Path Planning on Grids: The Effect of Vertex Placement on Path Length

Transcription

1 Proceeding, The Elevenh AAAI Conference on Arificial Inelligence and Ineracive Digial Enerainmen (AIIDE-15) Pah Planning on Grid: The Effec of Verex Placemen on Pah Lengh Jame Bailey Craig Tovey School of ISYE Georgia Iniue of Technology Alana, Georgia, USA Tanel Ura Sven Koenig Deparmen of Compuer Science Univeriy of Souhern California Lo Angele, California, USA Alex Nah Norhrop Grumman Informaion Syem Redondo Beach, California, USA Abrac Video-game deigner ofen eellae coninuou - dimenional errain ino a grid of blocked and unblocked quare cell. The hree main way o calculae hor pah on uch a grid are o deermine ruly hore pah, hore verex pah and hore grid pah, lied here in decreaing order of compuaion ime and increaing order of reuling pah lengh. We how ha, for boh verex and grid pah on boh 4-neighbor and 8-neighbor grid, placing verice a cell corner raher han a cell cener end o reul in horer pah. We quanify he advanage of cell corner over cell cener heoreically wih igh wor-cae bound on he raio of pah lengh, and empirically on a large e of benchmark e cae. We alo quanify he advanage of 8- neighbor grid over 4-neighbor grid. Inroducion Video-game deigner ofen eellae coninuou - dimenional errain o form quare grid (Tozour 004). Example include Dawn of War (1 and ) and Company of Heroe (Champandard 010). We aume ha each cell i eiher compleely blocked (grey) or unblocked (whie) and ha he raveral co of unblocked cell are uniform, which i reaonable for many video game. One ough o udy he properie of pah planning on uch grid, o ha videogame deigner can make informed choice. For example, if he ize of game characer i mall relaive o he cell ize, hen i i reaonable o place he verice of grid graph a eiher he cell cener or cell corner. In hi paper, we herefore udy he effec of wo deign deciion, namely verex placemen (a eiher cell cener or cell corner) and grid conneciviy (eiher 4 or 8 neighbor) on one paricular meric of inere, namely he lengh of he reuling pah. The reearch a he Univeriy of Souhern California (Georgia Iniue of Technology) wa uppored by NSF under gran number and (gran number ) and via a Norhrop Grumman fellowhip o Alex Nah. The view and concluion conained in hi documen are hoe of he auhor and hould no be inerpreed a repreening he official policie, eiher expreed or implied, of he ponoring organizaion, agencie or he U.S. governmen. Copyrigh c 015, Aociaion for he Advancemen of Arificial Inelligence ( All righ reerved. The hree main way o calculae hor pah on grid are o deermine ruly hore pah (TSP) in he coninuou errain (Lozano-Pérez and Wele979; Michell and Papadimiriou 1991; Harabor and Graien 013), hore verex pah (SVP, omeime alo called hore anyangle pah ince many any-angle earch algorihm, uch a Block A* (Yap e al. 011), Thea* (Daniel e al. 010) and A* wih Po-Smoohed Pah (Millingon and Funge 009), rive o find uch pah alhough hey are no guaraneed o ucceed) and hore grid pah (), lied here in decreaing order of compuaion ime and increaing order of reuling pah lengh (Ura and Koenig 015). SVP or, repecively, are deermined on he grid graph ha are conruced from he errain by placing verice a eiher all cell cener or all cell corner and hen connecing any wo verice or any wo neighboring verice, repecively, if he line egmen beween hem i unblocked. See (Nah and Koenig 013) for more informaion abou TSP, SVP and and abou earch algorihm baed on A* (Har, Nilon, and Raphael 198) ha deermine hem. For a given verex placemen, canno (by definiion) be horer han he correponding SVP, and SVP canno be horer han he correponding TSP. However, i ha no previouly been known how large he raio among hee pah lengh can be. Our heoreical reul are a complee e of