International Journal of Foundations of Computer Science c World Scientic Publishing Company Searching a Pseudo 3-Sided Solid Orthoconvex Grid ANTONIO

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1 Intrnational Journal of Foundations of Computr Scinc c World Scintic Publishing Company Sarching a Psudo 3-Sidd Solid Orthoconvx Grid ANTONIOS SYMVONIS Bassr Dpartmnt of Computr Scinc, Univrsity of Sydny Sydny, N.S.W. 2006, Australia and SPYROS TRAGOUDAS y Dpartmnt of Computr Scinc, Southrn Illinois Univrsity, Fanr Hall Carbondal, IL 62901, U.S.A. Rcivd 7 April 1993 Rvisd 8 Dcmbr 1993 Communicatd by D. T. L ABSTRACT In this papr w xamin th dg sarching problm on psudo 3-sidd solid orthoconvx grids. W dn a similar problm which w call modid dg sarching, and w driv a rlation among th original dg sarching problm and th modid on. Thn, for th modid dg sarching problm, w show that thr ar sarching stratgis that possss svral proprtis rgarding th way th grid is sarchd. Ths stratgis allow us to obtain a closd formula that xprsss th minimum numbr of sarchrs rquird to sarch a psudo 3-sidd solid orthoconvx grid. From that formula and a rathr straight forward algorithm w can show that th problm is in P. W obtain a paralll vrsion of that algorithm that placs th problm in NC. For th cas of squntial algorithms, w driv an optimal algorithm that solvs th problm in O(m) tim whr m is th numbr of points ncssary to dscrib th orthoconvx grid. Anothr important fatur of our mthod is that it also suggsts an optimal sarching stratgy that consists of O(n) stps, whr n is th numbr of nods of th grid. Kywords: Graph Sarching, Sarch Numbr, Orthoconvx Grid, Algorithms. 1. Introduction Th dg-sarching problm was introducd by Parson. 1 An undirctd graph was givn and th objctiv was to clan its contaminatd dgs (or, in a dirnt statmnt of th problm, to captur a fugitiv hiddn in thm). Thr kinds of actions wr allowd in this claning opration: 1. plac(nod): This action placs a sarchr at th nod spcid as paramtr of th action. symvonis@cs.su.oz.au y spyros@buto.c-cs.siu.du 1

2 2. pick(nod): This action picks up a sarchr from th nod spcid as paramtr of th action. 3. mov(origin; dstination): This action movs a sarchr along th dg that conncts th origin and th dstination nods. For th action to b lgal, th two nods must b connctd by an dg and a sarchr must b initially locatd at th origin nod. Th sarch numbr of graph G, dnotd by s(g), was dnd by Mggido t. al. 2 as th minimum numbr of sarchrs that ar rquird in ordr to clan th graph (or captur th fugitiv that is hiddn in its dgs). In thir papr it was provn that th dcision problm \ Givn a graph G and an intgr k, can G b clard with k sarchrs?" is NP-Hard. Thy also pointd out that th problm would blong in th class NP if it was tru that rcontamination cannot hlp in sarching a graph. W say that a clan dg is rcontaminatd if it bcoms adjacnt with a contaminatd dg and no sarchr is placd at thir common nod. Rcontamination can start whn a sarchr that is positiond at a nod adjacnt to a clan dg and at last on contaminatd dg, lavs th nod (ithr by a pick or mov action) and allows th clan dg to b contaminatd again. W assum that rcontamination propagats at an innit spd, i.., if rcontamination occurs as a rsult of an action t of th sarching, thn, bfor action t + 1, all dgs that can bcom contaminatd again will do so. LaPaugh 3 provd that rcontamination dos not hlp to sarch a graph and thus th dg sarching problm was includd in th class NP. (Latr on, Binstock and Symour 4 gav a simplr proof.) Bsids its thortical importanc, this rsult is usful in th sns that allows us to assum that thr xists a stratgy that sarchs th graph using th minimum numbr of sarchrs and nvr allows rcontamination. A consqunc is that th graph can b sarchd in a nit numbr of actions. Aftr LPaugh's work a grat dal of ort was dvotd to th sarching problm. Most of th rsults rlatd th sarching problm with othr combinatorial optimization problms such as pbbling 5, cutwidth 6, graph sparators 7, intrval thicknss 8 and path-width 8;9;10. In this papr, w concntrat on th sarching problm on a spcial kind of graphs, namly, th psudo 3-sidd solid orthoconvx grids. W show how to dtrmin th sarch numbr of such graphs in optimal tim. W do that by dning a modid vrsion of th dg sarching problm which w call modid dg sarching. Thn, for th modid dg sarching problm, by proving that thr ar sarching stratgis that possss svral proprtis rgarding th way th grid is sarchd, w ar abl to obtain a closd formula that xprsss th minimum numbr of sarchrs rquird to sarch a psudo 3-sidd solid orthoconvx grid. From that formula w driv an algorithm that computs s() in polynomial tim. W also obtain a paralll vrsion of that algorithm that placs th problm in NC. For th cas of squntial algorithms, w driv an optimal algorithm that solvs th problm in O(m) tim whr m is th numbr of points ncssary to dscrib th orthoconvx grid. Anothr important fatur of our mthod is that it also suggsts an optimal sarching stratgy that consists of O(n) stps, whr n is 2

3 th numbr of nods of th grid. Prviously, w wr abl to dtrmin th sarching numbr in optimal tim only for th class of trs 2. W wr also abl to solv th dcision problm \Givn a graph G of n nods can w sarch G by using a constant numbr of k sarchrs? " in O(n 2 ) 11. W improv this rsult for th cas of psudo 3-sidd solid orthoconvx grids. Som work has alrady bn don for sarch problm in rctangular grids 12. Howvr, in that work th sarchrs ar mor powrful than th ons w us (i.., dirnt actions ar assumd) and contamination dos not propagat in an innit spd. Furthrmor, th rctangular grid that was assumd as th undrlying graph structur blongs in th class of th psudo 3-sidd solid orthoconvx grids. Th rst of th papr is organizd as follows: In Sction 2, w dn th modid dg sarching problm and prov that thr ar sarching stratgis that posss svral proprtis rgarding th way th sarchrs mov on th graph thy sarch. In Sction 3, w dn th class of th psudo 3-sid solid orthoconvx grids. W prov our main thorm that stats that thr xists a sarching stratgy for sarching psudo 3-sidd solid orthoconvx grids such that during th sarch thr xists only on contaminatd ara. This thorm nabls us to obtain a closd formula for th minimum numbr of guards that ar rquird to solv th modid dg sarching problm. W also show how, from that rsult, to obtain s(). Sction 4, is dvotd to algorithms (squntial and paralll) that comput s(). W show that s() can b computd in optimal tim squntially and that thr xists a paralll algorithm that computs s() in polylog tim using a polynomial numbr of procssors. W conclud in Sction A Nw Vrsion of th Sarching Problm In this sction, w dn a nw vrsion of th sarching problm which w call modid dg sarching. Th dirnc btwn th two sarching problms ar in th possibl actions that can tak plac during th sarch. Dnition 1 W say that w hav a modid dg sarching problm on a graph G if w ar allowd to sarch th graph using all 3 actions of th original dg sarching problm as wll as th additional 4 th on: 4. clan(nod1; nod2): It clans dg (nod1; nod2) or th path (nod 1 ; nod i ), (nod i ; nod 2 ) whr nod i is of dgr 2. For th action to b lgal, sarchrs must b placd on both nod1 and nod2. To undrstand th dirncis btwn th two problms, w analyz th ways in which an dg can b cland in ach of thm. In th dg sarching problm an dg can b cland only by a mov action. In that cas, ithr on guard is positiond at a nod that has all but on of its incidnt dgs clan and thn movs towards th nod at th othr sid of th dg, or, two guards ar positiond at a nod and on of thm movs towards th nod at th othr sid of th contaminatd dg. So, thr ar two dirnt situations in which an dg can b cland. In th modid dg sarching problm an dg can b clard by both of th two prviously mntiond ways as wll as with an additional third way that is dscribd 3

