
Cu
Cv
Experimental Results
Input images
small: 106 x 85, and large: 256 x 320
 enlarge them into 424 x 340 and 1024 x 1280

Running time
Heuristic 1: 0.06 sec. for the small image
0.718 sec. for the large image
on PC: DELL Precision 350 with Pentium 4.
Heuristic 2: 0.109 sec. for the small image
1.031 sec. for the large image
Output of Heuristic Algorithm 1(contracted discs)
Output of Heuristic Algorithm 1(discs of original sizes)
Voronoi diagram for the set of circle centers
Output of Heuristic Algorithm 2(contracted discs)
Output of Heuristic Algorithm 2(discs of original sizes)
Voronoi diagram for the set of circle centers
Fill out each Voronoi cell
according to the grey level
at the center point of the cell
by cubic interpolation
Output halftoned image
enlarged picture of a part