maching (hence igh) upper and lower bound wih repec o he wor-cae raio of he hree kind of pah. Our main conribuion i he analyi of he deign deciion o place verice a cell corner for boh grid conneciviie, epecially ince everal of he reuling bound urn ou o be non-rivial. Table 1 ummarize our heoreical reul. All wor-cae raio are igh (or aympoically igh), ha i, here exi pah-planning problem for which he wor-cae raio have he given value (or are wihin any ɛ > 0 of he given value) bu no pah-planning problem for which he wor-cae raio are larger. Value ha are only aympoically igh are indicaed by an aerik ( ). The able how ha he verex placemen can indeed dramaically affec he wor-cae raio of he lengh of an and he correponding TSP and he wor-cae raio of he lengh of an SVP and he correponding TSP. For example, we prove wor-cae raio of 1.7 (for verice a cell cener) veru 1.41 (for verice a cell corner) on 4-neighbor grid and 1.4 veru 1.08 on 8-neighbor grid. We alo prove ha he 108

2 wor-cae raio beween he lengh of an and he correponding SVP remain, perhap urpriingly, unaffeced by he verex placemen for boh grid conneciviie. Finally, we compare he influence of verex placemen and grid conneciviy empirically on a large e of benchmark cae and verify ha placing verice a cell corner raher han cell cener indeed lead o horer pah lengh. Thee reul provide valuable informaion o video-game deigner when i come o making deign deciion abou verex placemen and grid conneciviy, even if hey ue earch algorihm ha only rive o find TSP, SVP or bu are no guaraneed o ucceed. Overview and Relaed Work Terrain i modeled a a 4-fold ymmeric eellaion of a finie porion of he plane ino uni ide lengh quare cell, each of which i eiher compleely blocked or unblocked. The andard grid graph correponding o he errain ha verice a eiher all unblocked cell cener (reuling in a cener grid graph) or all unblocked cell corner (reuling in a corner grid graph). The edge of he graph connec every verex o each of i 4 or 8 neighboring verice in he main compa direcion, excep ha edge ha would have o queeze beween blocked cell are no permied. Conequenly, we diinguih four cenario, namely 4-corner, 4- cener, 8-corner and 8-cener grid graph. Cardinal direcion edge have lengh 1, and diagonal edge have lengh. We are inereed in compuing he wor-cae raio (,) (,) (hor: ) of he lengh of an and he correponding TSP (on he ame grid and wih he ame (,) ar and goal verice), he wor-cae raio (,) (hor: ) of he lengh of an SVP and he correponding (,) TSP and he wor-cae raio (,) (hor: ) of he lengh of an and he correponding SVP, maximized over all grid ize, ar verice, goal verice and arrangemen of blocked cell, for he four cenario. Some relaionhip are known or obviou: If here exi a TSP, here alway exi a piecewie linear TSP ha urn only a cell corner (Lee 1978; Lozano-Pérez and Wele979). Thu, TSP and SVP are equally long if he verice are placed a cell corner. Conequenly, b = j, d = l and f = h = 1 in Table 1. SVP and TSP do no depend on he grid conneciviy. Conequenly, e = g and f = h in Table 1. Thee relaionhip imply ha only he wor-cae raio a, c, g, i, j, k and l need o be deermined in Table 1. Previou reearch ha udied ome of hee wor-cae raio under differen aumpion. In paricular, wor-cae raio b and d were previouly known o be and 4, repecively, if he pah are allowed o pa hrough diagonallyouching blocked cell, which wa fir analyzed under he aumpion ha all cell are unblocked (Ferguon and Senz 00) and laer generalized o he cae where cell can be blocked or unblocked (Nah 01). Our heoreical reul how ha hee wor-cae raio do no change even under our aumpion ha pah are no allowed o pa hrough diagonally-ouching blocked cell, which i he more realiic aumpion for video game ince game characer have non-zero ize. Alo, ome