A (1+ )-Approximation Algorithm for 2-Line-Center

1. A (1+)-Approximation Algorithm for 2-Line-Center P.K. Agarwal, C.M. Procopiuc, K.R. Varadarajan Computational Geometry 2003

2. Outline • Introduction • Preliminaries • Approximation Algorithm • Conclusion

3.    1.Introduction: Projective clustering • Given a set S of n objects in Rd and two integers k < n and q  d, find k q-dimensional flats h1,...,hk and partition S into k subsets S1, ...,Sk so that • is minimized. • The k-line-center problem is the projective clustering problem for d =2 and q = 1. • Partition S into k clusters and each cluster Si is projected onto a line so that the maximum distance between a point p and its projection p* is minimized.

4.    1.Introduction:This paper • 2-line-center • Given a set S of n points in R2, cover S by two strips so that th maximum width of a strip is minimized • Projective clustering has recently received attention as a tool for creating more efficient nearest neighbor structures, as searching amid high dimensional point set is becoming increasingly important.

5.      1.Introduction: Previous Work • 2-line: • near-quadratic running time for exact version. • 1-line: width problem • (nlogn) for d =2 • (1+  )Approximation: • General: computing k projective clusters • Whether a set of n points in the plane can be covered by k lines is NP-Complete • Projective clustering is NP-Complete • Approximating the minimum width within a constant factor is NP-Complete.

6. 1.Introduction: This result • Let w* denote the minimum value so that S can be covered by two strips of width at most w*. • This paper present an algorithm that computes, for any  >0, a cover of S by two strips of width at most (1+ ) w*, in time • Strategy of this paper: • first presenting a 6-approximation algorithm • then derive a (1+ )-approximation algorithm

7.              2.Preliminaries r lpq p q (p,q, r) • Notations • Strip : the region lying between two parallel lines l1 and l2 • width of  : distance between l1 and l2 • direction of : direction of l1 • strip cover  of S: two strips that each point of S lies in one of the strips. • For any points p,q, • lpq: the line passing through p, q • (p,q, r): if r  lpq , is the same as lpq • (p,q; w): the strip having lpq as the median line of width 2w.

8.                2.Preliminaries • Notations • Optimal cover: * = {1*, 2*} of S, its width w* • Si* = S  i* • Anchor pair (p,q) of  : if d(p,q)  diam(S  ) p S   diam(S  )  q

9.                 2.Preliminaries • Proof • let  be the diameter of S* • : the smallest rectangle containing S*, the length of  is L, the width of  is w. • We choose rS* to be the point farthest away from lpq . Since r, d(r,lpq)  3w. • Moreover S* =S  *  (p,q,r), and the lemma follows • ’: parallel to lpq, thinnest strip contains , its width w’.

10.                3. Approximation Algorithm • Two phases • phase 1: computes a cover  of S by two strips of width at most 6w* • phase 2: Use  to compute a new cover by two strips of width at most (1+ )w*

11.                3.1 6-approximation cover • Suppose we have an anchor pair (p,q) of a strip in * • How to obtain such a pair will be described in 3.2 • WLOG, let (p,q) be an anchor pair of 1* • By Lemma 2.1 there exist r  S so that width((p,q,r)) 6w* and (S\ (p,q,r))  2* • Perform a binary search to find that r !! • Then compute a strip of width at most 2w* that contains the rest points, i.e. S\ (p,q,r)

12.                g(w) 3.1 6-approximation cover • Suppose we have an anchor pair (p,q) of a strip in * • f(w) • Proof: 2w f(w) wi wi+1 wn w

13.                2w f(w) wi wi+1 wn w g(w) 3.1 6-approximation cover • f(w) • Binary search over w • Proof:

14.                3.2 Computing an anchor pair • Compute a family F of at most 11 pairs of points that contains an anchor pair. • compute the diameter  of S, and let (p,q) be a diametral pair in S. Let Dp, Dq be the disks of radius /2, centered at p, respectively q.

