2 Fgure : Arrangement wth cells of depth, (shaded), and 2; maxmum depth of 3 occurs at 8 vertces and two edges. (Some lnes are curved to t all ntersec

Transcription

1 Ecent algorthms for maxmum regresson depth Marc van Kreveld Dept. Comp. Sc. Utrecht Unv. Joseph S. B. Mtchell y Dept. Appled Math. & Stat. SUNY Stony Brook jsbm@ams.sunysb.edu Peter Rousseeuw Dept. Math. and Comp. Unv. Instellng Antwerpen rousse@ua.ua.ac.be Mcha Sharr z School Math. Scences Tel Avv Unv. sharr@math.tau.ac.l Jack Snoeynk x Dept. Comp. Sc. Unv. Brtsh Columba snoeynk@cs.ubc.ca Bettna Speckmann { Dept. Comp. Sc. Unv. Brtsh Columba speckmann@cs.ubc.ca Abstract We nvestgate algorthmc questons that arse n the statstcal problem of computng lnes or hyperplanes of maxmum regresson depth among a set of n ponts. We work prmarly wth a dual representaton and nd ponts of maxmum undrected depth n an arrangement of lnes or hyperplanes. An O(n d ) tme and space algorthm computes drected depth of all ponts n d dmensons. Propertes of undrected depth lead to an O(n log 2 n) tme and O(n) space algorthm for computng a pont of maxmum depth n two dmensons. We also gve approxmaton algorthms for hyperplane arrangements and degenerate lne arrangements. Introducton Motvated by the study of robust regresson n statstcs [3, 9, 23, 2, 2, 24, 22, 27, 26], Peter Rousseeuw posed the queston of computng maxmum regresson depth n hs nvted talk at the 4th ACM Symposum on Computatonal Geometry: Gven n ponts P, the Supported n part by ESPRIT IV LTR Project No (GALIA). y Research largely conducted whle the author was a Fulbrght Research Scholar at Tel Avv Unversty. The author s partally supported by NSF (CCR-95492, CCR ), Boeng, Brdgeport Machnes, Sanda, Seagull Technology, and Sun Mcrosystems. z Supported by NSF Grants CCR and CCR , by grants from the U.S.-Israel Bnatonal Scence Foundaton, the G.I.F., the German-Israel Foundaton for Scentc Research and Development, and the ESPRIT IV LTR project No (CGAL), and by the Hermann Mnkowsk{MINERVA Center for Geometry at Tel Avv Unversty. x Supported n part by grants from NSERC, the Kllam Foundaton, and CIES. { Supported n part by a UBC Unv. Graduate Fellowshp. Rousseeuw also posed a combnatoral queston, recently resolved by Amenta et al. [], who show that for any set of n ponts n R d, there exsts a hyperplane wth regresson depth at least dn=(d )e. regresson depth of a lne s the mnmum number of ponts that must be removed from P to allow the lne to rotate to vertcal about a pvot pont on the lne to a vertcal poston wthout ever contanng a pont of P. (Ths denton s gven more generally n the next secton.) A lne (or hyperplane) of maxmum depth has statstcal propertes that are desrable as a robust regresson estmator [28]. The expermental nvestgaton of these propertes has been hampered by the necency of the straghtforward algorthms for computng maxmum depth. These requred (n 3 ) tme n the plane [9] and (n 2d log n) tme n dmensons d 3 [2, 22]. In the next secton, we dene an equvalent dual problem, computng undrected depth n an arrangement of lnes or hyperplanes. The propertes of undrected depth wll lead to an O(n d ) algorthm for computng regresson depth for all dmensons. In Secton 3, we focus on arrangements n the plane where addtonal propertes gve us our man result: an algorthm to compute one lne of maxmum regresson depth n O(n log 2 n) tme. In Secton 4, we study the combnatoral complexty of the set of all lnes (or hyperplanes) wth maxmum regresson depth and ts relatonshp to k-sets. In Secton 5, we comment on other algorthms for computng or approxmatng depth. 2 Dualty and undrected depth n arrangements Although regresson depth s dened for a lne or hyperplane among n ponts, t s easer to work wth a dualty transformaton that maps ponts to hyperplanes and vce versa. We use the dualty from Edelsbrunner's book [8]: an nverson about the unt parabolod x d = x 2 x 2 2 x 2 d that maps a pont (p ; p 2 ; : : : ; p d ) to the hyperplane x d = 2p x 2p 2 x 2 2p d x d p d and maps a hyperplane x d = a x a 2 x 2 a d x d b to the pont (a =2; a 2 =2; ; a d =2; b). Ths dualty preserves pont/lne ncdence and above/below relaton-

2 2 Fgure : Arrangement wth cells of depth, (shaded), and 2; maxmum depth of 3 occurs at 8 vertces and two edges. (Some lnes are curved to t all ntersectons on the page) shps. Note that the dualty mappng wll nether accept nor produce vertcal hyperplanes, whch have equatons that do not nvolve the varable x d. All rotatons of a hyperplane h can be generated as follows. Choose a set of d ponts Q that dene h; that s, each pont n Q satses the hyperplane equaton of h and together they determne the coecents of ths plane equaton. (Equvalently, h s the ane hull of Q.) Move one of the ponts q 2 Q by ncreasng ts last coordnate toward nnty. If the ponts Q are stll taken to dene h, then h rotates toward the vertcal about the (d )-at dened by ponts of Q n fq g. The dual of a rotaton s easy to nterpret. The ponts of Q map to hyperplanes through a common pont h D. Hyperplane q D moves parallel to