wor-cae raio become infinie if pah were allowed o pa hough diagonally-ouching blocked cell. For example, Figure 1 how a pah-planning problem on a 4-neighbor grid wih verice a cell cener. The ar verex i, and he goal verex i. The lengh of he TSP would be finie (a hown in Figure 1) while he lengh of he i infinie (ha i, here i no uch pah). The wor-cae raio a would be infinie inead of 1.7. More imporanly, i i quie common o place verice a cell cener, and hi cae ha no been udied before o he be of our knowledge, which i why we analyze wheher verex placemen maer for he wor-cae raio. Figure 1: If movemen beween diagonally-ouching blocked cell were permied, (, ) = bu could no be reached from by a grid pah on he 4-cener grid graph. Poin and Pah Poin are pair of coordinae. Their fir coordinae i he x-coordinae, which increae o he righ. Their econd coordinae i he y-coordinae, which increae o he op. We now define differen kind of pah. Definiion 1. An - pah i a pah in he plane from poin o poin uch ha (i) i i conained in he union of unblocked cell (where he region of a cell include i perimeer); and (ii) may no pa hrough diagonally-ouching blocked cell (ha i, if i pae hrough a cell corner c conained in wo blocked cell ha inerec only a c, i exi c o he ame unblocked cell from which i enered). Definiion. (, ), he lengh of a ruly hore pah (TSP), i he (Euclidean) lengh of a hore - pah. For all inance for which no TSP exi, he lengh of he TSP, SVP and are all infiniy, which i conien wih Table 1. We herefore conider from now on only inance for which TSP exi. For cener grid graph G = (V, E, U), he verex e V conain all cell cener of he unblocked cell. For corner grid graph, V conain all cell corner of he unblocked cell. A poin ha i he cell corner of more han one unblocked cell i repreened a a ingle elemen of V. For 4-neighbor grid, he edge e E i obained by connecing verex v in V wih coordinae (x, y) o each verex w a coordinae (x ± 1, y) and (x, y ± 1) if and only if he line egmen vw beween hem i a v-w pah. For 8-neighbor grid, E conain addiional edge ince v i alo conneced o each verex w a coordinae (x ± 1, y ± 1) if and only if he line egmen vw beween hem i a v-w pah. The decripion of G alo include he li of unblocked cell U. 109

3 Wor-Cae Raio 4-Neighbor Grid 8-Neighbor Grid Verice a Cener Verice a Corner Verice a Cener Verice a Corner (4-Cener Grid Graph) (4-Corner Grid Graph) (8-Cener Grid Graph) (8-Corner Grid Graph) hore grid pah/ruly hore pah ( ) a = /( + ) b = c = d = hore verex pah/ruly hore pah ( SV P ) e = f = 1 g = h = 1 hore grid pah/hore verex pah ( SV P ) i = j = k = l = Table 1: Theoreical reul. Definiion 3. (, ), he lengh of a hore - grid pah (), i he (Euclidean) lengh of a hore - pah in grid graph G. We wan o compare he lengh of - wih hoe of he correponding - TSP and - SVP. The laer are defined in erm of - pah on a differen graph: Definiion 4. Le G = (V, E, U) be a grid graph. The verex graph G = (V, E, U) i defined by edge e E coniing of all (v, w) : v V, w V uch ha he line egmen beween hem i a v-w pah. (, ), he lengh of a hore - verex pah (SVP), i he (Euclidean) lengh of a hore - pah in graph G. For breviy, we omi mo proof of he correcne of he following definiion and properie. All omied proof coni of an enumeraion of he poible e of unblocked cell inciden on a verex, or he reamen of pah parallel o an axi eparae from pah no parallel o an axi. For any verice and, if here exi muliple - TSP, we may elec one of hoe pah and reain in U only a malle e of cell ha permi i o be an - pah. The TSP i hen unique. The addiional blocked cell do no change (, ) and canno decreae (, ) or (, ). Thu, we aume ha he TSP i unique when proving upper bound on he wor-cae raio and. Definiion 5. Le x 1, x,..., x n be he poin where he - TSP urn. For