Optimal Interleaving on Tori

Transcription

1 Opimal Inerleaving on Tori Anxiao (Andrew) Jiang, Mahew Cook, and Jehoshua Bruck California Insiue of Technology Parallel and Disribued Sysems Laboraory MC - Pasadena, CA, U.S.A. {jax,cook,bruck}@paradise.calech.edu Absrac This paper sudies -inerleaving on wo-dimensional ori, which is defined by he propery ha every conneced subgraph of order in he orus is labelled by disinc inegers. This is he firs ime ha he -inerleaving problem is solved for graphs of modular srucures. -inerleaving on ori has applicaions in disribued daa sorage and burs error correcion, and is closely relaed o Lee meric codes. We say ha a orus can be perfecly -inerleaved if is -inerleaving number he minimum number of disinc inegers needed o -inerleave he orus mees he sphere-packing lower bound. We prove he necessary and sufficien condiions for ori ha can be perfecly - inerleaved, and presen efficien perfec -inerleaving consrucions. The mos imporan conribuion of his paper is o prove ha when a orus is large enough in boh dimensions, is -inerleaving number is a mos one more han he sphere-packing lower bound, and o presen an opimal and efficien -inerleaving scheme for such ori. Then we prove bounds for he -inerleaving numbers of he remaining cases, compleing a general characerizaion of he -inerleaving problem on -dimensional ori. Index Terms Burs, chromaic number, cluser, error-correcing code, mulidimensional inerleaving, -inerleaving, orus. I. INTRODUCTION Inerleaving is an imporan echnique used for error burs correcion and nework daa sorage. A mos common example is he inerleaving of n codewords in he form of n n for combaing one-dimensional error burss in communicaion channels []. The concep of one-dimensional error burs was generalized o high dimensions by Blaum, Bruck and Vardy in [], where an error burs of size is a se of errors confined o a conneced subgraph of order in a muli-dimensional array. (The order of a graph is defined o be he number of verices in ha graph.) Accordingly, he concep of -inerleaving was defined in [], which is a scheme o label he verices of a muli-dimensional array wih inegers in such a way ha every conneced subgraph of order is labelled by disinc inegers. -inerleaving schemes on wo- and hree-dimensional arrays were presened in [], wih applicaions in combaing error burss in holographic sorage sysems and opical recording sysems. Subsequen work on -inerleaving includes [], where -inerleaving on circulan graphs wih wo offses was sudied, and [], where a dual problem of -inerleaving on wo-dimensional arrays was explored. The problem of wo-dimensional inerleaving wih repeiions was inroduced in [] by Blaum, Bruck and Farrell, and was exensively sudied in [] by Ezion and Vardy. Tha problem is o inerleave inegers on a wo-dimensional mesh (array or is variaion) in such a way ha in every conneced subgraph of order, each ineger appears a mos r imes. Here and r are given parameers, and he concep of inerleaving wih repeiions is a generalizaion of -inerleaving. More work on inerleaving wih repeiions includes [] and []. Inerleaving schemes on wo-dimensional arrays achieving he Reiger bound were sudied by Abdel-Ghaffar in [], where error burss of boh recangular shapes and arbirary conneced shapes were considered. More examples of inerleaving for coping wih error burss include [] and [], where he error burss are respecively of circular ypes and recangular shapes. As o inerleaving schemes for nework daa sorage, in [], an algorihm was presened o

2 inerleave N inegers on a ree whose edges have lenghs, in such a way ha for every poin of he ree (including a verex or a poin on an edge), he smalles ball cenered a he poin ha conains a leas N inegers conains all he N disinc inegers. Tha algorihm is useful for disribued daa sorage in hierarchical neworks ha minimizes daa rerieval delay. A relaed inerleaving algorihm aiming a he graceful degradaion of daa-sorage performance in fauly environmens was presened in []. In [], a scheme called muli-cluser inerleaving was sudied, which is a scheme o inerleave inegers on a pah or a cycle such ha every m disjoin inervals of lengh L in he pah or cycle ogeher conain a leas K disinc inegers, where K > L. Muli-cluser inerleaving can be used for daa sorage on array-neworks, ring-neworks or disks where daa are accessed hrough muliple access poins. In his paper, we sudy -inerleaving on wo-dimensional ori. I is he firs ime ha he -inerleaving problem on graphs of modular (wrapping-around) srucures is solved. Torus is an imporan nework srucure for parallel and disribued sysems [], [], [], []. -inerleaving on ori has applicaions in boh burs error correcion and disribued daa sorage, in he same way as inroduced in [], [], [], [] and []. (Specifically, for disribued daa sorage, a -inerleaving on a - dimensional orus ensures ha for every verex, he inegers assigned wihin hops are all disinc.) -inerleaving on ori is also closely relaed o a research opic in coding heory called Lee meric codes [], [], [], [], [], [], [], [], [], [], [], [], [], []. In