tself up the x d axs, so the pont common to all hyperplanes moves from h D toward nnty along a ray that s contaned n the duals of the statonary ponts. Gven n prmal ponts P, the number that must be removed to allow a partcular rotaton are the number that are passed over by the rotaton, plus the number that are on the nal vertcal plane (whch our rotaton never reaches). Ths number can be counted n the dual as the number of hyperplanes dual to ponts n P that are crossed by the ray correspondng to the rotaton, plus the number of hyperplanes parallel to the ray. Therefore, for an arrangement of n hyperplanes A, we dene the undrected depth, or just depth, of a pont p to be the mnmum number of hyperplanes ntersected by some ray from p, countng parallel hyperplanes as ntersectng at nnty. Hyperplanes contanng p are counted for all rays. For the rest of ths paper we focus on computng depth of a pont n an arrangement of n lnes or hyperplanes. Snce all ponts n the same cell C of an arrangement have the same depth, we can use the notaton depth(p) or depth(c) for the value of undrected depth. (In ths paper, unless otherwse stated, we use the word cell to refer to a full-dmensonal cell n an arrangement.) Fgure shows a two dmensonal example wth labels for some cells of depth,, and 2; the maxmum depth of 3 occurs at 8 vertces and two edges. The drectons for a cell C are the drectons of rays that ntersect depth(c) lnes or hyperplanes of the arrangement. We can call such rays wtnesses that the cell has a certan depth. We next observe three smple lemmas about depth by translatng wtness rays n the arrangement of hyperplanes n R d : ) depth of lower dmensonal features n the arrangement can be determned from depth of d-dmensonal cells, 2) drectons are dsjont for adjacent cells of the same depth, and 3) drectons determnng depth are nherted from adjacent cells of lower depth. Lemma In an arrangement of hyperplanes, let p be a pont on k hyperplanes, and let be the mnmum of the depths of cells whose closure contans p. Then depth(p) = k. Proof: Frst we can observe that depth(p) k: a ray that starts n the cell and crosses hyperplanes can be translated to start at p at the cost of crossng all hyperplanes through p that t dd not cross before. Second, by takng a ray, not contaned n a hyperplane ncdent on p, that wtnesses depth(p) and translatng ts startng pont nntesmally nto the rst cell entered by the ray, we can observe that there s an adjacent cell wth depth depth(p) k, whch s therefore the mnmum cell depth. Lemma 2 In an arrangement of hyperplanes, let h be a hyperplane that separates a cell B of depth from a cell A of depth at least. No wtness ray for B crosses h. Proof: Let be a ray from B that crosses h, and let be a translaton of ths ray that begns n A. Translated ray ntersects the same hyperplanes as, except for h. But snce ntersects at least hyperplanes, ntersects at least hyperplanes and s not a wtness ray for B. Lemma 3 (Inhertance lemma) The drectons for a cell of depth are the unon of the drectons for the adjacent cells of depth. Proof: We prove that the drectons for a cell wth depth(a) = contan the unon. For any adjacent

3 cell B of depth, let ray be a wtness for B. By Lemma 2, translatng to start n A adds at most one (and, therefore, exactly one) ntersecton, and provdes a wtness that A nherts the drecton of. To prove the other ncluson, take a wtness that depth(a) =. We can choose the start pont of so that does not pass through any vertex of the arrangement. By clppng to start n an adjacent cell B, we obtan a wtness that depth(b). But the depth of B cannot be less than, snce depth(a) = and we already know that A nherts all drectons for B wth only one more ntersecton. Thus, the drectons for A are contaned n the unon. As a corollary of Lemma 3, the depth of all ponts wth respect to a set of hyperplanes can be computed by constructng the arrangement of hyperplanes [9, ] and labelng cells n a breadth-rst search. The unbounded cells are labeled wth ther depth zero. Then, for =, 2,..., all cells wth label cause ther adjacent, unlabeled cells to be labeled. Fnally, lower-dmensonal cells can be labeled accordng to Lemma. Corollary 4 For n hyperplanes n R d, the depth of each cell can be computed n O(n d ) tme by buldng the arrangement and traversng the graph of adjacent cells. Lemma 5 (Wedge lemma) Let p be a pont, possbly on one lne of L, and let u and v be drectons of rays from p that each cross at most other lnes. No cell ntersectng the convex wedge (cone) dened by these rays from p has depth greater than. Proof: Consder the lnes that ntersect the unon of rays from p n drectons u and v. There are at most 2 ntersectons, f we count the lne contanng p only once. If we translate ths unon wthn the wedge, although we may lose ntersectons wth lnes that ntersect both rays, we wll not gan ntersectons. Thus, f the apex s nsde a cell of the lne arrangement, one of the translated rays wll wtness that the depth s at most. The wedge lemma can be helpful for dentfyng maxmum depth cells, as n the followng corollary. Corollary 6 Suppose that a cell C has three drectons u, v, and w that span the plane by postve lnear combnatons and wtness the value of depth(c). Then C s a deepest cell. Proof: Apply the wedge lemma to the three wedges dened by pars of drectons. 