15.                  3.2 Computing an anchor pair • Case 1 • If S\(Dp Dq)  , • let rS\(Dp Dq). Return F ={(p,q),(q,r),(p,r)} • Correctness: • At least two points among p,q, and r must be in the same strip subset. Since d(p,q) =  and d(p,r), d(q,r)   /2. • all these 3 are greater than diam(S)/2, and is also greater than any diam/2 of any subset. • At least one of these 3 pairs must be an anchor pair. (of an optimal strip) • (Recall the definition of anchor pairs)

16.                  P, Q are points 3.2 Computing an anchor pair • Case 2 • else, S\(Dp Dq) =  • Let P =S  Dp and Q = S  Dq . • conv(P) and conv(Q) be their convex hulls, these two hulls do not intersect • Compute l1 and l2, the inner common tangent lines of conv(P) and conv(Q) • let p1 P, q1 Q be the points lying on l1. Respectively p2, q2 • let p3, p4 be a diametral pair in P, and q3,q4 be a diametral pair in Q • Return F = { (p,q), (p3,p4), (q3,q4), • (p,q1), (p,q2), (p,q3), (p,q4), • (q, p1), (q,p2), (q,p3), (q,p4)}

17.                  S12* S21* P, Q are points 3.2 Computing an anchor pair • Correctness of Case 2 • Suppose on the contrary that no pair of F is an anchor pair. • This implies p,q is neither an anchor pair of 1* nor of 2* , so S12* contains either p or q but not both. • WLOG, let p  S12* and q  S21*. Since d(p,qi), d(q,pi)  /2 (different disk), pi  S12 and qi  S21*, for i = 1,2,3,4 • S12* Q  . because otherwise S12* P, and (p3,p4) is an anchor pair, a contradiction. Similarly S21* P   • Therefore there exist point p’ S12* Q, and q’  S21* P  

18.                  x P, Q are points 3.2 Computing an anchor pair • Correctness of Case 2 • 1* |2* • p1~p4 | q1~q4 • p | q • p’ | q’ • x • let s be the intersection point of l1 and l2. Since strip q1, q2, and q’, it also contains the triangle q1q2s. Hence, p’ q1q2s • But p’ lies in the wedge. • therefore p1p2p’ intersects the segment q1q2(green) .Let x be a point on this segment. Since 1* contains p1p2p’, it also contains x. • But q1 q2 do not lie inside 1* ,so 1* separates q1 and q2. 2* separates p1 and p2.

19.                   3.3 (1+  )-Approximation z • Strategy • We have a 6-approximation cover. • Within this region, we try to “guess” the optimal *. We guess its direction(), displacement(by z) and its width w* (by w) • The result of our guess is ’ , an (1+ )-approximation of *, and totally contains *. For the points not covered by ’ , we run the known PTAS width algorithm to find the second strip covering them. 4d(p,q) 3w’

20.                      (ε/2)．(w~/6) - 2w~ w~/6 3.3 (1+  )-APX • Detail • R as shown. Z , , and W • Let  = C, where C is a constant to be specified later. • Z: grid of “positions” along the boundary, so that there are • grid points on each side of R •  : grid of “directions”. • W: grid of the value of “width”

21.                    3.3 (1+  )-APX • Existence of z’, ’, w’ • Assuming we know z’, ’, we can perform binary search on w’.(By computing the width of the “rest”) • Since we don’t know z’, ’, we try all possible pairs of them.

22.                    3.3 (1+  )-APX s • Proof of correctness • Can we find by guessing? • Also by Lemma 2.1, we know • So R is “big enough”. • The remaining question is whether the grids are “dense enough”? • 1.First we prove there exists a “good”  • 2. Then we prove there exists a good z.

23.                    3.3 (1+  )-APX s • Proof of Lemma 3.6 • 2. There is a good z (together with the previous ,) such that there exists a strip  such that S*   , width() (1+  /2)w*

24.                    3.3 (1+  )-APX s • Proof of Lemma 3.6 • 1.There is a good , such that there exists a strip  such that S*   , width() (1+  /4)w* • If ½ (w~) /d(p,q), assuming   2/3, • If ½ >(w~) /d(p,q) , which implies  <30°

25. Conclusion • We have an simple and efficient 2-line-center approximation algorithm. • k-line-center for fixed k, to higher dimensions • hyper-strips