In this section, we present a 4-factor approximation algorithm for the GMDS problem.
The algorithm is an in-place algorithm and the worst case running time is O(n6logn).
We partition the regionR containing the point set P into regular hexagonal cells of side length 12 such that no point of P lies on the boundary of any cell (see Figure 3.1 (a)). The following observation is trivial as the length of a longest diagonal in any cell is of unit length.
Observation 3.3.1. All the points inside a cell can be covered by a unit radius disk centered at any point inside that cell (see Figure 3.1 (b)).
1
√3/2
(a)
pi 1
(b)
Figure 3.1: (a) Hexagonal partition of the rectangular region containing the points, and (b) a unit disk centered at a point pi circumscribes the cell in which pi lies.
Definition 3.3.2. A septa-hexagon is a combination of 7 adjacent cells such that one cell is surrounded by six other cells (see Figure 3.2 (a)).
(a)
A A A A
A A A
A A A
A A
A A A
A A A
A A A A
A A
A A A A
A A A
A A B B
B B B
B B
B B
B B B
B B
B B B B B
B B B B B
B
C C
C C C
C C
C C
C C C
C C
C C
C C C C C
C C C
C C C
C C C
C C
C D D
D D
D D DD
D D D D D D
D D DD
D D DD
D D D D D D D D DD D
D D D D D D D D D DD
D D
D D
D D
p
q
>2
(b)
Figure 3.2: (a) A septa-hexagon, and (b) a septa-hexagonal partition of R and its 4- coloring scheme.
Let us consider a septa-hexagonal partitioning of R such that no point of P lies on the boundary of any septa-hexagon, and a 4 coloring scheme of it (see Figure 3.2 (b)). Consider a septa-hexagon H which is assigned the color A. Its adjacent six septa- hexagons are assigned three colors, namely B, C, and D, such that pair of opposite septa-hexagons adjacent toH are assigned the same color. LetH0 andH00 be two septa- hexagons having the same color (say B) adjacent to H. Since no point of P lies on the boundary of H0 and H00, the minimum distance between two points p∈ H0∩ P and q ∈ H00∩P is greater than 2 (see Figure 3.2 (b)). This leads to the following observation.
Observation 3.3.3. If H0 and H00 are two septa-hexagons having the same color, then a single unit disk cannot cover points in H0 and H00 simultaneously.
The basic idea of the proposed algorithm is, consider each septa-hexagon H, and compute a subset of points in P of minimum size that covers all the points in P ∩ H. We refer this problem as a single septa-hexagon GMDS problem. Finally, we report the set T which is the union of the solutions for all septa-hexagons.
Observation 3.3.3 says that ifOP TA denotes a minimum size subset of points in P such that OP TA covers all the points of P lying in the septa-hexagons colored with A, then OP TA is the union of the optimum solutions of the GMDS problems for all the septa-hexagons colored A. The same result holds for the other three colors also.
Lemma 3.3.4. |T | ≤4|OP T|, where OP T is a subset of P of minimum size such that OP T covers P.
Proof. The result follows from the facts that (i) T =OP TA∪OP TB∪OP TC ∪OP TD, and (ii) |OP Ti| ≤ |OP T| for all i=A, B, C, D.
3.3.1 Computing an optimum solution for a single septa-hexagon GMDS problem
Consider a septa-hexagon H. Let S1 and S2 be two subsets of P such that S1 contains the points lying in H and S2 contains all the points that are coverable from the points
in S1. That is, S1 ={p |p ∈ P ∩ H} and S2 = {p| p∈ P and δ(p)∩S1 6=∅}. Surely, S1 ⊆S2. The following lemma is due to Observation 3.3.1.
Lemma 3.3.5. If OP TH is a subset ofS2 of minimum size such that OP TH covers S1, then |OP TH| ≤7.