3 An ecent algorthm for maxmum depth n the plane Undrected depth n two dmensons satses some addtonal propertes that allow an ecent algorthm to compute a 2-dmensonal cell of maxmum depth. Suppose that we are gven a set L of n lnes n the plane, whch we may assume are not vertcal. For the moment, let us also assume that they are n general poston we wll relax ths assumpton n Subsecton 5.2. Our goal s to nd, among all the ponts of the plane that do not le on lnes of L, a pont p whose depth s maxmum. Note that vertces of the arrangement A(L) may attan greater depth than p we return to these n Subsecton 4.. We wll use a bnary search on x-coordnates of vertces of the arrangement A(L), wth a test for whch sde of a vertcal lne contans a maxmum depth cell. Subsecton 3. establshes propertes that allow a sdedness test; Subsecton 3.2 descrbes a tournament data structure needed to mplement the sdedness test. 3. A sdedness test In the plane, we use two concepts to determne whch sde of a vertcal test lne can have cells of maxmum depth: a \wedge lemma" and the noton of \top drectons." Fgure 2: Drectons (shaded) and top drectons We can order the wtness rays for a cell C by ncreasng slope to the rght of C and decreasng slope to the left. We call the two extreme drectons for wtness rays the top drectons for the cell C. There wll be a sngle top drecton when one sde of the lne has no wtness rays, or when the ray upward s a wtness. Fgure 2 llustrates a cell wth two top drectons. If we assume that we have the top drectons for each cell that ntersects the vertcal lne `, then we can use the wedge lemma to determne whether a maxmum depth cell occurs to the left or rght of `. We gve an algorthmc proof of the followng lemma, snce t becomes part of our procedure for computng maxmum depth. Lemma 7 Gven a vertcal test lne ` that does not pass through any vertex n an arrangement of n lnes n the plane, and gven a top drecton for each cell ntersected by `, one can determne one sde of ` that ntersects a maxmum depth cell.

4 Proof: Let be the maxmum depth of the cells ntersected by `. We slde a pont p up the lne `, stoppng as soon as we show that one sde of the lne ` cannot contan a cell of depth greater than. We mantan a top drecton u wth the nvarant that the wedge below the ray from p n drecton u ntersects no cells of depth greater than. See Fgure 3. For an ntal pont p n the lowest cell, we can choose one of the top drectons for the cell and apply the wedge lemma to establsh the nvarant. Move the pont p up the lne `. Whle p remans n a sngle cell the top drecton does not change; applyng the wedge lemma n that cell establshes the nvarant for the enlarged wedge. When p crosses a lne of the arrangement, we may obtan a new top drecton v. Let W denote the wedge wth apex p and drectons u and v. Applyng the wedge lemma to W, we see that no cell of depth greater than les n W. Fgure 3: Invarant wedge Now, how does W le wth respect to the prevous nvarant wedge? If the new top drecton s on the same sde of `, then ether t s above the old, and W adds to the nvarant wedge, or below the old and W removes from the nvarant wedge. If the new top drecton s on the opposte sde, then ether W contans the downward drecton and s thus the new nvarant wedge, or W contans the vertcally upward drecton and we are done. Snce the upward drecton s the top drecton for the uppermost cell, the algorthm must termnate. As an asde, one can use a smlar argument along a curved path to show that the maxmum depth cells are connected. Corollary 8 In an arrangement of lnes n the plane, the closure of the cells of depth at least s smply connected. Proof: Consder a connected component of the unon of the closures of cells of depth, and draw a path n the neghborng cells (whch have depths and 2). Applyng the wedge lemma as one traverses the path wll show that no cell of depth les outsde the path. 3.2 Computng top drectons In ths secton we descrbe a data structure that can determne the top drectons for a sequence of adjacent cells n an arrangement of n lnes usng logarthmc tme per cell, after O(n log n) preprocessng. Preprocessng takes lnear tme f the lnes of the arrangement are sorted by slope. Let us contnue to assume that no lne s vertcal and let l ; l 2 ; : : : ; l n be the lnes ordered by ncreasng slope. We can dentfy a cell C n the arrangement wth ts bt strng b(c) = b : : : b n, where bt b = f lne l s above the cell C, and b = otherwse. Notce that the number of bts n b(c) s exactly the number of lnes crossed by a ray from C n the downward drecton. Consder rotatng the ray from C counter-clockwse. The set of lnes crossed by does not change untl ray reaches the drecton of the lne l then bt b s complemented, snce wll begn to ntersect or cease to ntersect l. We therefore consder an