a -inerleaved n-dimensional orus, every se of verices labelled by he same ineger is a Lee meric code of lengh n whose minimum disance is ; and he se of Lee meric codes corresponding o differen inegers pariion he whole code space. Below we presen he definiions. -inerleaving was originally defined in [] for arrays. We generalize is noion for general graphs sraighforwardly. Definiion.: Le G be a graph. We say ha G is inerleaved (or here is an inerleaving on G) if every verex of G is labelled by one ineger. We say ha G is -inerleaved (or here is a -inerleaving on G) if every conneced subgraph of G of order is labelled by exacly disinc inegers. The classic verex coloring problem is clearly also a -inerleaving problem, where =. On he oher hand, -inerleaving a graph G is he same as verex-coloring he power graph G. Deermining he chromaic number of a power graph is difficul in general. To he bes of our knowledge, no resul on he ype of graphs we are ineresed in has appeared in he lieraure. Definiion.: A wo-dimensional l l orus is a graph conaining l l verices and l l edges. We denoe is verices by (i, j) for i l and j l, in he way shown in he figure below: (, ) (, ) (, l ) (, ) (, ) (, l ) (l, ) (l, ) (l, l ) Each verex (i, j) is inciden o four edges, which connec i o is four neighbors ((i ) mod l, j), ((i + ) mod l, j), (i, (j ) mod l ) and (i, (j + ) mod l ). Now we can define he problem of -inerleaving on ori. Definiion.: Given a -inerleaved orus G, he number of disinc inegers used o label he verices of G is called he degree of his given -inerleaving scheme. The minimum degree of all he possible -inerleaving schemes for G is called he -inerleaving number of G. A -inerleaving on a orus whose degree equals he orus -inerleaving number is called an opimal -inerleaving. Example.: The following orus is -inerleaved wih degree.

3 S S S S S S Fig.. Six examples of spheres. If we replace he wo inegers wih, we will ge a -inerleaving wih degree. Consider he verex (, ) and is four neighbors (, ), (, ), (, ) and (, ), and noice ha any wo of hem are conained in a conneced subgraph of order herefore any -inerleaving scheme has o label hose verices wih disinc inegers. So he -inerleaving number of his orus acually equals. Our objecive is o find opimal -inerleaving. To do ha, i is imporan o obain he -inerleaving numbers of ori. A universal lower bound of hem, for ori ha have a leas rows and columns, can be obained as follows. Figure shows six subgraphs of a orus, which we call spheres S, S,, S, respecively. In general, for any, he sphere S is obained by aaching o he sphere S all he verices adjacen o i. Any wo verices in S are conneced by a pah of a mos edges, so a -inerleaving needs o label hem wih differen inegers. So he order of S, which we shall denoe by S, ses a universal lower bound for he -inerleaving number. This argumen was originally proposed in [] for sudying -inerleaving on arrays. A direc calculaion ells us ha S = + when is odd, and S = when is even. We define perfec -inerleaving o be a -inerleaving whose degree equals S, he universal lower bound, on a orus ha has a leas rows and columns. (A orus ha does no saisfy ha condiion has only a very limied number of rows or columns; in his paper, we do no discuss he perfecness of inerleaving for hose ori.) We will show ha a orus can be perfecly inerleaved if and only if is sizes in boh dimensions are muliples of a cerain funcion of. Then wha abou ori of oher sizes? Our main resul will show ha when a orus is sufficienly large in boh dimensions, is -inerleaving number exceeds he lower bound S by a mos one. A more deailed descripion of our resuls is as follows: We prove ha an l l orus can be perfecly -inerleaved if and only if he following condiion is saisfied: when is odd (respecively, even), boh l and l are muliples of + (respecively, ). We reveal he close relaionship beween perfec -inerleaving and perfec sphere packing, and presen he complee se of perfec sphere packing consrucions. Based on ha, we obain a se of efficien perfec -inerleaving consrucions, which include he laice inerleaver scheme presened in [] as a special case. We prove ha for any orus ha is sufficienly large in boh dimensions, is -inerleaving number is eiher S or S + ha is, a mos one more han he degree of perfec -inerleaving. More specifically, here exis ineger pairs (θ, θ ) such ha whenever l θ and l θ, he -inerleaving number of an l l orus is a mos S +. Here θ and θ depend on, and naurally, here is a radeoff beween hem if θ akes a greaer value, hen he minimum value ha θ can ake decreases or remains he same, and vice versa. We find a sequence of valid values for θ and θ, which are shown in Theorem and Theorem. We presen opimal -inerleaving consrucions for ori whose sizes exceed he found pairs (θ, θ ). (And we commen ha hose consrucions, as a general inerleaving mehod, can also be used o opimally -inerleave ori of many oher sizes.) We sudy upper bounds for -inerleaving numbers, and show ha every l l orus -inerleaving number is S +O( ).