all 1 i n, x i mu be on he cell corner of exacly one blocked cell, denoed F i. There exi a unique unblocked cell B i (he one diagonally-ouching F i ) uch ha F i B i = x i. Define c Bi o be he cell cener of B i. Define x 0 = and x n+1 =. We now inroduce he key idea ha enable u o prove upper bound on he wor-cae raio. We define a rericed cla of grid pah, called rericed grid pah (RGP), ha ay cloe o he TSP. Since any RGP i a grid pah, an i a lea a hor a a hore RGP. Hence any upper bound on he lengh of he laer i an upper bound on he lengh of he former. Informally, an RGP ravere only cell ha he TSP ravere, ravere all verice ha he TSP ravere and alway urn in a cell where he TSP urn. Formally: Definiion. For any verex v, le P (v) be he e of cell ha conain v. Similarly, for any line egmen vw, le P (v, w) be he e of cell ha inerec vw. For 4-corner, 8-corner and 8-cener grid graph, le y i = x i for all i, where x i i a in Definiion 5. For 4-cener grid graph, le y 0 = x 0 =, y n+1 = x n+1 = and y i = c Bi for 1 i n. An - rericed grid pah (RGP) i an - pah in G ha aifie he following condiion: If he TSP conain v V, he RGP mu alo conain v. The RGP divide ino n + 1 y i -y i+1 pah, repecively conained in P (x i, x i+1 ). SRGP (, ) i he (Euclidean) lengh of a hore - rericed grid pah (SRGP). The SRGP and he can be very differen in boh hape and lengh a hown in Figure for a 4-neighbor grid wih verice a cell cener. The dahed pah i he TSP, he doed pah i he, and he olid pah i he SRGP. Poin and y are hoe required in he definiion of he RGP. Figure : The and SRGP have differen hape and lengh for hi 4-cener grid graph. We omi he raighforward proof ha an SRGP alway exi and make repeaed ue of he following imple formula for SRGP (, ) ha demonrae ha, under he condiion given, an can ay cloe o he TSP, reuling in he SRGP and being equally long. An example for he conrucion ha prove Lemma 1 for 8-cener grid graph (Lemma for 4-corner grid graph) i hown in Figure 3 (Figure 4). Lemma 1. For any 8-cener (8-corner) grid graph G, le and be poin in he inerior of cell A and B, repecively, uch ha i an - pah. Le v A and v B be he cell cener (lower lef and upper righ corner, repecively) of A and B, repecively. Le v B v A = (p, q). If p q 0, hen SRGP (v A, v B ) = q + p q. Proof. The lower bound follow ince hi i he lengh of an if here are no blocked cell. By ranlaion aume ha v A = (0, 0) and v B = (p, q) where p q 0. Le b(i, j) be he cell wih cell cener (upper righ corner) a (i, j) where i, j Z. For all j {1,..., q}, le f(j) be he malle ineger uch ha ener he inerior of b(f(j), j) and le g(j) be he maximum of j and f(j). Since p q 0, ener he inerior of b(g(j), j) and mu alo ener he inerior of b(g(j) 1, j 1). The grid pah v A = (0, 0) y 110

4 = (f(j), j) = (g(j), j) Figure 3: SRGP of Lemma 1 for 8-cener grid graph. = (f(j), j) = (g(j), j) Figure 4: SRGP of Lemma for 4-corner grid graph. (g(1) 1, 0) (g(1), 1) (g() 1, 1) (g(), )... (f(q), q) (p, q) = v B i a valid RGP wih lengh q + p q. Lemma. For any 4-cener (4-corner) grid graph G, le,, A, B, v A, v B be a in Lemma 1. If p q 0, hen SRGP (v A, v B ) = p + q. In general, SRGP (v A, v B ) = p + q. Proof. The general reul i obained by ranforming he general cae via roaion, reflecion and ranlaion o he pecial cae p q 0. Therefore, aume ha p q 0. The proof i hen he ame a for Lemma 1 wih he following addiion: For 4-corner grid graph, each (g(i) 1, i 1) (g(i), i) in he pah for 8-corner grid graph i replaced by (g(i) 1, i 1) (g(i), i 1) (g(i), i), which i poible ince b(g(i), i) mu be unblocked. For 4-cener grid graph, each (g(i) 1, i 1) (g(i), i) in he pah for 8-cener grid graph i replaced by eiher (g(i) 1, i 1) (g(i) 1, i) (g(i), i) or (g(i) 1, i 1) (g(i), i 1) (g(i), i), which i poible ince pah are no allowed o queeze