extended bt strng B(C) = b(c)b(c)b(c), whch s the bt strng for C, followed by ts complement, and the bt strng agan. The extended strng B(C) has 2n subsequences of length n; we drop the last, snce t equals the rst. The counts of the number of bts n these 2n subsequences gve the number of lnes ntersected by a ray from C to the unbounded cells of the arrangement n the correspondng 2n drectons. The mnmum of these counts s the value depth(c). Wth a relatvely-smple tournament, we can mantan the mnmum of the counts and nformaton about drectons n whch the mnmum occurs. We use a statc, balanced, bnary tree that stores n the leaves the sequence of 2n counts. The leftmost leaf stores the count for the upward drecton. Each nternal node stores three ntegers: the sze of ts subtree, the mnmum count of the leaves n ts subtree, and a correcton value. The correcton value s a postve or negatve nteger that should be added to the counts of all leaves n the subtree. It s processed as follows: before the count of a node s nspected, the correcton value s added to the count and to the correcton values of the two chldren nodes, then set to zero. Snce tree operatons wll process nodes from root to leaf, the value of nspected nodes wll always be properly corrected. The tree supports two operatons: a query and an update. The query asks for the leaf wth mnmum count; n case of equal counts we want both the leftmost leaf and the rghtmost leaf wth these counts these gve the top drectons for the cell C. Snce each nternal node stores the mnmum count n ts subtree, such a query s easy to perform n O(log n) tme by followng two paths n the tree. The update operaton corresponds to movng from a

5 cell C to a cell C by crossng some lne l. Ths means that the bt strng of b(c ) ders from b(c) n the -th bt. In the extended strng B(C ), three bts change to ther complements. Snce the 2n counts for a cell are obtaned by addng n consecutve bts, every count changes f b changes from to, then the rst counts ncrease by one, the next n counts decrease by one, and the nal n counts ncrease by one. Thus, we should not update the counts n the leaves explctly, snce ths would take lnear tme; nstead we update correcton values. We follow the two paths n the tree to the -th leaf and the ( n)-th leaf usng the sze-of-subtree ntegers stored at the nternal nodes. The paths partton the tree nto three parts. For all hghest nodes left of the search path to the -th leaf we ncrement the correcton value (or decrement, f b changes from to ). Ths s done too for the hghest nodes rght of the search path to the (n)-th leaf. For the hghest nodes between the search paths we decrement (or ncrement) the correcton value. Snce there can be at most O(log n) hghest nodes left (or rght) of any path n the tree, only O(log n) correcton values are updated. Because the structure of the tree s statc, we mplemented t by ndexng nto a xed array, and subtree szes were calculated rather than stored. Lemma 9 Usng the data structure descrbed above, one can determne the top drectons for a sequence of adjacent cells n an arrangement of n lnes usng logarthmc tme per cell, after O(n log n) preprocessng. 3.3 Bnary search for a maxmum depth cell It s probably no surprse that we use the sdedness test n a bnary search on x-coordnates of vertces of the arrangement A(L). A Java prototype can be seen at maxdepth. Standard results on slope selecton [2, 5, 4, 6] allow us to consder the porton of the arrangment A(L) that les between two vertcal lnes, and to generate the vertex of medan x coordnate n O(n log n) tme. We based our mplementaton on a randomzed algorthm of Dllencourt, Mount, and Netanyahu [7]. At a vertcal test lne ` through ths medan vertex, we sort the ntersectons wth the lnes of L and use the tournament descrbed n Subsecton 3.2 to compute the depth of each pont on the test lne ` and the top drectons n O(n log n) tme. Lemma 7 then allows us to dscard one sde of the lne `, and to contnue the search on the other sde. The search termnates when there are no ntersecton ponts remanng, whch occurs after at most log(n 2 ) = 2 log n steps. Thus, we clam the followng result. Theorem A cell of maxmum undrected depth n an arrangement of n lnes can be computed n O(n log 2 n) tme and O(n) space. 4 The structure of depth n the plane and hgher dmensons Although our bnary search dentes a deepest cell, we know from Lemma that the maxmum depth n an arrangement wll always occur at a vertex. In statstcal analyss, we may also wsh to know the set of all lnes wth maxmum regresson depth, whch corresponds to the set of all ponts at maxmum depth. In ths secton, we characterze the set of ponts at maxmum depth n the plane. We also establsh relatonshps wth k-sets n all dmensons. We defer most of the computatonal problems to Secton Fndng a deepest vertex n a non-degenerate arrangement Fgure showed an example n whch edges and solated vertces attan the maxmum depth, but no cell does. Once we have found a pont n a cell of maxmum depth, we stll must determne whether there s a vertex wth greater