4 by action clan(nod1; nod2). Th sam holds for a path of lngth 2 providd that th intrnal nod of th path is of dgr 2. In th following w will dnot by ms(g) th minimum numbr of sarchrs that ar rquird in ordr to solv th modid dg sarching problm on graph G. Binstock and Symour 4 and Takahashi, Uno and Kajitani 13 indpndntly dnd th mixd sarching problm which is quit similar to th modid dg sraching problm w dn in this papr. Th dirnc btwn th two problms is that in mixd sarching th clan action can clan only on dg whil in th modid dg sarching it can clan a chain of two dgs. Takahashi, Uno and Kajitani 13 also dnd th propr path-width of a graph (similar to path-width d- nd by Robrtson and Symour 14 ) and stablishd that th propr path-width of a graph is qual to its mixd sarch numbr. Kirousis and Papadimitriou 5 dnd a similar sarching problm which thy calld nod sarching. In thir vrsion of th gam only plac, pick and clan actions wr allowd and th clan action could clan only on dg. Dnition 2 A sarching stratgy S(G) is a squnc < a 1 ; a 2 ; : : : ; a i ; : : : ; a m > of actions such that whn applid on a graph G which has all of its dgs contaminatd has th ct to clan th dgs of G. Action a i is any action allowd in th sarching problm. A sarching stratgy is said to b optimum whn thr is no othr stratgy that uss a smallr numbr of sarchrs and also sarchs th graph. Two sarching stratgis ar quivalnt if thy both sarch th sam graph with th sam numbr of sarchrs. Givn a sarching stratgy S(G) =< a 1 ; a 2 ; : : : ; a i ; : : : ; a m > w can dn a nw sarching stratgy S?1 (G) =< a?1 m ; a?1 m?1 ; : : : ; a?1 m?i ; : : : ; a?1 1 > whr plac?1 (v) = pick(v), pick?1 (v) = plac(v), mov?1 (v 1 ; v 2 ) = mov(v 2 ; v 1 ) and clan?1 (v 1 ; v 2 ) = clan(v 2 ; v 1 ). Obviously, th two sarching stratgis ar quivalnt. W say that a nod is clan if all of its adjacnt dgs ar clan. A nod is dirty if all of its adjacnt dgs ar contaminatd. If it is nithr clan nor dirty w say that it is partially clan. A mov(origin; dstination) action is uslss if it nithr clans an dg nor rsults into contamination. A uslss mov action occurs whn th sarchr movs from a clan nod, or, from a partially clan nod to a clan on and is not causing rcontamination, or, from a dirty nod that has no othr sarchr. An action a i cannot b postpond if no action a j ; j > i, can b xcutd on th partially clard graph that rsults from th xcution of < a 1 ; : : : ; a i?1 > on graph G. Dnition 3 A sarching stratgy S 1 (G) is (t 1 ; t 0 )?suprior of sarching stratgy S 0 (G) if aftr t 1 actions of S 1 (G) and t 0 actions of S 0 (G): 1. Th st of dgs that S 1 (G) lavs contaminatd is a subst of th st of dgs that S 0 (G) lavs contaminatd, and 2. Th st of nods of G that contain sarchrs placd by S 1 (G) is a subst of th st of nods containing sarchrs placd by S 0 (G). 4

5 Dnition 4 W say that a sarchr is usful for action a t (or simply that it is usful at tim t) if its rmoval will caus rcontamination, or it will prvnt action a t of happning. Othrwis, w say that th sarchr is uslss. Lmma 1 Thr is an optimum sarching stratgy for th (modid) dg sarching problm on graph G that contains only usful mov actions. Proof. Assum an optimal sarching stratgy S(G) that sarchs graph G and has k uslss movs. W will show how to liminat thm, on aftr th othr. Start with th rst such action in S(G). Say that it is action a i1. So, a i1 = mov(n 1 ; n 2 ). Rplac a i1 by th following two actions: < pick (n 1 ); plac(n 2 ) >. Now th nw schdul S 1 (G) is (i 1 + 1; i 1 )-suprior of S(G) and also quivalnt with it. By induction, aftr m; m < k; rplacmnts w will hav that S m (G) is (i m + 1; i m )-suprior of S m?1 (G) and also quivalnt with it. It follows that aftr k rplacmnts w will gt th sarching schdul S k (G) that contains only usful mov actions and is quivalnt with S(G). Not that th lmma is tru for both th original and th modid dg problms. 2 Lmma 2 Thr is an optimum sarching stratgy for th modid dg sarching problm on graph G that has th proprty that no uslss sarchr is on G immdiatly bfor any plac, mov, or clan action. Proof. By Lmma 1 w can assum that w hav an optimum sarching stratgy S M (G) that contains no uslss movs. Working in th sam lins with th proof of Lmma 1 w will construct an quivalnt sarching stratgy SM 1 (G) that posssss th statd proprty. A uslss guard can b cratd by a plac, mov, or clan action. A plac action can crat xactly 1 uslss sarchr, a clan action can crat at most 2 uslss sarchrs ( th ons that ar positiond at th nods that dn th dg it clans), whil a mov action can mak uslss any numbr of sarchrs (this happns whn th mov action causs rcontamination). Considr th rst action that crats a uslss guard. Say that it is action a i. W hav th following 3 cass: Cas 1 a i is a plac action. W follow th guard in th sarching stratgy and w locat th nxt action that nds it (if any). If thr is such an action thn w postpon th plac action to b xcutd just bfor th action that nds th guard. If thr in no action that nds th sarchr in th futur, w simply cancl th plac action as wll as th pick action that rmovs it from th graph. Cas 2 a i is a clan action. A clan action action can crat up to 2 uslss guards. Again w can follow thm to locat th actions (if any) that nd thm in th futur. W thn immdiatly pick thm, on aftr th othr, and (if ndd) plac thm just bfor th actions that nd thm in th futur. Cas 3 a i is a mov action. W trat this cas similarly to cass 1 and 2. Obviously, w will gt an quivalnt stratgy that has th proprty that th rst action that crats uslss sarchrs is immdiatly followd by actions that pick thm from th graph. Also obsrv that in th nw sarching stratgy no uslss mov actions wr introducd. Now, th rst of such actions that crat uslss sarchrs can b rmovd by induction. 2 5

