Guillotine subdivisions approximate polygonal subdivisions: Part III { Faster polynomial-time approximation schemes for

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1 Guillotine subdivisions approxiate polygonal subdivisions: Part III { Faster polynoial-tie approxiation schees for geoetric network optiization Joseph S. B. Mitchell y April 19, 1997; Last revision: May 29, Introduction In this note, we show how a odication to our earlier results on guillotine subdivisions leads to an n O(1) tie (deterinistic) PTAS for Euclidean versions of various geoetric network optiization probles on a set of n points in the plane. This iproves the previous n O(1=) tie algoriths of Arora [1] and Mitchell [10]. Arora [2] has recently obtained even better bounds than our own: He obtains a randoized algorith with expected running tie O(n log O(1=) n). However, our new theore on approxiating guillotine subdivisions ay be interesting in its own right, and ay lead to yet further iproveents or generalizations. Our ethods are based on the concept of an \-guillotine subdivision", which were introduced by Mitchell [9, 10]. Roughly speaking, an \-guillotine subdivision" is a polygonal subdivision with the property that there exists a line (\cut"), whose intersection with the subdivision edges consists of a sall nuber (O()) of connected coponents, and the subdivisions on either side of the line are also -guillotine. The upper bound on the nuber of connected coponents allows one to apply dynaic prograing to optiize over -guillotine subdivisions, as there is a succinct specication of how subprobles interact across a cut. Key to our ethod is a theore showing that any polygonal subdivision can be converted into an appropriate -guillotine subdivision by adding a set of edges whose total length is sall: at ost c ties that of the original subdivision (where c = 1; p 2, depending on the etric). Key to our iproveent over previous results on approxiating with guillotine subdivisions is the notion of a \grid-rounded" -guillotine subdivision, in which each connected coponent is also required to contain one of a sall nuber of regularly spaced grid points. (These notions are ade precise in the next section.) Then, exactly as in [10], we use dynaic prograing to optiize over an appropriate class of -guillotine subdivisions, resulting in, for any xed, (1 + c )-approxiation algoriths that run in polynoial-tie (O(n O(1) )), for various network optiization probles. Related Work. There has been an abundance of work on the probles studied here, both on instances of the probles in graphs and on geoetric instances. We refer the reader to soe standard textbooks, such as [4, 6, 12]. For the particular proble of the TSP, there is a survey book edited by Lawler et al. [7], and for results on approxiation theory and algoriths, there is the recent book edited by Hochbau [5]. Manuscript available at y jsb@as.sunysb.edu, Dept. of Applied Matheatics and Statistics, State University of New York, Stony Brook, NY Supported in part by Hughes Research Laboratories and NSF Grant CCR

2 While the geoetric optiization probles considered here are known to be NP-hard, polynoialtie approxiation algoriths have been known previously that get within a constant factor of optial. Further, polynoial-tie approxiation schees were discovered last year, by Arora [1] and by Mitchell [9, 10]. This paper represents a continuation of our previous work on guillotine subdivisions ([9, 10, 11]), which in turn is based on the concept of \division trees" introduced by Blu, Chalasani, and Vepala [3, 11], and the guillotine rectangular subdivision ethods of Mata and Mitchell [8]. Here, we obtain substantially better bounds than before, and we generalize our previous results to apply to a uch broader class of proble instances that include those weighted graphs whose edge lengths correspond to shortest path lengths aong obstacles in the plane. 2 Grid-Rounded -Guillotine Subdivisions Denitions We follow ost of the notation of [9, 10]. We consider a polygonal subdivision (\planar straight-line graph") S that has n edges (and hence O(n) vertices and facets). Let E denote the union of the edge segents of S, and let V denote the vertices of S. We can assue (without loss of generality) that S is restricted to the unit square, B (i.e., E int(b)). Then, each facet (2-face) is a bounded polygon, possibly with holes. The length of S is the su of the lengths of the edges of S. Assue, without loss of generality, that no two vertices of S have a coon x- or y-coordinate. A closed, axis-aligned rectangle W is a window if W B. In the following denitions, we x attention on a given window, W. We let E W denote the subset of E consisting of the union of segents of E having at least one endpoint inside (or on the boundary of) W. The cobinatorial type (with respect to E) of a window W is an ordered listing, for each of the four sides of W, of the identities of the line segents of E W that intersect it. We say that W is inial (with respect to E) if there does not exist a W 0 W, strictly contained in W, having the sae cobinatorial type as W. By standard critical placeent arguents, we see that, since it has four degrees of freedo, a inial window is deterined by four contact pairs, dened by a vertex v 2 V in contact with a side of W or by a corner of W in contact with a segent of E W. (Thus, there are at ost O(n 4 ) inial windows.) For any window W containing at least one vertex of E, we let W denote a inial window, contained within W, having the sae cobinatorial type as W. (At least one such W ust exist.) A line ` is a cut for E (with respect to W ) if ` \ int(w ) 6= ;. The intersection, ` \ (E \ int(w )), of a cut ` with E \ int(w ) (the restriction of E to the window W ) consists of a discrete (possibly epty) set of subsegents of `. (Soe of these \segents" ay be single points, where ` crosses an edge.) The endpoints of these subsegents are called the endpoints along ` (with respect to W ). (The two points where ` crosses the boundary of W are not considered to be endpoints along `.) Let be the nuber of endpoints along `, and let the points be denoted by p 1 ; : : :; p, in order along `. For a positive integer, we dene the -span, (`), of ` (with respect to W ) as follows. If 2(? 1), then (`) = ;; otherwise, (`) is dened to be the (possibly zero-length) line segent, p p?+1, joining the th endpoint, p, with the th-fro-the-last endpoints, p?+1. Given a line segent = pq (p 6= q) and a positive integer M, consider the set of subsegents obtained by cutting pq into M equal-length segents; we dene the M-grid of = pq to be the set of M + 1 endpoints of these subsegents. (In particular, the M-grid contains the two points p and q.) A line ` is an (; M)-perfect cut with respect to W if (`) E, and each connected coponent of ` \ E contains an M-grid point of the 1-span, 1 (`). In particular, if 2(? 1), then ` is 2