depth. For arrangements of lnes n general poston, ths s not dcult to do. When the maxmum depth of a cell s, then the maxmum depth of a vertex s,, or 2, as llustrated n Fgure 4. These cases can be detected by postprocessng after computng the maxmum depth cell. When the maxmum depth vertex v has depth 2 n a non-degenerate arrangement, then the two lnes crossng at v form four quadrants contanng ncdent cells at depth. Lemma 2 says that the drectons for these cells are contaned n the respectve quadrants. Durng the bnary search, test lnes to the rght of the vertex wll elmnate ther rght sde and those to the left wll elmnate ther left sde. Thus, there s at most one such vertex and the bnary search wll nd t. When the maxmum depth vertex has depth the same as the maxmum depth cell then each such vertex s ncdent on one cell of depth, two of, and one of 2. Snce cells are convex and the maxmum depth s connected, there s only one cell that attans the maxmum. Once we have computed a maxmum depth cell, therefore, we can construct the cell as the ntersecton of halfplanes contanng the cell that are dened by lnes of the arrangement. Ths s equvalent to convex hull computaton, and takes O(n log n) tme. Then we can use the tournament to check the depth of all vertces, also n O(n log n) tme.

6 A: X: : 2 2: I: V: Fgure 4: Cases for maxmum vertex depth 4.2 Deepest ponts n non-degenerate arrangements It s natural to ask for all ponts of maxmum undrected depth, whch correspond to all lnes that have maxmum regresson depth. Ths appears to be a more dcult queston. We can characterze the maxmum depth ponts as follows: Lemma If the maxmum cell depth s, then the maxmum depth ponts form ether. a sngle pont of depth 2, 2. a convex polygon whose vertces, edges, and nteror all have depth, or 3. a sngle chan of sze O(n) and some solated ponts of depth. Proof: The rst and second cases are dscussed n the prevous Subsecton; we establsh the structure of the thrd by consderng the conguratons of Fgure 4 that gve vertces and edges of depth. It s clear that conguratons A and X gve solated vertces of depth, that I gves the end of a chan, and that V gves the mddle of a chan. We need to show that there s at most one chan. A: I: X: V: Fgure 5: Wedge lemma appled If we consder the wtness drectons for cells of depth n these cases, and apply the wedge lemma, we can make the followng observatons: In I there s a wedge dened by drectons for the two cells of depth that ncludes a ray on the lne separatng these two cells. In A, there are two such wedges. The wedge lemma mples that cells n these wedges are of depth at most. Ths mmedately mples that all edges n the wedge have depth at most. In fact, vertces n the wedge also have depth at most, snce the only way for a vertex to have depth would be to have four ncdent cells of depth, but then would be the maxmum depth cell n the arrangement. In X there s a wedge that contans one of the two ncdent cells of depth. We can extend the wedge lemma to observe that cells and edges n ths wedge have depth at most. (The argument for edges s that there are at most 2 lnes that can be ntersected by translates of the two rays of the wedge and there s a bonus of for startng the rays on the edge. Therefore, one of the rays ntersects at most =2 lnes, showng that the depth of the edge s at most.) The full paper uses these observatons n an nducton proof that shows that there s a sngle chan. Unfortunately, there are close connectons between ponts wth gven undrected depth and k-sets that mply superlnear bounds on the number of solated ponts n the thrd case. 4.3 Connectons wth k-sets In ths secton we observe the connectons between the complexty of ponts wth gven undrected depth and the concept of k-sets n a conguraton of ponts. There has been consderable attenton n computatonal geometry devoted to k-sets, and the dual concept of k- levels n an arrangement of lnes or hyperplanes; see, e.g., [4, 6, 8, 8, 25]. The k-level of an arrangement A for a partcular drecton conssts of all ponts p such that a ray from p n drecton ntersects exactly k hyperplanes. (Usually, hyperplanes contanng p are not counted.) In the

7 dual, the k ntersected hyperplanes become a k-set: k ponts that can be separated from the conguraton by an open halfspace bounded by a hyperplane, namely p D. Note that pont p has undrected depth at most k (assumng that p does not le on any hyperplane) and that the hyperplane p D has regresson depth at most k as shown by rotaton about any lne outsde the convex hull of the dual ponts. The combnatoral complexty of k-levels and algorthms to compute them have been ntensvely studed, although many open problems reman. In a smlar manner, we dene the k-envelope n an arrangement A to be the unon of all ponts wth undrected depth k. Examples can be seen back n Fgure. There have been some results on -envelopes of lnes [, 5], but we know of no deeper results. We show that the worst-case combnatoral complexty of k-envelopes s asymptotcally the same as the worst-case complexty of a k-level n any xed