6 Lmma 3 If thr is a sarching stratgy that solvs th modid dg sarching problm on graph G using k sarchrs thn thr is a sarching stratgy that solvs th dg sarching problm on graph G using ithr k or k + 1 sarchrs. Proof. Lt S M (G) b a sarching stratgy that solvs th modid dg sarching problm on graph G using k sarchrs. In th cas that S M (G) contains no clan actions, it is also a valid stratgy for th dg sarching problm that uss k sarchrs. Considr th cas whr S M (G) contains clan actions and lt a i1 = clan(n 1 ; n 2 ) b th rst clan action in S M (G). If th dg (n 1 ; n 2 ) xists w can rplac th clan action by th squnc of actions < plac(n 1 ); mov(n 1 ; n 2 ); pick(n 2 ) >. Sinc 3 actions wr usd to rplac th on clan action, thn th nw stratgy SM 1 (G) is (i 1 + 2; i 1 )-suprior of S M (G), quivalnt with it and no action a j ; j i is a clan action. If (n 1 ; n 2 ) dos not xist, for th clan action to b lgal, thr must xist a nod n i such that dgr(n i ) = 2 and (n 1 ; n i ), (n i ; n 2 ) xist. In this cas w can rplac th clan action by th squnc of actions < plac(n 1 ); mov(n 1 ; n i ); mov(n i ; n 2 ); pick(n 2 ) >. Th nw stratgy SM 1 (G) will b (i 1 + 3; i 1 )-suprior of S M (G), quivalnt with it and no action a j ; j i is a clan action. By induction, w can liminat all clan actions by using only 1 xtra sarchr. Now th nw sarching stratgy, say S(G), is a valid stratgy for th dg sarching problm on graph G that uss k + 1 sarchrs. Not that, in th cas that S M (G) is an optimum sarching stratgy that uss k sarchrs, any sarching stratgy for th original problm will nd at last k sarchrs. This is bcaus th st of actions allowd in th modid problm is a suprst of th st of actions allowd in th original sarching problm Sarching stratgis for Psudo 3-Sidd Solid Orthoconvx Grids In this sction w dn th family of psudo 3-sidd solid orthoconvx grids. W also dn th notions of clan and dirty (or contaminatd) aras during th sarch procss. Thn, w prov that thr xist an optimum sarching stratgy for th modid dg sarching problm on any psudo 3-sidd solid orthoconvx grid D in which during th sarch thr is only on dirty ara. W xprss ms(d) by a closd formula and w show how to dtrmin s(d) from ms(d) Dnitions Lt G 1 (V 1 ; E 1 ) b th innit undirctd graph whos nod st V 1 consists of all points of th plan with intgr coordinats and in which two vrtics ar connctd by an dg in E 1 if and only if th Euclidan distanc btwn thm is qual to on. Lt G i (V i ; E i ) b a nit nod-inducd subgraph of G. A grid graph D(V; E) is a subgraph of G i whr, V = V i and E E i. In th following discussion w will considr only graphs that ar connctd and all of thir nods hav dgr gratr than 1. W say that a grid graph D(V; E) is solid if it has no hols. If w color black all unit squars in G 1 that ar surroundd by th dgs of a 6

7 solid grid graph D, w will divid th plan into two rgions, on black and on whit. A nod v that blongs into a solid grid graph and is adjacnt to both th black and th whit rgion is said to b a boundary nod. Th st of all boundary nods of a solid grid D is said to constitut th boundary of D. Assum a solid grid graph D and its corrsponding black and whit rgions. D is said to b orthoconvx if and only if th intrsction of any lin paralll to any coordinat axis with th black rgion consists of at most on lin sgmnt. A nod v at th boundary of a solid grid graph is said to b a convx boundary nod if it is of dgr 2 and a concav boundary nod if it is of dgr 4. A nod v at th boundary of a solid grid graph is said to b a turning boundary nod if it is ithr a convx or a concav boundary nod. Othrwis, it is calld a simpl boundary nod. From th abov dnitions it is obvious that a solid grid graph can b compltly dnd by its turning boundary nods givn in th ordr that w mt thm whn w travrs th boundary in th clockwis or countr clockwis dirction. In th rst of th papr, w assum that th grid undr considration is rprsntd by its turning boundary nods givn in th ordr thy appar if w travrs its boundary in th clockwis dirction. An arbitrary nod is slctd to b th start of th travrsal. During our travrs of th boundary of a solid grid graph and assuming that w always look towards th nxt boundary nod, w hav to mak svral turns. So, th travrsal of a solid grid can b rprsntd by a word which has lngth qual to th numbr of turning boundary nods ovr an alphabt of two lttrs namly, L for lft and R for right. W call that word th coding of th solid orthoconvx grid. Sinc w can start our travrsal of th boundary of th grid from any turning point, it is usful to think of th coding as a circular word whr th rst and th last charactrs wrap around. Sinc w can rturn to th point from which w startd, it is obvious that w hav 4 mor R's than L's. Also obsrv that in an orthoconvx grid w nvr hav two conscutiv L's in its coding. By cancling ach L and th R that follows it in th coding of a solid orthoconvx grid, w ar lft with 4 R's. Ths corrspond to 4 convx boundary nods. Ths points dn th sids of th grid. In that sns, all orthoconvx grids ar 4-sidd. Figur 1 shows a solid orthoconvx grid, its coding, and its sids. Howvr, w can rlax that dnition for th spcial cas of th solid orthoconvx grids that contain th pattrns RRRR or RRRRR. In ths two cass, w can combin 2 sids togthr and thus, considr th orthoconvx to b composd by a bas, a rising rgion that is immdiatly aftr th bas in th clockwis dirction, and a falling rgion that follows th rising rgion in th clockwis dirction. For this rason, w call all th solid orthoconvx grids that fall into that catgory psudo 3-sidd. In th following w will rfr to th boundary nods that lay btwn any two sids (or psudo sids) as cornrs. Figur 2 shows th two typs of psudo 3-sidd solid orthoconvx grids and thir sids. A cord is any path (possibly a closd on) intrnal to th grid that consists of 7