3 trivially an (; M)-perfect cut (since (`) = ;). Siilarly, if = 2? 1, then ` is -perfect (since (`) is a single point). Otherwise, if ` is -perfect, and 2, then = 2. In the reainder of this paper, we x M = (? 1) and assue that 2. Finally, we say that S is a grid-rounded -guillotine subdivision with respect to window W if either (1) V \ int(w ) = ;; or (2) there exists an (; M)-perfect cut, `, with respect to a inial window, W W, such that S is grid-rounded -guillotine with respect to windows W \ H + and W \ H?, where H +, H? are the closed halfplanes induced by `. (Note that, since W is inial, necessarily windows W \ H + and W \ H? will each have a cobinatorial type dierent fro that of W.) We say that S is a grid-rounded -guillotine subdivision if S is grid-rounded -guillotine with respect to the unit square, B. The Approxiation Theore The theore below shows that grid-rounded -guillotine subdivisions can approxiate arbitrary subdivisions arbitrarily closely (as a function of ). Its proof directly follows that of [9, 10], with relatively inor changes to incorporate the concept of (; M)-perfect cuts, which allow us to strengthen the requireents fro that of -perfect cuts to include the eect of rounding to the M-grid of the 1-span. Theore 1 Let S be a polygonal subdivision, with edge set E, of length L. Then, for any positive integer, there exists a grid-rounded -guillotine subdivision, S G, of length at ost (1 + 2p 2 )L whose edge set, E G, contains E. Proof. We will convert S into a grid-rounded -guillotine subdivision S G by adding to E a new set of horizontal/vertical edges whose total length is at ost 2p 2 L. The construction is recursive: at each stage, we show that there exists a cut, `, with respect to the current window W (which initially is the box B), such that we can aord to add the following set of segents to E: (\red" segent) the -span, (`); and (\blue" segents) a line segent on ` connecting each of the endpoints of ` \ (E [ (`)) to a point of the M-grid of 1 (`). By construction, once we add these segents to E, ` becoes an (; M)-perfect cut with respect to W. The sense in which we can \aord" to add these segents is that we can charge o the lengths of the constructed segents to a portion of the length of the original edge set, E. First, note that if an (; M)-perfect cut (with respect to W ) exists, then we can siply use it, and proceed, recursively, on each side of the cut. Thus, we assue that no (; M)-perfect cut exists with respect to a given window, W. We say that a point p on a cut ` is -dark with respect to ` and W if, along `? \ int(w ), there are at least endpoints (strictly) on each side of p, where `? is the line through p and perpendicular to `. 1 We say that a subsegent of ` is -dark (with respect to W ) if all points of the segent are -dark with respect to ` and W. The iportant property of -dark points along ` is the following: Assue, without loss of generality, that ` is horizontal. Then, if all points on subsegent pq of ` are -dark, then we can charge the length of pq o to the bottos of the rst subsegents, E + E, of edges that lie above pq, and the tops of the rst subsegents, E? E, of edges that lie below pq (since we know that there are at least edges \blocking" pq fro the top/botto of W ). We charge pq's length half to E + (charging each of the levels of E + fro below, with 1 units of charge) and 2 1 We can think of the edges E as being \walls" that are not very eective at blocking light light can go through? 1 walls, but is stopped when it hits the th wall; then, p on a line ` is -dark if p is not illuinated when light is shone in fro the boundary of W, along the direction of `?. 3