dmenson. The exact asymptotc worst-case complexty of a k-level s stll unknown [6, 8]. In the plane, t known to be between (n log n) and O(n 4=3 ). We begn wth the lower bounds that show that the complexty of a k-envelope s at least as great as that of a k-level. Lemma 2 The worst-case complexty of the k- envelope of an arrangement of n hyperplanes s at least as large as the worst-case complexty of a k-level n an arrangement of n dk hyperplanes, for k < n=d. Proof: Consder the k-level n an arrangement of n kd > hyperplanes, none of whch are parallel to the x d axs. There s a unque unbounded cell n ths arrangement that contans the vertcally-downward drecton,. In ths cell we can construct a smplex wth one horzontal face such that all rays through the horzontal face from the opposte vertex reman nsde the cell. Scale and translate untl contans the full complexty of the k-level. Then add to the arrangement k perturbed copes of the hyperplanes through each of the d non-horzontal faces of. For ponts on the k-level, rays n the downward drecton ntersect k old hyperplanes and none of the new ones. Rays n drectons outsde the cell of the downward drecton ntersect at least k of the new hyperplanes. Thus, the k-level appears on the k- envelope. The constructon above says nothng about the complexty of the ponts wth maxmum depth of k n=d. Wth another constructon, llustrated n Fgure 6, we can show that the complexty of the ponts wth maxmum depth n the plane s lower bounded by the complexty of a medan level. Fgure 6: Medan level to maxmum depth Lemma 3 The worst-case complexty of the set of ponts wth maxmum undrected depth n an arrangement of n lnes s at least as large as the worst-case complexty of the medan level n an arrangement of n=3 lnes. Proof: Consder any arrangment wth 2m lnes, none of whch s parallel to the vertcal y axs, and enclose t n a trangle wth a vertcal longest sde, and two other nearly-vertcal sdes. Add 2m lnes through the longest sde and m through each of the others, then perturb the new lnes to be n general poston. Unbounded cells n the orgnal arrangement now have undrected depth at most 2m by crossng only new lnes. Bounded cells n the orgnal arrangement also have undrected depth at most 2m by crossng m old lnes and m new wth a near-vertcal ray. The former medan level has undrected depth of exactly 2m, and thus contrbutes ponts of maxmum depth. The proofs of complexty for k-levels can be adapted to prove upper bounds for k-envelopes. For example, we can prove the followng n the plane. Lemma 4 In the plane, the worst-case complexty of the k-envelope s at most O(n 4=3 ). Proof: We can adapt Dey's proof [6] for the complexty of a k-level. Detals are gven n the full paper. 4.4 The wedge lemma cannot extend to R 3 The soluton for the planar case was based on the wedge lemma, whch allowed us to argue that certan regons of the plane could not contan a cell of maxmum depth. When thnkng about the extenson to 3-dmensons, one would rst try to generalze the wedge lemma: that for a pont p whose depth s wtnessed by three vectors ~u, ~v and ~w, the cone dened by ~u, ~v and ~w does not

8 z = 4 y = (2; ; ) (2; 2; 2) 2 2 Fgure 7: Cross-secton of nne planes wth the plane x = 2, wth depth values for drectons n ths plane. Plusses denote a possble ncrement by one due to perturbaton. contan a cell of depth greater than. The followng constructon shows that ths s not true. Let pont p be the orgn of the coordnate system. We construct an arrangement of fteen planes such that the postve x-axs, the postve y-axs, and the postve z-axs each wtness that depth(p) = 2, the pont q = (2; 2; 2) wll have depth(q) = 3. There are sx planes that ntersect the postve octant: Planes x = 4, y = 4, and z = 4 are parallel to the coordnate planes. Planes 5 = x y 5z, 5 = x 5y z, and 5 = 5x y z pass through a common pont (5=3; 5=3; 5=3), and each ntersect one of the postve coordnate axes. Note that the rst ntersects the z axs at (; ; ) and the x and y axes at ( 5; ; ) and (; 5; ). Note that f the coordnate frame s translated from the orgn to q = (2; 2; 2), then each postve axs ntersects three of these sx planes, whch already shows that the argument used to prove the 2-dmensonal wedge lemma does not hold n the 3-dmensonal case. The remanng nne planes are chosen to make sure that only drectons n or near the postve octant can gve depth counts below three for all cells n the postve octant. They are perturbed versons of x =, x = 2, x = 3 and smlarly, y; z = ; 2; 3. The perturbatons are such that none of the planes ntersect the postve octant. The common ntersecton of the half-spaces bounded by these planes and contanng the orgn can be seen as the perturbed postve octant. These make sure that for any pont n the postve octant, and any drecton outsde the postve octant by a small angle, the depth count n that drecton wll be at least three. Let us rst consder the number of planes ntersected by rays from q nsde the postve quadrant of the plane x = 2 (whch tself s not one of the sx planes). By constructng a gure of ths