8 Sid 1 R 2 R 1 Sid 2 Sid 4 Sid 3 R 3 R 4 Coding : RLRLRLRRLRLRLRLRLRRLRLRRLRLRLR (W mak th first turn at R 1 ) Fig. 1. A solid orthoconvx grid, its coding, and its sids. Sid 3 Sid 2 Sid 3 Sid 2 Sid 1 (Bas) Sid 1 (Bas) Typ RRRRR Typ RRRR Coding: RRLRLRLRLRLRRR Coding RLRLRLRRLRLRLRLRRR Fig. 2. Th two typs of psudo 3-sidd solid orthoconvx grids. grid dgs as wll as lin sgmnts that connct nods that ar of distanc 2 and diagonally opposit of ach othr. If it is not a closd on it must hav its ndpoints on th boundary. A diagonal is a cord that has its nd-nods on two dirnt sids. W mak th convntion that a convx boundary nod that sparats two sids blongs in both of thm and, in that sns, it is considrd to b a diagonal as wll. It is obvious that th nods that blong into a nontrivial diagonal that touchs ach of its adjacnt sids at most onc form a cut-st for th solid grid. During th cours of th sarching thr ar rgions of th grid that ar clan and othrs that ar considrd to b contaminatd (or dirty). Dnition 5 Assum a sarching stratgy S(G) on a graph G. Lt Dc t b th graph that is inducd by rmoving all cland dgs and all nods that hav no incidnt contaminatd dgs aftr t stps of S(G). Conn(Dc) t is th numbr of connctd componnts of Dc t and it dnots also th numbr of contaminatd rgions at tim t. W say that a graph can b sarchd in such a way that w hav at most contaminatd rgions if and only if max t>0 Conn(Dc) t Sarching Stratgis for Psudo 3-Sidd Solid Orthoconvx Grids In this sction w show that thr xists an optimum sarching stratgy for th modid dg sarching problm on a psudo 3-sidd solid orthoconvx grid D in which during th cours of th sarching thr xists only on contaminatd rgion. Basd on that, w comput ms(d) and w show how from it to driv s(d). 8

9 Lmma 4 Assum a solid orthoconvx grid and a cord that has its nd-nods on th sam sid of th grid. W can sarch th rgion that is boundd from that sid and th cord using th modid dg sarching with a numbr of sarchrs qual to th cord points and in such a way that on sarchr nds up at vry cord nod. Proof. Without loss of gnrality assum that th bas is paralll to th X-axis. Lt P 1 = (x 1 ; y 1 ) and P 2 = (x 2 ; y 2 ) b th two points of th cord with th smallst and th largst x-coordinat, rspctivly. W can projct all points of th cord on a lin sgmnt that is paralll to th X-axis, lis outsid th grid(abov it if w ar considring th falling or th rising sid and bllow if w ar considring th bas) and xtnds from x = x 1 to x = x 2. Thn, w can sarch th ara of th grid by moving sarchrs from that lin sgmnt, paralll to th Y-axis and towards th cord. During th sarch, whn w rach a cord point, w lav a sarchr thr. W hav nough sarchrs to do so, sinc, all cord points wr projctd on th lin sgmnt and thus, for vry cord point thr is a corrsponding sarchr that is moving on th lin paralll to th Y-axis and passs through it. 2 Corollary 1 Assum a solid orthoconvx grid and a cord that has its nd-nods on th sam sid of th grid. W can sarch th rgion that is boundd from that sid and th cord using th modid dg sarching with a numbr of sarchrs qual to th cord points and in such a way that on sarchr starts from ach cord nod. Proof. By Lmma 4 and by obsrving that if a sarching stratgy S sarchs graph G so dos S?1. 2 Lmma 5 Assum a psudo 3-sidd solid orthoconvx grid D and a diagonal. W can sarch th rgion of D that is boundd from th diagonal and th part of th boundary that contains xactly 1 cornr using th modid dg sarching with a numbr of sarchrs qual to th diagonal points and in such a way that on sarchr nds up at vry diagonal nod. Proof. For simplicity, assum that th diagonal touchs th sam sid just onc. Thn, th nods that lay on it form a cutst and thy divid th grid in xactly two rgions. Sinc thr ar only 3 cornrs in a psudo 3-sidd solid orthoconvx grid, on of ths two rgions contain xactly 1 cornr. Without loss of gnrality, assum that th bas of th grid is paralll to th X axis. W will only considr th cas whr th diagonal is btwn th rising and th falling sids. Th othr cass can b handld in a similar way. Lt th two nd points of th diagonal, say P 1 and P 2 hav coordinats (x 1 ; y 1 ) and (x 2 ; y 2 ) corrspondingly. Thn, x 1 6= x 2 and assum that x 1 > x 2. Obsrv now that any boundary nod that is in th rgion undr considration has X coordinat x that satiss th rlation x 1 x x 2. Th rst of th proof is thn similar to that of Lmma 4. Cass whr th diagonal touchs th sam sid mor than onc ar asy to b handld sinc that diagonal can b dcomposd to on that touchs ach of th two sids xactly onc and in svral cords that touch th sam sid. 2 Corollary 2 Assum a psudo 3-sidd solid orthoconvx grid D and a diagonal. W can sarch th rgion of D that is boundd from th diagonal and th part of th boundary that contains xactly 1 cornr using th modid dg sarching with 9

10 a numbr of sarchrs qual to th points of th diagonal and in such a way that on sarchr starts from ach point of th diagonal. Proof. By Lmma 5 and by obsrving that if a sarching stratgy S sarchs graph G so dos S?1. 2 In a way similar to that of Lmmata 4 and 5 w can prov th following Lmma: Lmma 6 Any solid grid D that has b boundary nods can b sarchd with b sarchrs in such a way that on sarchr starts at vry boundary nod, or, on sarchr nds at vry boundary nod. Thorm 1 Thr is an optimal sarching stratgy for th modid dg sarching problm on a psudo 3-sidd solid orthoconvx grid which has th proprty that during th sarch thr is only on contaminatd ara. Proof. Assum any psudo 3-sidd solid orthoconvx grid D and an optimum stratgy S(D) that solvs th modid dg sarching problm on it. Furthrmor, assum that during th cours of th sarching thr ar mor than on contaminatd aras. Obviously, at th bginning of th sarch thr is only on contaminatd ara. By xamining th sarching stratgy w can locat th action that, for th rst tim, maks Dc t to consist of two connctd componnts, for som t. This action has to b a mov or a clan action. W mak th following obsrvations: If Dc t?1 contains connctd componnts and at stp t of th sarching stratgy a mov or a clan action is taking plac, thn Dc t + 1: Th rst tim w hav two contaminatd aras w can idntify two disjoint cords that form th boundaris btwn th two contaminatd aras and th clan ara that sparats thm. Th dg that was last cland has sarchrs at its nd-nods. Furthrmor, that dg conncts th two cords idntid in th prvious obsrvation. In th rst of th proof w assum that w hav idntid th action that cratd for th rst tim two contaminatd aras as wll as th two cords that bound thm. Thn w show how to us that information as wll as th initial optimal sarching stratgy to obtain a nw stratgy that liminats that action. Inductivly, w can liminat all such actions, and thus, driv an optimum sarching stratgy that during th sarch maintains only on contaminatd rgion. Lt th cords that wr idntid in th prvious obsrvations b dnotd by C 1 and C 2 and b th dg which's sarching cratd th two contaminatd aras. W considr thr cass: 1. C 1 and C 2 do not hav a common dg with th boundary. Th two contaminatd aras will look lik Figur 3. W can driv a nw stratgy that sarchs th clan rgion togthr with rgion Dirty 2 as Lmma 6 indicats, and, at th nd, it placs guards at vry point of C 1. Thn, th sarching continus as in th original schdul and by ignoring all actions that sarchd rgion Dirty 2. 10