beween wo diagonally-ouching blocked cell and hu b(g(i) 1, i) or b(g(i), i 1) mu be unblocked. Tigh Bound for Wor-Cae Raio for 4-Cener Grid Graph The proof of wo of he bound repored here each require muliple cae-by-cae analye. Raher han ummarizing each, we give a fairly deailed proof for he wor-cae raio a in Table 1. Many of he principle employed here can alo be ued o prove he bound for wor-cae raio c in Table 1. Theorem 3 (Wor-Cae Raio a in Table 1). For 4-cener grid graph, he wor-cae raio i a= +. Figure 5: Wor-cae raio for 4-cener grid graph. We fir prove ha, for 4-cener grid graph, he wor-cae raio SRGP i +. Figure 5 eablihe by example he lower bound α = + SRGP on he wor-cae raio. To prove ha hi lower bound i igh (ha i, o prove an upper bound of α), we repeaedly uilize he lower bound. The lower bound i helpful becaue, hroughou he proof, we SRGP (,) (,) conider any inance where α for i ar verex and goal verex ince only uch an inance can poibly achieve he wor-cae raio. We hen ry o move eiher or o anoher verex o reduce (, ) by a lea b > 0 and SRGP (, ) by a > 0. Thi procedure i called α-worening if and only if a b α becaue of he SRGP (,) following propery: Given ha he raio (,) of he inance wa originally a lea α, he procedure ha made he raio no maller while decreaing (, ) by b > 0. Thu, he inance a hand canno be an inance ha come wihin ɛ of he wor-cae raio wih minimal (, ), called a malle inance. Evenually, we will characerize he unique malle inance (up o ymmery) a he one hown in Figure 5. SRGP (,) (,) Conider any inance where α. We proceed by breaking he problem ino cae baed on he number of urn of he - TSP for he inance. Zero Turn Suppoe ha and are he repecive cell cener of cell A and B uch ha = (p, q), and ha he TSP i he line egmen. Then, (, ) = p + q and, by Lemma, a con-, SRGP (, ) = p + q. Thu, radicion. SRGP (,) (,) One or More Turn Le x i indicae he i h urn ha he TSP make, and le y i be a decribed in Definiion. The pair compriing an x i -x i+1 pah and y i -y i+1 pah i a egmen. The pair compriing he -x 1 and - pah i he head. If x n i he la verex where he TSP make a urn, hen he pair compriing he x n - and y n - pah form he ail. Head and Tail Lemma 4. Subjec o roaion and reflecion, every malle inance for 4-cener grid graph mu have he ail hown in Figure if he TSP for he inance ha a lea one urn. Similarly, every malle inance for 4-cener grid graph mu have he head correponding o he ail hown in Figure under he ame condiion. Proof. By revering he role of and i uffice o analyze only he ail. Through roaion and reflecion, we may 111

5 x n y n Figure : Tail for malle inance. aume ha he la urn, x n, i he upper righ cell corner of he blocked cell F n ha caue x n and ha here i a righ urn a x n. For any ail, we may move in order o creae he ail hown in Figure ince every ail require he unblocked cell hown in Figure. Suppoe ha x n = (p, q). Our aumpion imply ha p 1 and q 1, which mean ha p, q 1. (x n, ) = p + q. Conider he poin reached by moving ɛ along x n, and le A and B be he cell conaining and, repecively. Then, SRGP (y n, ) = p + q ince boh SRGP (y n, ) SRGP (v A, ) + 1 = SRGP (v A, v B )+1 = ( p 1/)+( q 1/)+1 = p + q by Lemma and SRGP (y n, ) (y n, ) p + q. By moving o he verex hown in Figure, he TSP i horened by p + q 1 and he SRGP i horened by p + q 1. Thi i α-worening if p + q ince hen p + q 1 < p +q 1 + = α. If p + q = 1, we have he ail hown in Figure. A a reul, a malle inance mu ue only he ail hown in Figure and he head hown in Figure 7. Segmen When examining egmen i, decribed by he x i -x i+1 pah and y i -y i+1 pah, we aume ha here i a righ urn a x i and ha x i i locaed a he upper righ cell corner of F i. We begin by analyzing egmen 1: Lemma 5. Segmen 1, if i exi, in a malle inance for 4-cener grid graph mu be a