4 half to E? (charging each of the levels of E? fro above, with 1 2 units of charge). We refer to this type of charge as the \red" charge. In addition to charging o the length of the -dark portion of `, in order to round to the M-grid of 1 (`), we are also going to charge o (1=)th of the 1-dark portion of `: If pq is 1-dark, then we charge (1=)th of pq's length, by charging half of this length (i.e., (1=2)th of the length of pq) o to the level of E that lies above pq, and half of it to the level of E that lies below pq. We refer to this type of charge as \blue" charge. The chargeable length of a cut ` is dened to be the length of the -dark portion of `, plus (1=) ties the length of the 1-dark portion of `. The cost of a cut, `, is dened to be the length of the segents we ust add to ake the cut (; M)-perfect. Thus, the cost of a cut ` is at ost the length, j (`)j, of the -span \red" segent, (`), plus the lengths of the \blue" segents on ` connecting each of the endpoints of ` \ (E [ (`)) to a point of the M-grid of 1 (`). Since there are at ost 2 endpoints of ` \ (E [ (`)), and two of these (the endpoints of 1 (`)) are already at M-grid points of 1 (`), the total nuber of blue segents is at ost 2? 2. Further, each blue segent is at ost one half of j1(`)j M, where j 1(`)j is the length of the 1-span of `. Thus, the overall cost of a cut ` is at ost j (`)j + (2? 2) 1 2 j 1(`)j M = j (`)j + 1 j 1(`)j; where we have used our choice o M = (? 1). We call a cut ` favorable if the chargeable length of ` \ W is at least as long as the cost of the cut. The lea below shows that a favorable cut always exists. For a favorable cut `, we add its -span to the edge set (charging o its length, as above), and recurse on each side of the cut, in the two new windows. After a portion of E has been charged red on one side, due to a cut `, it will be within levels of the boundary of the windows on either side of `, and, hence, within levels of the boundary of any future windows, found deeper in the recursion, that contain the portion. Thus, no portion of E will ever be charged red ore than once fro each side, in each of the two directions (horizontal/vertical), so no portion of E will ever pay ore than twice its total length, ties 1=, in red charge ( 1 fro each side, for each of the two directions). Siilarly, no portion of 2 E will ever be charged blue ore than once fro each side, in each of the two directions, and when it is charged blue, it is charged at the rate of only 1=2 per unit length (per side, per direction); thus, no portion of E will ever pay ore than its total length, ties 2=, in blue charge. So far, this charging schee gives rise to a total charge of at ost 4 L. This factor can be iproved slightly by noting that each side of an inclined segent of E ay be charged red (resp., blue) twice, once vertically and once horizontally, so the red (resp., blue) charge assigned to a segent is at ost 1 ties the su of the lengths of its x- and y-projections, i.e., at ost p 2 ties its length. This gives the overall charge of 2p 2L, as claied. It is also iportant to note that we are always charging portions of the original edges set E: The new edges added are never theselves charged, since they lie on window boundaries and cannot therefore serve to ake a portion of soe future cut -dark or 1-dark. (Note too that, in order for a cut ` to be favorable, but not (; M)-perfect, there ust be at least one vertex of V in each of the two open halfplanes induced by `; thus, the recursion ust terinate in a nite nuber of steps.) ut We now prove the lea that guarantees the existence of a favorable cut. The proof of the lea uses a particularly siple arguent, based on eleentary calculus (reversing the order of integration). It is based on the siilar lea that appears already in [9, 10], but we include its details here for copleteness: Lea 1 For any subdivision S, and any window W, there is a favorable cut. 4