11 Dirty_2 C 1 Sid 3 Sid 2 Clan C2 Dirty_1 Sid 1 (bas) Fig. 3. Th situation for cas 1 of Thorm Only on of th two idntid cords has its nd-nods adjacnt to a clan boundary dg. Assum that C 1 is th cord that touchs th boundary in th dscribd way. Thr ar thr subcass to considr: (a) Both of th nd-nods of C 1 blong to th sam sid. Th two contaminatd aras will look lik Figur 4. By Lmma 4, w can sarch rgions Clan and Dirty 2 togthr, and, at th nd, hav sarchrs placd at vry point of C 1. Thn, th sarching continus as in th original schdul and by ignoring all actions that sarchd rgion Dirty 2. Dirty_2 Clan C 1 Sid 3 C2 Sid 2 Dirty_1 Sid 1 (bas) Fig. 4. Th situation for cas 2a of Thorm 1. (b) Th two nd-nods blong to dirnt sids and th contaminatd ara adjacnt to th boundary contains on cornr. Th two contaminatd aras will look lik Figur 5. A nw sarching stratgy is th following: Follow th original stratgy up to stp that clans dg.(do not prform that stp.) Clan rgion Dirty 2, thn dg and, nally, swp th rst of th grid as Corollary 2 suggsts. (c) Th two nd-nods blong to dirnt sids and th contaminatd ara adjacnt to th boundary contains two cornrs. 11

12 Dirty_2 Sid 3 Dirty_1 Sid 2 C 1 C2 Clan Sid 1 (bas) Fig. 5. Th situation for cas 2b of Thorm 1. Th two contaminatd aras will look lik Figur 6. A nw sarching stratgy is th following: By Lmma 5, w can sarch rgions Clan and Dirty 2 togthr, and, at th nd, hav sarchrs placd at vry point of C 1. Thn, th sarching continus as in th original schdul and by ignoring all actions that sarchd rgion Dirty 2. Dirty_2 C2 Sid 3 Sid 2 Clan C 1 Dirty_1 Sid 1 (bas) Fig. 6. Th situation for cas 2c of Thorm Both of th two idntid cords hav thir nd-nods adjacnt to a clan boundary dg. Thr ar svn subcass to considr: (a) Thr xist on sid that contains all th nd-nods of th cords. Th two contaminatd aras will look lik Figur 7. W can crat a schdul that clans rgions Dirty 1 as wll as th alrady cland rgion by swping thm, as Lmma 4 suggsts, and thn continus with th original schdul. (b) Thr is on sid that contains thr nd-nods and th clan ara contains on cornr. Th two contaminatd aras will look lik Figur 8. W can trat this cas in a way similar to cas 3a. (c) Thr is on sid that contains thr nd-nods and th clan ara contains two cornrs. 12

13 Dirty_1 Clan Sid 3 C 1 Dirty_2 Sid 2 C2 Sid 1 (bas) Fig. 7. Th situation for cas 3a of Thorm 1. Dirty_1 Sid 3 C 1 Dirty_2 Sid 2 Clan C2 Sid 1 (bas) Fig. 8. Th situation for cas 3b of Thorm 1. Th two contaminatd aras will look lik Figur 9. Sinc th clan ara contains two cornrs, th contaminatd ara has to contain th third on. Obsrv that thr is anothr cord dnd that consists of part of C 1, and part of C 2 and it surrounds rgion Clan 1. Th guards that ar placd on this cord ar nough to swp th rgion that contains th third cornr. A nw stratgy can b obtaind from th original on by ignoring all movs that clan rgion Clan 2. Thn, w clan and nally w swp th rst of th grid. Dirty_1 Clan_2 Sid 3 Dirty_2 Sid 2 C 2 C 1 Clan_1 Sid 1 (bas) Fig. 9. Th situation for cas 3c of Thorm 1. (d) Thr ar two sids that ach contains two nd-nods of th sam cord. 13

14 Th two contaminatd aras will look lik Figur 10. W can trat this cas in a way similar to cas 3c. Dirty_1 Sid 3 Sid 2 C 1 Dirty_2 Clan C2 Sid 1 (bas) Fig. 10. Th situation for cas 3d of Thorm 1. () Thr ar two sids that ach contains on nd-nod from ach cord. Th two contaminatd aras will look lik Figur 11. Th nw schdul consists by swping th cland rgion as wll as rgion Dirty 1. At th nd of th swping, sarchrs ar placd at vry nod of C 2. Thn, w continu with th initial sarching stratgy and by ignoring movs that clan alrady cland ara. Dirty_1 Dirty_2 C2 Sid 3 Sid 2 Clan C 1 Sid 1 (bas) Fig. 11. Th situation for cas 3 of Thorm 1. (f) Evry sid contains at last on nd-nod and thr is not a sid that contains both ndpoints of th sam cord. Th two contaminatd aras will look lik Figur 12a. This is th most intrsting cas. Lt b th last clard dg and L 1 b th th numbr of guards that blong to cord C 1 btwn th intrsction with (not including 's nd-nods) and th sid that both cords touch. Similarly w dn L 2. Assum that L 1 6= L 2 and without loss of gnrality that L 1 > L 2. Lt jcj dnot th numbr of nods in diagonal C. A nw sarching stratgy is th following: Using jc 1 j? L 1 + L jc 1 j w sarch th ara dnotd by Clan 1 and Dirty 1. Thn, by using jc 2 j? L 2 + L 1 jc 2 j w can sarch th ara dnotd by Dirty 2 such 14

15 that w nvr crat a scond dirty ara. Now th rst of th grid can b swpt by th sarchrs. Th cas that L 1 = L 2 nds som spcial tratmnt. W cannot us xactly th sam argumnt sinc w hav availabl xactly jc 2 j? 1 sarchrs to sarch rgion Dirty 2 and w ar on sarchr sort. Howvr, w can sav on sarchr if w pay mor attntion to th way th sarchrs ar placd around. Thr ar two possibl arrangmnts that ar shown in Figur 12b. In th arrangmnt to th lft w can sav th sarchr on nod B, by rmoving dgs AB and BC from th cord and by introducing dg AC. In th arrangmnt to th right again w can spar th sarchr on B by rmoving dgs AB and BD from th cord and by introducing dg AD. Clan_1 L 2 Sid 3 Dirty_2 Dirty_1 Sid 2 L 1 C 2 C 1 Sid 1 (bas) a. Th gnral cas whr L 1 > L 2 A A B C 2 B D C 2 C C C 1 C 1 b. Th cas whr L = L 1 2 Fig. 12. Th situation for cas 3f of Thorm 1. (g) Evry sid contains at last an nd-nod and thr is a sid that contains both ndpoints of th sam cord. Th two contaminatd aras will look lik Figur 13. W can trat this cas in a way similar to cas 3f. By idntifying th situation that occurs whn, for th rst tim, two contaminatd aras ar cratd, w can always liminat th action which cratd ths two aras and, thus, driv an quivalnt sarching stratgy. By applying this argumnt inductivly, w can gt a sarching stratgy that sarchs any psudo 3-sidd solid orthoconvx grids by maintaining only on dirty ara during th sarch. 2 Lt d(n 1 ; s) b th lngth of a shortst diagonal from th boundary nod n 1 to sid s. If n 1 is on sid s thn d(n 1 ; s) = 0. If nod n 1 is a cornr thn lt s(n 1 ) dnot th sid which is opposit of it. Lt dnot a boundary dg (n 1 ; n 2 ) or a path (n 1 ; n i ); (n i ; n 2 ) whr n i is of dgr 2. Lt S a () to dnot th sid that is 15