hown in Figure 8. x 1 Figure 7: Prefix of pah up o x 1. x 1 x y Figure 8: Prefix of pah up o x. Proof. For all egmen excep he one hown in Figure 8, we give an α-worening procedure. By Lemma 4,, x 1 and are locaed a hown in Figure 7. Cae 1, x 1 x i no parallel o he y-axi: Suppoe ha x x 1 = (p, q). We have (x 1, x ) = p + q and SRGP (, y ) = p + q by Lemma. By moving uch ha, x and y form he head hown in Figure, we eenially remove egmen 1 and hereby reduce he TSP by p + q and he SRGP by p + q. Thi procedure i α- worening, implying ha hee ype of egmen canno occur in a malle inance. Cae, Lef urn a x and x 1 x i parallel o he y-axi: Suppoe ha x 1 x = (0, q) and (x 1, x ) = q. Since x 1 x i parallel o he y-axi, he -y SRGP i able o ue only he column conaining or. The -y SRGP ar in he righ column and may zig ino he lef column and zag ino he righ column in order o avoid blocked cell. Since y i in he lef column, he pah mu zig one more ime han i zag. Le m be he number of ime he -y SRGP zig ino he lef column. Then SRGP (, y ) = q + m. If z w and z w+1 denoe wo conecuive urn in he SRGP where z w and z w+1 are in he ame column, hen SRGP (z w, z w+1 ) and hu m q Furhermore, q 3 ince oherwie we canno ravel in a raigh line from x 1 o x. I now follow ha i i α-worening o move uch ha, x and y form he head hown in Figure, implying ha hee ype of egmen canno occur in a malle inance. Cae 3, Righ urn a x and x 1 x i parallel o he y-axi: Suppoe ha x 1 x = (0, q) and (x 1, x ) = q. Again, he -y SRGP i only able o ue he column conaining or. There now mu be an equal number of zig and zag. Uing he ame noaion a before, SRGP (, y ) = q +m+1. Again, SRGP (z w, z w+1 ) and hu m q 1 4, where q 1. I i α-worening o move uch ha, x and y form he head hown in Figure if q, implying ha hee ype of egmen canno occur in a malle inance. If q = 1, he egmen i a hown in Figure 8 and he lemma i proven. Lemma. Segmen, if i exi, in a malle inance for 4-cener grid graph mu be geomerically imilar o egmen 1; in paricular, SRGP (y, y 3 ) = and (x, x 3 ) = 1. The proof of hi lemma follow in he ame fahion a he proof of Lemma 5. The key difference i ha here, in eence, we remove boh egmen 1 and. Then, for any poible egmen, excep he one hown in Figure 9, we are able o give an α-worening procedure. Lemma 7. There i no egmen 3 in a malle inance for 4-cener grid graph. Proof. By Lemmaa 4, 5 and,, x 1,, x, y, x 3 and y 3 are locaed a hown in Figure 9. The cener righ cell canno be blocked ince here could no be a righ urn a x 3 oherwie. In addiion, he lower righ cell hown mu be blocked ince oherwie here i a horer pah from o x 3. Cae 1, x 3 x 4 i no parallel o he y-axi: Suppoe ha x 3 x 4 = (p, q). We have (x 3, x 4 ) = p + q and SRGP (y 3, y 4 ) = p + q by Lemma. I i α-worening o move uch ha, x 4 and y 4 form he head hown in Figure, implying ha hee ype of egmen canno occur in a malle inance. 11

6 y x x 1 x 3 y 3 Figure 9: Prefix of pah up o x 3. Cae, x 3 x 4 i parallel o he y-axi: Becaue he lower righ cell mu be blocked (a argued above), hi cae canno occur in a malle inance. Conrucion of Smalle Inance Under he juified aumpion ha he upper bound of he wor-cae raio SRGP i a lea α = +, we have proved ha, if he TSP ha zero urn, he wor-cae raio i a mo < α. a conradicion. If he TSP ha only one urn, by Lemma 4, he unique malle inance (up o ymmery) can coni of only a head and ail a hown in Figure 10, which reul in he TSP having zero urn a conradicion. If he TSP ha wo urn, by Lemmaa 4 and 5, he unique malle inance (up o ymmery) i a hown 4 in Figure 11, reuling in a wor-cae raio of 1+ < α again a conradicion. If he TSP ha hree urn, hen, by Lemmaa 4, 5 and, he unique malle inance (up o ymmery) i a hown in Figure 5, reuling in a worcae raio of + = α. By Lemma 7, he TSP canno have four or