Given an array containing the points inP, inO(n) time we can arrange these points such that the points in S1 are placed at the beginning followed by the points inS2\S1. As suggested in Lemma 3.3.5, we consider all possible combinations of points in S2
of sizei= 1,2, . . . ,6, respectively in this order. For each combination, we check whether all the points in S1 are covered or not. While considering the number i, if there exists a subset of S2 of size ithat covers all the points in S1, then that subset is reported and execution stops. If this fails for all i = 1,2, . . . ,6, then the optimum solution of the septa-hexagon H consists of any one point from each non-empty cell of H. This trivial algorithm needs O(n7) time.
We can reduce the time complexity as follows: let the points ofS2 are stored at the beginning of the array containing the points in P such that the points in each cell of H are stored consecutively. As earlier, we generate different combinations of points of size i= 1,2, . . . ,5 from S2 in this order. If any one of these combinations can produce a solution, the process terminates. Otherwise, we again consider different combinations of size 5. For each combination, say X, the uncovered points of S1 in each cell of H are moved at the beginning of the array, and the convex hull of the uncovered points is computed in an in-place manner in O(nlogn) time [15]. Now, each pointp inS2\X is chosen, and the farthest uncovered point q in S1 is identified. Note that q is a vertex of the convex-hull computed. If d(p, q) ≤1, return the solution X together with q and terminate. This needsO(logn) time for each point inS2\X. If the covering is not done yet, we chose one point from each non-empty cell of H.
Lemma 3.3.6. For a given set P of n points and a septa-hexagon H, a GMDS for P ∩ H can be computed using the points in S2 in O(m6logm) time using O(1) extra space, where m=|S2|.
Theorem 3.3.7. The proposed 4-coloring scheme gives a 4-factor approximation algo- rithm for the GMDS problem in O(n6logn) time using O(1) extra space, where n is the size of the input.
Proof. The approximation factor follows from Lemma 3.3.4. The time complexity result of the theorem follows from Lemma 3.3.6 and the fact that each point inP can participate in the computation for at most a constant number of septa-hexagons.
3.3.2 A
143-factor approximation algorithm
We now show that a slight modification of the earlier algorithm produces a 143 -factor approximation result for the GMDS problem inO(n5logn) time. First we find a feasible solution for a single septa-hexagon, then we use the union of these solutions to provide a feasible solution for the GMDS problem. We prove that the bound on the ratio of this feasible solution to an optimum solution is 143 .
For a single septa-hexagon H, we try to find a solution of size at most 5, if exists.
This can be done by considering all the possible combinations of size i = 1,2,3,4, respectively in this order from S2 as in the previous section. For each combination X, the uncovered points in S1 are examined to be covered by a single point in S2 \X. If there is a point q∈S2\X which covers all the uncovered points inS1, thenX together with q is reported. After considering all possible combinations of 4 points in S2, if the covering is not done, then we arbitrarily choose one point from each non-empty cell of H and report them. Thus, we have the following lemma.
Lemma 3.3.8. For a given set P of n points and a septa-hexagon H, a set SH ⊆ S2
of size at most 5 points such that SH covers all the points in P ∩ H, can be tested in O(m5logm) time using O(1) extra space, where m=|S2|.
Theorem 3.3.9. The proposed 4-coloring scheme gives a 143-factor approximation algo- rithm for the GMDS problem in O(n5logn) time using O(1) extra space, where n is the input size.
Proof. If the above algorithm cannot obtain a solution of size at most five for a septa- hexagon H, then |OP TH| ≥ 6. If |OP TH| ≥ 6, our algorithm returns a solution, say SOLH, of size 7. Thus, for any sept-hexagon H, if|OP TH| ≥6 implies|SOLH|= 7, and hence the approximation factor is |OP T|SOLH|
H| ≤ 76 . LetSOLA,SOLB,SOLC, and SOLD be the union of the solutions of all the septa-hexagons coloredA, B, C, andD, respectively.
Since |SOLA|=P
|SOLH|, where the sum is taken over all the septa-hexagons colored A, implies, |SOLA| ≤ 76|OP T|. Similarly, the bound also holds for the remaining three colors B, C, and D. Thus the theorem follows.