16 Dirty_1 C2 Sid 3 Dirty_2 Sid 2 C 1 Clan Sid 1 (bas) Fig. 13. Th situation for cas 3g of Thorm 1. following th on that lis on if w mov from n 1 to n 2, and S b () to dnot th sid that is following that lis on if w mov from n 2 to n 1. In ordr for S a () and S b () to b wll dnd, must not b a path of lngth 2 that contains a cornr. In that cas ( lis on two sids) w dn S a () (= S b ()) to b th third sid of th psudo 3-sidd solid orthoconvx grid. Basd on Thorm 1 w prov: Thorm 2 Th minimum numbr of sarchrs rquird to solv a modid dg sarching problm on a psudo 3-sidd solid orthoconvx grid D is givn by: ms(d) = min(cornr distanc; diagonal pair distanc) whr, cornr distanc = minfd(c; s(c))g ovr any cornr c (out of 3 possibl), and diagonal pair distanc = minfd(n 1 ; S b ()) + d(n 2 ; S a ())g ovr any boundary dg = (n 1 ; n 2 ) or any path =< (n 1 ; n i ); (n i ; n 2 ) > whr n i is of dgr 2. Proof. Th rst thing that w hav to prov is that th numbr of sarchrs statd in th thorm is sucint to sarch th grid. Obsrv that if a diagonal that conncts any cornr of th grid with its opposit sid contains k points, thn k sarchrs can sarch th grid. Th sarchrs will swp th part of th grid that contains on of th othr cornrs and will b placd on th points of th diagonal as Lmma 5 suggsts. Thn, ths sarchrs can sarch th rst of th grid as Corollary 2 indicats. A similar obsrvation can b mad whn th availabl sarchrs ar nough to ll all th points of two diagonals that touch all sids of th grid and also hav two of thir ndpoints connctd by an dg or by a path of th form < (n 1 ; n i ); (n i ; n 2 ) > whr n i is of dgr 2. As Lmma 5 indicats, sarchrs can nd up at th points of th diagonals aftr sarching th ara that contains on cornr and is boundd btwn ach of thm and th boundary. Thn, by using Corollary 2 th guards on ach diagonal can clan th contaminatd ara that is adjacnt to both of thm. Now that w hav stablishd that th numbr of sarchrs statd in th thorm is sucint to sarch th grid, w hav to show that th grid cannot b sarchd by lss guards. To prov that, w will show that thr is an optimal schdul in which, during th sarch, guards ar ithr forming a pair of adjacnt diagonals that touch all sids or, thy form a diagonal btwn a cornr and its opposit sid. 16

17 Assum an optimal schdul that has th proprty that during th sarch only on dirty ara is maintaind. In Thorm 1 w provd that such a sarching stratgy xists. Considr th last tim in th sarching stratgy that th dirty ara contains boundary dgs from all sids and th action that clans th last dg on som sid. This action can b ithr a mov or a clan action. In th cas whr th action that cland th last dg of that sid was a mov action w can dnot that dg by and assum that a sarchr is at only on ndpoint of. If this wr not th cas, w can substitut th mov action by a clan. So, th sarchr that clans movs to a boundary nod that has no guard on it and thus, has all of its incidnt dgs contaminatd. This implis that th sarchr rachs a cornr. If not, dg wouldn't b th last on on it's sid to b cland. So, a diagonal is formd that conncts a cornr with th sid opposit of it. Th xistnc of such diagonal is guarantd by th fact that only on dirty ara is maintaind during th sarching. In th cas whr a clan action happnd, thn, ithr on or two dgs wr cland. Dnot that dg or pair of dgs by and assum that if is a path of lngth 2 it dos not contain a cornr. Thn, w can idntify two diagonals that hav two of thir ndpoints connctd by and bound th dirty ara. Sinc th othr two sids still hav contaminatd dgs ths diagonals must xist. In th cas whr contains a cornr w can idntify a diagonal from on of th ndpoints of to sid S a (). can b considrd as a scond diagonal but th sum of thir lngths will b at last as larg as th lngth of th diagonal from th cornr to th opposit sid. 2 Up to now, w hav dtrmind ms(d), th minimum numbr of sarchrs that ar rquird in ordr to solv th modid dg sarching problm on a psudo 3-sidd solid orthoconvx grid. By Lmma 3 w know that at most ms(d) + 1 sarchrs can solv th original dg sarching problm. So, w hav a way to approximat s(d) within 1 from th optimum. Unfortunatly, as Figur 14 dmonstrats, thr ar grids that can b sarchd optimally with th sam numbr of sarchrs on both problms. So, in ordr to gt an algorithm that computs s(d) w hav to idntify ths grids. Dirty Clan Fig. 14. A grid D for which ms(d) = s(d) = 5. Dnition 6 W say that at a givn tim t a sarchr is of typ R i if it is at a nod that, at tim t, it has xactly i contaminatd incidnt dgs. Clarly, i is lss than th maximum dgr of th sarchd graph (4 for grids). Lmma 7 If thr is a sarching stratgy that solvs th dg sarching problm on grid D that contains no path of th form (n 1 ; n i ); (n i ; n j ); (n j ; n 2 ) whr dgr(n i ) = dgr(n j ) = 2 using k sarchrs, k > 2; thn thr is a sarching stratgy that solvs th modid dg sarching problm on grid D using at most k? 1 sarchrs. 17

18 Proof. Lt S(D) b a sarching stratgy that solvs th dg sarching problm on grid D using k > 2 sarchrs and in which no rcontamination vr taks plac. Such a stratgy xists 3. This sarching stratgy is also a valid sarching stratgy for th modid dg sarching problm that uss k sarchrs. W can driv from S(D) anothr sarching stratgy for th modid dg sarching problm by insrting aftr ach plac and mov action all th clan actions that ar possibl to occur. In othr words, in th nw stratgy w clan vry dg immdiatly, whn it is possibl. W can also rn this schdul such that it contains no uslss mov actions and no uslss guards bfor any plac, mov or clan actions (as indicatd by Lmmata 1 and 2). Call th nw stratgy S M (D). Obsrv that, no rcontamination occurs in S M (D) sinc no rcontamination occurrd in S(D). Obviously, som actions that cland dgs in S(D) will b liminatd in S M (D) (bcaus th dgs ar alrady clan) and othr will surviv. Each action a i of S(D) that survivd has its corrsponding (an idntical action) a f (i) in S M (D), whr f() is som mapping function. Th two stratgis ar quivalnt and S M (D) is (f(i); i)-suprior of S(D). W claim that S M (D) uss at most k? 1 sarchrs (th sarchrs that w can sav wr pickd up as uslss). To prov it, assum to th contrary that thr ar tims that S M (G) uss k sarchrs. Considr th procss that crats S M (D) and th rst tim that k sarchrs ar usd. Obviously, if w continu from that point th sarch using S(D), w hav a valid stratgy for th modid dg sarching problm. Th action that causs k sarchrs to b on D for th rst tim is a plac action. Furthrmor, w can assum that a plac action cannot b postpond (othrwis w can gt an quivalnt stratgy that \pushs" th plac action as clos as possibl to th nd). Th plac action that put th k th guard on th grid was followd in th initial schdul by a mov action. So, it is placd on a nod that alrady has a sarchr (sinc w assum no uslss sarchrs) and at th nxt stp th nw sarchr will clan an dg. W mak th following obsrvations: 1. Th nw sarchr is not pickd up immdiatly aftr th mov action. If this wr th cas, ithr th cland dg had sarchrs in both of its ndpoints and thus was alrady cland, or on of its ndpoints was a nod of dgr 1 (such nods do not xist on th grid). 2. Th nw sarchr was not placd on a nod containing a sarchr of typ R 1. If this wr th cas, w can gt an quivalnt stratgy that movs th R 1 guard along its adjacnt contaminatd dg and thus, sav on sarchr. 3. Th nw sarchr was placd on a nod that is connctd by a contaminatd dg with a nod of dgr 2. To s that, lt a b th nod that th nw sarchr was placd on and assum that all th nods that ar connctd with a through a contaminatd dg ar of dgr gratr than 2. Obsrv that all incidnt dgs of ths nods ar contaminatd. In this cas th nw sarchr will mov to such a nod and from thr it will not b abl to continu its movmnt sinc, if it dos, it will 18