more urn. Thi complee he proof of he upper bound + SRGP on he wor-cae raio. Noe ha Figure 10 and 11 illurae he wor-cae raio if he TSP ha fewer han wo urn and wo urn, repecively, bu we don prove hi propery ince i i no required for he proof of Theorem 3 due o he aumpion ha he upper bound of he wor-cae raio i a lea α. Figure 5 eablihe by example alo he lower bound + on he wor-cae raio. An upper bound on he wor-cae raio SRGP i alo an upper bound on he worcae raio. Thu, he wor-cae raio i i + (a illuraed in Figure 5, which complee he proof of Theorem 3. Figure 10: Smalle inance for 4-cener grid graph if he TSP ha fewer han wo urn. Figure 11: Smalle inance for 4-cener grid graph if he TSP ha wo urn. Tigh Bound for Oher Wor-Cae Raio Theorem 8 (Wor-Cae Raio c and g in Table 1). For 8- cener grid graph, he wor-cae raio c = g = 3 + = 3 3. and are Proof. The proof of he upper bound on he wor-cae raio follow from he ame ep a he proof of Theorem 3. The upper bound on he wor-cae raio i alo an upper bound on he wor-cae raio ince SVP are never longer han he correponding. The lower bound for boh wor-cae raio i hown in Figure 1 and coincide wih he upper bound. Figure 1: Wor-cae raio graph. and for 8-cener grid For Theorem 9 and 10, we aler Definiion 5 and and replace every inance of TSP wih SVP. Specifically, we require in he proof of Theorem 9 and 10 ha he SRGP ravere only cell ha he SVP ravere, ravere all verice ha he SVP ravere and alway urn in a cell where he SVP urn. Theorem 9 (Wor-Cae Raio i and j in Table 1). For 4- cener and 4-corner grid graph, he wor-cae raio i i = j =. Proof. The - SVP of an inance ha come wihin ɛ of he wor-cae raio wih minimal (, ), called a malle inance, canno urn. If i urned a ome verex u, hen he SVP and he SRGP would inerec a SRGP (,) SRGP (,u)+srgp (u,) u. However, = implie SRGP (,u) (,u) (,) SRGP (u,), (,u)+ (u,) SRGP (,) (,) max{ (u,) }, and he inance a hand canno be malle. Conequenly, he - SVP of a malle inance mu be he line egmen. Through roaion, reflecion and ranlaion, we may aume ha = (0, 0) and = (p, q) where p q 0. If q = 0, i a valid grid pah, yielding a wor-cae raio of 1. If q 1, by Lemma, (,) (,) SRGP (,) (,) = p + q p +q. An upper bound on he wor-cae raio SRGP i alo an upper bound on he wor-cae raio. We achieve hi upper bound if all cell are unblocked and p = q. Theorem 10 (Wor-Cae Raio k and l in Table 1). For 8- cener and 8-corner grid graph, he wor-cae raio i aympoically igh a k = l = 4. Proof. A in he proof of Theorem 9, aume ha he - SVP i he line egmen wih endpoin = (0, 0) and = (p, q) where p q 0. If q = 0, i a valid grid 113

7 4-Neighbor Grid 8-Neighbor Grid Game Random Game Random (cener) average (corner) (cener) average (cener) (corner) average (corner) (cener) wor-cae (cener) (corner) wor-cae (corner) % (cener) > (corner) 57.% 7.34% 59.58% 74.43% % (corner) > (cener) 0.00% 0.00% 0.00%.71% Table : Experimenal reul. pah, yielding a wor-cae raio of 1. If q 1, by Lemma (,) SRGP (,) 1, (,) (,) = p q+p q = +(β 1) β where +q +1 β = p/q. Seing he derivaive o zero, we find ha he raio i maximized a β = 1+, wih value = 4. An upper bound on he wor-cae raio SRGP i alo an upper bound on he wor-cae raio. We ge arbirarily cloe o hi upper bound if all cell are unblocked and p = ( + 1)q a q grow large. Experimenal Reul We compare he lengh of pah on boh random and game map in Nahan Surevan repoior. For cener grid graph, he ar and goal verice are placed a cell cener. For corner grid graph, he ar and goal verice are hifed o he upper lef cell corner. Table repor he average raio of he lengh of an on a cener grid graph and he on he correponding corner grid graph, he average and wor-cae raio of he lengh of he and he correponding TSP (eparaely for cener and corner grid graph), and he percenage of inance