5 Proof. We show that there ust be a favorable cut that is either horizontal or vertical. Let f(x) denote the cost of the vertical line, `x, through x; then, Then, A x = R 1 0 R () x f(x) = j (`x)j + 1 j 1(`x)j: f(x)dx is siply the area, A() x = R 1 of points of B that are -dark with respect to horizontal cuts, plus (1=) ties the area, 0 j (`x)jdx, of the (x-onotone) region A (1) x = R 1 j 0 1(`x)jdx, of the (x-onotone) region R (1) x of points of B that are 1-dark with respect to horizontal cuts. Siilarly, dene g(y) to be the cost of the horizontal line through y, and let A y = R 1 g(y)dy. 0 Assue, without loss of generality, that A x A y. We clai that there exists a horizontal favorable cut; i.e., we clai that there exists a horizontal cut, `, such that its chargeable length (i.e., length of its -dark portion, plus (1=) ties the length of its 1-dark portion) is at least as large as the cost of ` (j (`)j + 1 j 1(`)j). To see this, note that A x can be coputed by switching the order of integration, \slicing" the regions R x () and R (1) x horizontally, rather than vertically; i.e., A x = R 1 h(y)dy = R 1 h R 0 0 (y)dy h 0 1(y)dy, where h(y) is the chargeable length of the horizontal line through y, and h (i) (y) is the length of the intersection of R x (i) with a horizontal line through y. (i.e., h () (y) (resp., h (1) (y)) is the length of the -dark (resp., 1-dark) portion of the horizontal line through y.) Thus, since A x A y, we get that R 1 h(y)dy R 1 g(y)dy 0. Thus, it cannot 0 0 be that for all values of y 2 [0; 1], h(y) < g(y), so there exists a y = y for which h(y ) g(y ). The horizontal line through this y is a cut satisfying the clai of the lea. (If, instead, we had A x A y, then we would get a vertical cut satisfying the clai.) ut Algoriths The dynaic prograing algoriths of Mitchell [9, 10] carry over alost verbati to the new setting of grid-rounded -guillotine subdivisions. The ain dierence is in the coplexity analysis. A subproble in the dynaic prograing recursion is specied now by a rectangle (O(n 4 ) choices), and, on each of the four sides, a segent corresponding to the 1-span (O(n 2 ) choices per side), and a set of up to 2 M-grid points within each segent that specify the attachent points between this subproble and neighboring subprobles. (Depending on the proble instance, other inforation, of constant size for xed, is also specied for a subproble; see [10].) The key to the iproveent given in this paper is that there are now only? M 2 = O( 4 ) choices for these grid points on any one side, and this nuber is constant for xed. (Copare this to the O(n 2 ) choices of crossing points in [10].) Thus, there are overall O(n 12 ) subprobles. We then optiize over all O(n) choices of cuts, O(n 2 ) choices of 1-spans along the cut, and O( 4 ) choices of grid points on the cut. The overall coplexity of the dynaic prograing algorith is therefore O(n 15 ). By rounding the 1-span intervals up to be intervals of lengths that are power-of-two factors saller than the diensions of the window, it is not hard to iprove this coplexity to O(n 10 log 5 n), without signicantly changing the approxiation factor. Corollary 1 Given any xed positive integer, and any set of n points in the plane, there is an O(n O(1) ) algorith to copute a Steiner spanning tree (or Steiner k-mst), or a traveling salesperson tour, whose length is within a factor (1 + c ) of iniu, for constant c. Acknowledgeents I thank Avri Blu and Santosh Vepala for useful discussions on the subject of this paper. 5

6 References [1] S. Arora, Polynoial tie approxiation schees for Euclidean TSP and other geoetric probles, Manuscript, March 30, Appears in Proc. 37th Annu. IEEE Sypos. Found. Coput. Sci.(1996), pp. 2{12. [2] S. Arora, More ecient approxiation schees for Euclidean TSP and other geoetric probles, Unpublished anuscript, January 9, [3] A. Blu, P. Chalasani, and S. Vepala, A constant-factor approxiation for the k-mst proble in the plane, Proc. 27th Annu. ACM Sypos. Theory Coput. (1995), pp. 294{302. [4] T. H. Coren, C. E. Leiserson, and R. L. Rivest, Introduction to Algoriths, The MIT Press, Cabridge, Mass., [5] D. Hochbau, editor, Approxiation Probles for NP-Coplete Probles, PWS Publications, [6] E. Lawler, Cobinatorial Optiization: Networks and Matroids, Holt, Rinehart and Winston, New York, [7] E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shoys, editors, The Traveling Salesan Proble, Wiley, New York, NY, [8] C. Mata and J. S. B. Mitchell, Approxiation algoriths for geoetric tour and network design probles, Proc. 11th Annu. ACM Sypos. Coput. Geo. (1995), pp. 360{369. [9] J. S. B. Mitchell, Guillotine subdivisions approxiate polygonal subdivisions: A siple new ethod for the geoetric k-mst proble, Proc. 7th ACM-SIAM Sypos. Discrete Algoriths (1996), pp. 402{408. [10] J. S. B. Mitchell, Guillotine subdivisions approxiate polygonal subdivisions: Part II A siple polynoial-tie approxiation schee for geoetric k-mst, TSP, and related probles, SIAM J. Cop., to appear. [11] J. S. B. Mitchell, A. Blu, P. Chalasani, and S. Vepala, A constant-factor approxiation for the geoetric k-mst proble in the plane, SIAM J. Cop., to appear. [12] C. H. Papadiitriou and K. Steiglitz, Cobinatorial Optiization: Algoriths and Coplexity, Prentice Hall, Englewood Clis, NJ,

Related Work.. Over the last few decades, there has been a wealth of research

Related Work.. Over the last few decades, there has been a wealth of research GUILLOTINE SUBDIVISIONS APPROXIMATE POLYGONAL SUBDIVISIONS: A SIMPLE POLYNOMIAL-TIME APPROXIMATION SCHEME FOR GEOMETRIC TSP, K-MST, AND RELATED PROBLEMS JOSEPH S. B. MITCHELL Abstract. We show that any