19 caus rcontamination. Th sam holds for th sarchr that was originally in a. So, S M (D) cannot sarch th whol grid. But this is a contradiction sinc w assumd that S M (D)can sarch it. 4. All th cycls on a grid ar of siz at last 4. So, w can safly assum that th nw sarchr was placd on a nod that has a contaminatd incidnt dg that lads to a path containing xactly on nod of dgr 2 and movs toward this path. Sinc th nod of dgr two dos not contain a sarchr th nw sarchr has to clan its othr dg too. Th nod that th sarchr currntly is, dos not contain a scond sarchr. If this wr th cas, th path would b clan. Morovr, that nod is of dgr gratr than 2, and thus, th sarchr cannot procd. Sinc ach cycl on a grid is of lngth at last 4, th nod that th nw sarchr is locatd is not connctd by an dg with th nod it was originally placd. Thrfor, th sarchr that was in that nod at th bginning cannot mov to join it and triggr a pick action. Thus, th sarchrs that ar on th grid cannot procd to sarch th grid. This is a contradiction sinc w assumd that thy can. 2 Lmma 8 Assum a psudo 3-sidd solid orthoconvx grid D that contains a path of th form (n 1 ; n i ); (n i ; n j ); (n j ; n 2 ) whr dgr(n i ) = dgr(n j ) = 2 and n j is a cornr, and th diagonal from n j to th opposit sid has ms(d) points. Thn, ms(d) = s(d) if and only if thr ar two points in th diagonal with th sam X or Y -coordinat. Proof. \(=" Without loss of gnrality lt dg (n i ; n j ) b paralll to th Y -axis. Dnot th diagonal from n j by d and lt a b th nod it shars with th sid opposit of n j. Assum that thr ar two points p 1 and p 2 on d that hav th sam Y -coordinat. (Th cas whr thy agr on th X-coordinat can b handld similarly.) For simplicity assum that p 1 and p 2 ar adjacnt points on d. (S Figur 15.) Y a X p 1 p 2 b n 1 n i n 2 n j Fig. 15. Th diagonal in th proof of Lmma 8. By rducing th Y -coordinat of th points of d that li btwn p 1 and n j by at most 1, w can crat a nw diagonal from n j to a that contains points n 2 and b. (b = (x n2? 1; y n2 + 1) whr x n2 and y n2 ar th X and Y coordinats of n 2, rspctivly.) W ar abl to do so bcaus, th fact that thr ar two points in th diagonal that hav th sam X or Y -coordinat implis that thr is a sarchr 19

20 of typ R 1 that can triggr th dsird movmnt. Now a sarching stratgy for th original dg sarching problm that sarchs grid D with s(d) = ms(d) sarchrs is th following: 1. Sarch th rgion bllow th diagonal d 0 with nd points a and n 2 using ms(d) sarchrs. Sinc d 0 has ms(d)? 1 points, ms(d) sarchrs ar sucint to solv th original dg sarching problm for th ara bllow it as Lmma 3 indicats. 2. Plac th ms(d) th sarchr on nod n 2 and thn mov it along th path (n 2 ; n j )(n j ; n i )(n i ; n 1 ) to clan it. Now, this sarchr is placd in nod n 1 that is opposit of nod n 2. Nod n 2 is of typ R 1 whil nod n 1 is of typ R 2 or R Clan dg (n 2 ; n 1 ) by moving th sarchr that is in n 2 towards n 1. Now nod n 1 contains 2 sarchrs and w can pick on of thm. 4. Finally, clan th rgion abov th diagonal d 00 from n 1 to a using ms(d) sarchrs. This can b don bcaus d 00 contains ms(d)? 1points. \=)" Rcall th proof of Lmma 7. Th last sarchr that was placd on th grid was placd in a nod n 1 that alrady had a sarchr. n 1 had an adjacnt nod n i of dgr 2 and th sarchr movd to it along dg (n 1 ; n i ). Thn th sarchr lft n i along th othr dg and arrivd at nod n j whil claning dg (n i ; n j ). Sinc th dgr of n j couldn't b 2, th sarchr couldn't mov and th sarching couldn't proccd. Now, it is possibl that dgr(n j ) = 2. In this cas n j is a cornr and th sarchr procds to clan dg (n j ; n 2 ) and arrivs at nod n 2. It bcoms a sarchr of typ R 2 or R 3 and thus it cannot mov. Edg (n 1 ; n 2 ) is contaminatd but th sarch can procd sinc w hav that ms(d) = s(d). Th only way to do so is by moving th sarchr that initially was at nod n 1 along dg (n 1 ; n 2 ). This action will clan th dg and will triggr a pick action. But, in ordr to b abl to mov th sarchr in n 1 w must hav that it is of typ R 1. This implis that at last th nighbor b of n 1 that is symmtrically opposit of n i contains a sarchr. So, b and n 1 blong to th diagonal and hav ithr thir X or Y coordinats qual. 2 Thorm 3 Assum that th minimum numbr of sarchrs that ar rquird to solv th modid dg sarching problm on a psudo 3-sidd solid orthoconvx grid D is ms(d). Thn, in th cas whr thr is a diagonal from a cornr c of D to th opposit sid of lngth ms(d) such that c is nxt to a dgr 2 nod and th diagonal has two points with th sam X or Y -coordinat, s(d) = ms(d). Othrwis, s(d) = ms(d) + 1. Proof. Follows from Thorm 2 and Lmmata 7, 8 and Algorithms for dtrmining s(d) In this sction w prsnt algorithms to dtrmin s(d). Th algorithms ar basd on Thorms 2 and 3. Obsrv that ths thorms suggst also an optimum 20