where he on a cener grid graph i ricly longer or horer han he on he correponding corner grid graph. All raio in Table how ha, on average and in he wor-cae, on cener grid graph are longer han on he correponding corner grid graph. In fac, hey are on average 0.193%-3.791% longer, which make ene ince hey canno hug obacle formed by blocked cell a cloely. They are never ricly horer, excep for.71% of he inance for 8-neighbor random map (which i due o he lighly differen ar and goal verice on he correponding cener and corner grid graph). However, hey are ricly longer for 57.%-74.43% of he inance. The random map conain many mall obacle formed by blocked cell, while he game map coni of large empy pace (ince he game map model he errain bu no building or uni), which migh explain why he diadvanage of an on a cener grid graph over he on he correponding corner grid graph i ypically larger for random map han game map. We expec he gap o decreae a he game map are populaed wih building and uni. Similarly, all raio in Table how ha, on average and in he wor-cae, on 4-neighbor grid are longer han on he correponding 8-neighbor grid, which make ene ince hey ravere 1 hp://movingai.com/benchmark/ obacle-free area le effecively and canno hug obacle formed by blocked cell a cloely. For all combinaion of verex placemen and grid conneciviy, he experimenal wor-cae raio on random map agree o 4 ignifican digi wih he correponding heoreical wor-cae raio in Table 1. Concluion We have deermined how he wor-cae raio beween he lengh of a ruly hore pah, a hore verex pah and a hore grid pah vary. Our heoreical reul (Table 1) quanify he advanage of placing verice a cell corner raher han cell cener when finding hore grid pah on 4- and 8-neighbor quare grid. Our experimenal reul (Table ) confirm hee qualiaive relaionhip. I i fuure reearch o exend our reul o oher eellaion, uch a riangle, hexagon and exe (Björnon e al. 003). Reference Björnon, Y.; Enzenberger, M.; Hole, R.; Schaeffer, J.; and Yap, P Comparion of differen grid abracion for pahfinding on map. In Proceeding of he Inernaional Join Conference on Arificial Inelligence, Champandard, A Peronal Communicaion. Daniel, K.; Nah, A.; Koenig, S.; and Felner, A Thea*: Any-angle pah planning on grid. Journal of Arificial Inelligence Reearch 39: Ferguon, D., and Senz, A. 00. Uing inerpolaion o improve pah planning: The Field D* algorihm. Journal of Field Roboic 3(): Harabor, D., and Graien, A An opimal any-angle pahfinding algorihm. In Proceeding of he Inernaional Conference on Auomaed Planning and Scheduling, Har, P.; Nilon, N.; and Raphael, B A formal bai for he heuriic deerminaion of minimum co pah. IEEE Tranacion on Syem Science and Cyberneic 4(): Lee, D.-T Proximiy and Reachabiliy in he Plane. Ph.D. Dieraion, Univeriy of Illinoi a Urbana-Champaign. Lozano-Pérez, T., and Weley, M An algorihm for planning colliion-free pah among polyhedral obacle. Communicaion of he ACM (10): Millingon, I., and Funge, J Arificial Inelligence for Game. Morgan Kaufmann, econd ediion. Michell, J., and Papadimiriou, C The weighed region problem: Finding hore pah hrough a weighed planar ubdiviion. Journal of he ACM 38(1): Nah, A., and Koenig, S Any-angle pah planning. Arificial Inelligence Magazine 34(4): Nah, A. 01. Any-Angle Pah Planning. Ph.D. Dieraion, Univeriy of Souhern California. hp://idm-lab.org/projec-o.hml. Tozour, P Search pace repreenaion. In Rabin, S., ed., AI Game Programming Widom. Charle River Media Ura, T., and Koenig, S An empirical comparion of anyangle pah-planning algorihm. In Proceeding of he Sympoium on Combinaorial Search. Yap, P.; Burch, N.; Hole, R.; and Schaeffer, J Any-angle pah planning for compuer game. In Proceeding of he Conference on Arificial Inelligence and Ineracive Digial Enerainmen. 114