4 Boundary curve of Region I l Region I Region II Region III l Fig.. A qualiaive illusraion of he -inerleaving numbers. Tha upper bound is igh, even if l + or l +. When boh l and l are of he order Ω( ), he -inerleaving number of an l l orus is S + O(). The resuls can be illusraed qualiaively as Fig.. (The figure is no quaniaive. The coordinaes of poins, such as he shape of he curve, are no exac.) Fig. shows for any given, how he l l ori can be divided ino differen classes based on heir -inerleaving numbers. The uniform laice of dos in Fig. are he sizes of all he ori ha can be perfecly -inerleaved. The region labelled as Region I consiss of all he ineger pairs (θ, θ ). The boundary curve of Region I is non-increasing, and symmeric wih respec o he line l = l. We know he exac -inerleaving number of every orus in his region S if i is one of he laice dos, and S + oherwise. The mos imporan conribuion of his paper is o prove he exisence of Region I, and presen he corresponding opimal inerleaving consrucions. Region II is he region where l = Ω( ) and l = Ω( ), in which he ori s -inerleaving numbers are upper-bounded by S + O(). Region III includes every orus, where he -inerleaving number is upper-bounded by S + O( ). Tha upper bound for Region III is igh, even if l or l approaches +. (So increasing a orus size in only one dimension does no help reduce he -inerleaving number very effecively in general.) The res of he paper is organized as follows. In Secion II, we show he necessary and sufficien condiions for ori ha can be perfecly -inerleaved, and presen perfec -inerleaving consrucions based on perfec sphere packing. In Secion III, we presen a -inerleaving mehod, wih which we can -inerleave large ori wih a degree wihin one of he opimal. In Secion IV, we improve upon he -inerleaving mehod shown in Secion III, and presen opimal -inerleaving consrucions for ori whose sizes are large in boh dimensions. As a parallel resul, he exisence of Region I is proved. In Secion V, we prove some general bounds for he -inerleaving numbers. In Secion VI, we conclude his paper. II. PERFECT -INTERLEAVING In his secion, we show he close relaionship beween perfec -inerleaving and perfec sphere packing, and use i o prove he necessary and sufficien condiion for ori o have perfec -inerleaving. We presen he complee se of perfec sphere packing consrucions. Based on hem, we derive efficien perfec -inerleaving consrucions.