21 sarching stratgy that consists of O(n) actions whr n is th numbr of nods of th grid D. It is customary to xprss th complxity of an algorithm that dtrmins th minimum numbr of sarchrs that ar rquird to sarch a graph as a function of n, th numbr of nods of th graph. Howvr, for a grid w can dn two nw quantitis: th numbr of boundary nods b and th numbr of turning points m. Obviously m turning nods (givn in clockwis or countr clockwis ordr) can compltly dn th grid and thus it is dsird to xprss th tim complxity of th algorithm that dtrmins s(d) as a function of m. Obsrv that thr ar grids with m = o(b) and b = o(n). A quantity that w hav to comput is th distanc from a boundary nod v to som sid of th grid. In th rst of th papr w assum that th bas of th grid is paralll to th X axis. Informally, w dscrib how this can b don whn th boundary nod is on th rising sid and w want to comput its distanc to th falling sid. All othr cass ar handld in a similar way. W mov from nod v diagonally up (on a lin paralll with l : x = y) until w hit th boundary. If w hit th wantd sid w ar don. If not, w mov to th right at th nxt concav turning point. W thn mov diagonally up, and so on. From th abov discussion, it bcoms obvious that w nd to comput th distanc from any concav turning point to any sid. Th following lmma is important for driving cint algorithms: Lmma 9 Assum a psudo 3-sidd solid grid D that has its bas paralll to th X axis. In ordr to dtrmin ms(d) as Thorm 2 indicats, it is sucint to xamin diagonals that start from i) turning nods, ii) boundary nods that ar adjacnt to turning nods, and iii) th nods that ar at th intrsction of th bas and all lins that pass from concav turning nods on th rising sid and ar paralll with l : y =?x. Proof. If in quation ms(d) = min(cornr distanc; diagonal pair distanc) th trm cornr distanc is th minimum, thn th lmma trivially holds. Th diagonal that contains th minimum numbr of sarchrs sucint to sarch th grid starts from a turning nod (th cornr). So, w will prov th lmma for th cas that ms(d) is obtaind by th diagonal pair distanc trm. In this cas two things may happn. An dg lis on som sid of th grid and two diagonals ar formd that ach starts from on nd-nod of dg, or, a path of two dgs that contains on convx turning nod lis on som sid of th grid and two diagonals ar formd that ach starts from on nd-nod of path. W hav to considr only th rst cas bcaus in th scond cas th diagonals start from nods that ar adjacnt to turning boundary nods. W considr thr cass dpnding on th sid that dg lis on. 1. lis on th rising sid. Lt = (n 1 ; n 2 ): Thr ar two cass to considr: (a) lis on a horizontal sgmnt of th rising sid. 21

22 In this cas is adjacnt to a concav turning nod. If this was not th cas, thn, for th dg 0 = (n 0 1 ; n0 2) that blongs to th sam horizontal sgmnt and is adjacnt to th right nd-nod of that sgmnt, w should hav: d(n 1 ; S b ()) + d(n 2 ; S a ()) > d(n 0 1 ; S b( 0 )) + d(n 0 2 ; S a( 0 )) sinc, d(n 2 ; S a ()) > d(n 0 2 ; S a( 0 )) and d(n 1 ; S b ()) = d(n 0 1 ; S b( 0 )) (Figur 16a). n 2 n 1 n 2 n 1 n 1 n 2 n 3 " a. b. n 2 n 1 n 3 c. n 2 n 1 n 3 d. n 3 n 2 Fig. 16. Th cass in th proof of Lmma 9. n 1 (b) lis on a vrtical sgmnt of th rising sid. W will show that dg 0 = (n 0 1 ; n0 2) which blongs to th sam vrtical sgmnt and is adjacnt to th lowr nd-nod of that sgmnt, satiss th following: d(n 1 ; S b ()) + d(n 2 ; S a ()) d(n 0 1 ; S b( 0 )) + d(n 0 2 ; S a( 0 )) Th abov rlation trivially holds for = 0. So, w can assum that 6= 0. Considr th dg 00 = (n 3 ; n 1 ) that is adjacnt with and bllow it. In th cas that th diagonal from n 2 hits rst th rising sid at a nod that is not a concav turning nod, and thn th falling on, w obsrv that if w considr 00 instad of w hav that: d(n 1 ; S b ()) = d(n 3 ; S b ()) + 1 and d(n 2 ; S a ()) = d(n 1 ; S a ())? 1 (Figur 16b). This implis that dg 00 can b considrd instad. Applying this argumnt inductivly, w will rach th lowr nod of th vrtical sgmnt lis on, or a nw dg, say 1 = (a 1 ; a 2 ) that has th proprty that th diagonal from th nod with th largst Y-coordinat hits th falling sid. 22

23 2 If by applying th abov argumnt w rach th lowr nd-nod, w ar don. Othrwis, w can assum that th diagonal from a 2 to th falling sid dos not hit th rising sid. Obsrv that it cannot hit a non-concav nod of a vrtical sgmnt of th falling sid. If this was th cas, for th dg 0 1 = (a 3 ; a 1 ) w would hav that d(a 1 ; S b ( 1 )) = d(a 3 ; S b ( 0 1)) + 1 and d(a 2 ; S a ( 1 )) = d(a 1 ; S a ( 0 1)) and thus 0 1 would b usd to computs ms(d) instad of 1. So w can safly assum that 1 hits a horizontal sgmnt of th falling sid. Now obsrv that if w considr 0 1 instad of 1 w will gt th sam rsults for ms(d) sinc th summation of th lngths of th two diagonals that start from th two dgs is qual. (Figur 16c). Th proof is similar to th cas whr th diagonal hits th rising sid rst. By applying th abov argumnts inductivly, w gt that it is nough to considr, out of all dgs of th rising sid, only thos that ar adjacnt to turning nods. 2. lis on th falling sid. This cas is tratd in a symmtric way with th prvious on. 3. lis on th bas. Lt = (n 1 ; n 2 ): Considr th diagonal from n 2 to th rising sid (Figur 16d). First w considr th cas whr it hits a vrtical sgmnt of th rising sid. W can considr in th computation of ms(d) th dg immdiatly to th lft of and hav th sam rsults. This is bcaus th distanc to th rising sid is dcrasd by on whil th distanc to th falling sid may b incrasd by on. In th cas whr th diagonal from n 2 to th rising sid hits a horizontal sgmnt of th rising sid w can considr in th computation of ms(d) th dg immdiatly to th right of and hav th sam rsults. This is bcaus th distanc to th rising sid rmains th sam whil th distanc to th falling sid may b dcrasd by on. Th abov argumnts prov that in th computation of ms(d) w hav to considr only th nods that ar at th intrsction of th bas and all lins that pass from concav turning nods on th rising sid and ar paralll with l : y =?x. It is trivial to comput th minimum diagonal btwn any boundary nod and th bas of th grid sinc on minimum diagonal will b paralll to th Y axis. Th nontrivial part is to comput th th minimum diagonal from a concav turning nod that lis on th th rising (falling) sid to th falling (rising) sid. In th following high lvl dscription w assum that th concav turning nod is on th rising sid. In a similar way w trat th cas whr that nod lis on th falling sction. Algorithm 1 /* It computs th lngth of th minimum diagonal from all turning nods v on th rising sid, to th falling sid. */ 23