cross-secton, one can easly verfy that all of these drectons gve rays ntersectng three or four planes, see Fgure 7. The perturbaton of the planes does not nuence the depth of the cell contanng q. Fnally, when we consder drectons from q where x; y; z-contrbutons are all strctly postve, we smply observe that any such drecton ntersects each one of x = 4, y = 4, and z = 4. Thus, the depth(q) = 3. Ths example also shows that the closures of cells of a partcular depth need not be connected. The proof of the wedge lemma does mply that the postve quadrants of the three coordnate planes do not ntersect cells of depth greater than two f we translate a par of postve coordnate axes wthn the quadrant that they dene, we do not gan new ntersectons. Thus, Corollary 8 cannot be extended beyond the plane. 5 Further algorthms for depth n the plane and three dmensons In ths secton we gve some further results on algorthms for computng depth n the plane and n hgher dmensons. In the plane, we show how to compute

9 the maxmum depth cells based on the characterzaton of Subsecton 4.2, then brey dscuss degenerate arrangments. In hgher dmensons, we use cuttngs to derve results on approxmatng depth and to obtan some space/tme tradeos for computng exact depth. 5. Output-senstve constructon for maxmum depth n non-degenerate planar arrangements The Overmars/van Leeuwen [7] algorthm for dynamc convex hulls, when appled to the duals of the lnes, allows us to mantan a descrpton of the current cell as we walk from cell to cell n the arrangement. Wth the characterzaton of the ponts of maxmum depth from Subsecton 4.2, ths allows us to compute a descrpton of the maxmum depth ponts n an output-senstve manner. Theorem 5 The set of all ponts at maxmum depth n an arrangement of lnes n general poston can be computed at the cost of O(log 2 n) per feature. Proof: Sketch: The key observaton s that there s only one canddate for the next solated pont n the cell contaned n the wedge of a X conguraton namely, the pont wth tangent parallel to the tangent of the wedge. Thus, solated ponts occur n strngs of X conguratons that end wth a A con- guraton, and we can use a bnary search n the Overmars/van Leeuwen data structure [7] to nd the next canddate and enter the next cell. 5.2 Depth of vertces n degenerate arrangements Fndng a deepest vertex n a degenerate arrangement already appears to be dcult n the plane. We can ef- cently nd a vertex whose depth s wthn an addtve term of o(n) from the maxmum depth vertex. Lemma 6 A pont whose depth s at most n= log n less than the maxmum can be found n O(n log 2 n) tme. Proof: Frst, compute the cell of maxmum depth n the arrangement. Then, usng an algorthm of Gubas et al. [2], nd all vertces V that are contaned n n= log n lnes n O(n log 2 n) tme. There are at most O(log 2 n) of these vertces, and ther depth can be tested n O(n) tme each once the lnes are sorted by slope. Ether a vertex of V has maxmum depth, or, by Lemma, a pont n the cell of maxmum depth s less than n= log n from the true maxmum value. One heurstc that nvolves less programmng s to symbolcally perturb the lnes of the arrangement to smulate general poston, and compute the cell of maxmum depth. In the orgnal arrangement ths cell may correspond to a vertex, n whch case we evaluate the depth of ths vertex, or to a cell, n whch case we construct the cell and evaluate the depth of all of ts vertces. From the wedge lemma t can be seen that the actual maxmum depth wll be at most double the computed depth. 5.3 Approxmatng depth Theorem 7 For any xed >, one can compute the ( )-approxmate depth, ~ (wth ~ ( )), of an arrangement of n lnes n the plane, n tme O( n log n) usng O(n) space. Proof: We compute a =r-cuttng (n tme O(nr)) of the set of lnes, where r = =. (A (=r)-cuttng of H s a partton of < d nto dsjont regons, each of whch ntersects at most n=r hyperplanes of H. A (=r)-cuttng of optmal sze O(r d ) can be found n determnstc tme O(nr d ) [3].) The cuttng has O(r 2 ) edges; we can assume that they are formed by an arrangement of O(r) lnes. We then use our \steppng" algorthm to compute the depth along each of these O(r) lnes. By the denton of cuttngs, the total (addtve) error s at most n=r. Ths requres tme O(n log n) per lne. 5.4 Reduced space complexty We can reduce the space complexty n < 3 by computng depths of cells along each plane of the arrangement. Note that the wtness rays are not conned to the gven plane durng the computaton we show that they can stll be determned n O(n 2 ) tme. Performng ths computaton for each of the n planes gves the followng result. Theorem 8 In tme O(n 3 ), usng O(n 2 ) space, one can compute the depth of all cells n an arrangement of n planes n < 3. Acknowledgments J. Mtchell and M. Sharr thank E. Arkn and S. Har- Peled for several helpful dscussons and suggestons. References [] N. Amenta, M. Bern, D. Eppsten, and S.-H. Teng. Regresson depth and center ponts. Manuscrpt, 998. [2] H. Bronnmann and B. Chazelle. Optmal slope selecton va cuttngs. Comput. Geom. Theory Appl., ():23{29, 998.