5 (a) S S S S S S (b) (,) S S (,) Fig.. Examples of he sphere S. A. Perfec -Inerleaving and Sphere Packing Definiion.: The Lee disance beween wo verices in a orus is he number of edges in he shores pah connecing hose wo verices. For wo verices in an l l orus G, (a, b ) and (a, b ), he Lee disance beween hem is denoed by d((a, b ), (a, b )). (Therefore, d((a, b ), (a, b )) = min{(a a ) mod l, (a a ) mod l } + min{(b b ) mod l, (b b ) mod l }.) Occasionally, in order o emphasize ha he wo verices are in G, we also denoe i by d G ((a, b ), (a, b )). Clearly, an inerleaving on a orus is a -inerleaving if and only if he Lee disance beween any wo verices labelled by he same ineger is a leas. The following is a more deailed definiion of spheres, compared o he one in he Inroducion secion. Definiion.: Le G be an l l orus where l + and l, and le (a, b) be a verex in G. When is odd, he sphere cenered a (a, b), S (a,b), is defined o be he subgraph induced by all hose verices whose Lee disance o (a, b) is less han or equal o. When is even, he sphere lef-cenered a (a, b), S(a,b), is defined o be he subgraph induced by all hose verices whose Lee disance o eiher (a, b) or (a, (b + ) mod l ) is less han or equal o. (a, b) is called he cener of S (a,b) if is odd, or he lef-cener of S (a,b) if is even. If we do no care where he sphere is cenered or lef-cenered, hen he sphere is simply denoed by S. The number of verices in he sphere is denoed by S. Example.: Fig. (a) shows he spheres S o S. Fig. (b) shows wo spheres, S (,) and S (,), in a orus. For any l l orus where l and l, is -inerleaving number is a leas S. We call S he sphere packing lower bound. The relaionship beween his bound and sphere packing will become clearer soon. Definiion.: A orus G is said o have a perfec packing of spheres S if spheres S are packed in G in such a way ha every verex of G lies in exacly one of he spheres. Lemma : () Le be odd. An inerleaving on an l l orus (where l and l ) is a -inerleaving if and only if for any wo verices (a, b ) and (a, b ) ha are labelled by he same ineger, he wo spheres cenered a hem, S (a,b) and S (a,b ), do no share any common verex. () Le be even. An inerleaving on an l l orus (where l and l ) is a -inerleaving if and only if for any wo verices (a, b ) and (a, b ) ha are labelled by he same ineger, he wo spheres wih hem as lef-ceners, S (a,b) and S (a,b), do no share any common verex and wha s more, b b or (a a ) ±( ) mod l. Proof: () Le be odd. Boh S (a,b) and S (a,b) are classic spheres wih radius. If he inerleaving is a -inerleaving, mus have no inersecion. hen he Lee disance beween (a, b ) and (a, b ) is a leas = +, so S(a,b ) and S (a,b ) The converse is also rue. () Le be even. We consider wo cases b = b and b b.

6 Firs consider he case b = b. In his case, S (a,b ) and S (a,b ) have no inersecion if and only if d((a, b ), (a, b )) ( )+ =. And d((a, b ), (a, b )) = if and only if (a a ) ±( ) mod l. So he Lee disance beween (a, b ) and (a, b ) is a leas if and only if S (a,b) and S (a,b) have no inersecion and (a a ) ±( ) mod l, which is he conclusion we wan. Now consider he case b b. In his case, he Lee disance beween (a, b ) and (a, b ) is a leas boh he Lee disance beween (a, (b + ) mod l ) and (a, b ) and he Lee disance beween (a, (b + ) mod l ) and (a, b ) are a (a,(b+) mod l) leas S does no inersec S (a,b) (a,(b+) mod l) and S does no inersec S (a,b) S (a,b) and S (a,b ) have no inersecion. (Noe ha S (a,b ) is he union of S (a,b ) and S (a,(b +) mod l ), and S (a,b ) is he union of S (a,b ) and S (a,(b +) mod l ).) So we ge he conclusion we wan. Theorem : For an l l orus where l and l, if an inerleaving on i is a perfec -inerleaving, hen for every ineger, he spheres S cenered or lef-cenered a he verices labelled by ha ineger form a perfec sphere packing in he orus. The converse is also rue when. Proof: Le s say ha he orus is inerleaved. We used I o denoe he se of disinc inegers used by he inerleaving. For any ineger i I, we use N i o denoe he number of verices labelled by i. Le s firsly prove one direcion. Assume ha he inerleaving is a perfec -inerleaving. Then I = S. By Lemma, for any i I, he spheres S cenered or lef-cenered a verices labelled by i do no overlap. By couning he number of verices in he orus and in each sphere S, we ge N i l l S for any i I. Since i I N i = l l, we ge N i = l l S for any i I. So for any ineger i I, he spheres S cenered or lef-cenered a he verices labelled by i form a perfec sphere packing in he orus. Now le s prove he converse direcion. Assume. And assume for every ineger, he spheres S cenered or lefcenered a he verices labelled by ha ineger form a perfec