10 [3] B. Chazelle. Cuttng hyperplanes for dvde-andconquer. Dscrete Comput. Geom., 9(2):45{58, 993. [4] B. Chazelle and F. P. Preparata. Halfspace range search: An algorthmc applcaton of k-sets. Dscrete Comput. Geom., :83{93, 986. [5] R. Cole, J. Salowe, W. Steger, and E. Szemered. An optmal-tme algorthm for slope selecton. SIAM J. Comput., 8(4):792{8, 989. [6] T. K. Dey. Improved bounds on planar k-sets and related problems. Dscrete Comput. Geom., 9:373{382, 998. [7] M. B. Dllencourt, D. M. Mount, and N. S. Netanyahu. A randomzed algorthm for slope selecton. Internat. J. Comput. Geom. Appl., 2:{27, 992. [8] H. Edelsbrunner. Algorthms n Combnatoral Geometry, volume of EATCS Monographs on Theoretcal Computer Scence. Sprnger-Verlag, Hedelberg, West Germany, 987. [9] H. Edelsbrunner, J. O'Rourke, and R. Sedel. Constructng arrangements of lnes and hyperplanes wth applcatons. SIAM J. Comput., 5:34{363, 986. [] H. Edelsbrunner, R. Sedel, and M. Sharr. On the zone theorem for hyperplane arrangements. SIAM J. Comput., 22(2):48{429, 993. [] D. Eu, E. Guevremont, and G. T. Toussant. On envelopes of arrangements of lnes. J. Algorthms, 2:{48, 996. [2] L. J. Gubas, M. H. Overmars, and J.-M. Robert. The exact ttng problem for ponts. Comput. Geom. Theory Appl., 6:25{23, 996. [3] M. Hubert and P. J. Rousseeuw. The catlne for deep regresson. J. Multvar. Analyss, 66:27-296, 998. [4] M. J. Katz and M. Sharr. Optmal slope selecton va expanders. Inform. Process. Lett., 47:5{22, 993. [5] M. Kel. A smple algorthm for determnng the envelope of a set of lnes. Inform. Process. Lett., 39:2{24, 99. [6] J. Matousek. Randomzed optmal algorthm for slope selecton. Inform. Process. Lett., 39:83{87, 99. [7] M. H. Overmars and J. van Leeuwen. Mantenance of conguratons n the plane. J. Comput. Syst. Sc., 23:66{24, 98. [8] G. W. Peck. On k-sets n the plane. Dscrete Math., 56:73{74, 985. [9] P. J. Rousseeuw and M. Hubert. Regresson depth. J. Amer. Statst. Assoc., to appear n June 999. [2] P. J. Rousseeuw and M. Hubert. Depth n an arrangement of hyperplanes. Dscrete Comput. Geom., to appear, 999. [2] P. J. Rousseeuw and I. Ruts. Constructng the bvarate Tukey medan. Statstca Snca, 8:827{ 839, 998. [22] P. J. Rousseeuw and A. Struyf. Computng locaton depth and regresson depth n hgher dmensons. Statstcs and Computng, 8:93{23, 998. [23] P. J. Rousseeuw, S. Van Aelst and M. Hubert. Rejonder to dscusson of \Regresson Depth". J. Amer. Statst. Assoc., to appear n June 999. [24] I. Ruts and P. J. Rousseeuw. Computng depth contours of bvarate pont clouds. Computatonal Statstcs and Data Analyss, 23:53{68, 996. [25] M. Sharr. k-sets and random hulls. Combnatorca, 3:483{495, 993. [26] A. Struyf and P. J. Rousseeuw. Halfspace depth characterzes the emprcal dstrbuton. J. Multvar. Analyss, to appear, 999. [27] A. Struyf and P. J. Rousseeuw. Hgh-dmensonal computaton of the deepest locaton. Manuscrpt, Dept. of Mathematcs and Computer Scence, Unversty of Antwerp, Belgum, 998. [28] S. Van Aelst and P. J. Rousseeuw. Robustness of Deepest Regresson. Manuscrpt, Dept. of Mathematcs and Computer Scence, Unversty of Antwerp, Belgum, 998.