sphere packing in he orus. Then N i = l l S for any i I. Since i I N i = l l, we ge I = S. Wha is lef o prove is ha he inerleaving is a -inerleaving. By Lemma, he inerleaving can fail o be a -inerleaving only if he following siuaion becomes rue: is even, and here exis wo verices (a, b ) and (a, b ) labelled by he same ineger such ha b = b and a a mod l. We will show ha such a siuaion canno happen. Assume ha siuaion happens. Then i is sraighforward o verify ha he following four verices (a ( ) mod l, b ), (a +( ) mod l, b ), (a ( ) mod l, b mod l ), (a +( ) mod l, b mod l ) are conained in eiher S (a,b) or S (a,b), while he following wo verices (a ( ) mod l, b mod l ) and (a + ( ) mod l, b mod l ) are neiher conained in S (a,b) nor in S (a,b). The wo verices, (a ( ) mod l, b mod l ) and (a + ( ) mod l, b mod l ), canno boh be conained in spheres S ha are lef-cenered a verices labelled by he same ineger which labels (a, b ) and (a, b ), because hey are verically adjacen, and he verices direcly above hem, below hem and o he righ of hem are all conained in wo spheres ha do no conain hem. (To see ha, observe he shape of a sphere.) Tha conradics ha fac ha all he spheres S lef-cenered a he verices labelled by he ineger which labels (a, b ) form a perfec sphere packing in he orus. So he assumed siuaion canno happen. By summarizing he above resuls, we see ha he inerleaving mus be a perfec -inerleaving. Theorem : For an l l orus where l and l, if i can be perfecly -inerleaved, hen he spheres S can be perfecly packed in i. The converse is also rue when. Proof: Le G be an l l orus. For any, Theorem has shown ha if G can be perfecly -inerleaved, hen he spheres S can be perfecly packed in i. Now we prove he oher direcion. Assume, and he spheres S can be perfecly packed in G. Le (x, y ), (x, y ),, (x n, y n ) be a se of verices such ha he spheres S cenered or lef-cenered a hem form a perfec packing in G. The proof of Theorem has essenially showed ha for any i and j (i j), he Lee disance beween

7 (x i, y i ) and (x j, y j ) is a leas. Now we can inerleave G is his way: label each sphere S wih S disinc inegers such ha every ineger is used exacly once in every sphere, and make all he spheres o be labelled in he same way (namely, all he spheres have he same inerleaving paern ). Clearly, for any wo inegers a and b, he wo ses of verices respecively labelled by a and b are coses of each oher in he orus herefore he Lee disance beween any wo verices labelled by he same ineger is a leas. So G has a perfec -inerleaving. B. Perfec -Inerleaving and Is Consrucion The following lemma is an imporan propery of perfec sphere packing. I will help us derive he necessary and sufficien condiion for perfec -inerleaving. Lemma : Le be even and. When spheres S are perfecly packed in an l l orus, here exiss an ineger a {+, }, such ha if here is a sphere lef-cenered a he verex (x, y), hen here are wo spheres respecively lefcenered a ((x ) mod l, (y a ) mod l ) and ((x + ) mod l, (y + a ) mod l ). Proof: Assume spheres S are perfecly packed in an l l orus, where and is even. Firsly, we will show ha l. Since is even, a sphere S spans rows. So l. Now we show why l. Fig. (a) shows wo examples he firs example shows a sphere S in a orus of rows, and he second example shows a sphere S in a orus of rows. (The verices in he wo spheres are indicaed by relaively large black dos in he figure.) Considering he shapes of he spheres, we can easily see ha he wo adjacen verices in each dashed circle canno be boh conained in non-overlapping spheres. Such a phenomenon always happens when l =. Since here spheres S are perfecly packed in he orus, we ge l. Clearly, one of he following wo cases mus be rue: Case : whenever here is a sphere lef-cenered a a verex (x, y), here are four spheres respecively lef-cenered a he four verices ((x ) mod l, (y ) mod l ), ((x ) mod l, (y + ) mod l ), ((x + ) mod l, (y ) mod l ) and ((x + ) mod l, (y + ) mod l ). Case : here exiss a sphere lef-cenered a a verex (x, y ), such ha here is no sphere lef-cenered a a leas one of he following four verices ((x ) mod l, (y ) mod l ), ((x ) mod l, (y + ) mod l ), ((x + ) mod l, (y ) mod l ) and ((x + ) mod l, (y + ) mod l ). If Case is rue, hen he conclusion of his lemma obviously holds. From now on, le us assume ha Case is rue. WLOG (wihou loss of generaliy), we assume ha here is one sphere lef-cenered a (x, y ), bu here is no sphere lef-cenered a ((x ) mod l, (y + ) mod l ). (All he oher possible insances can be proved wih he same mehod.) Since l, he verex ((x ) mod l, (y + ) mod l ) which we shall call verex A is no conained in he sphere lef-cenered a (x, y ). (An example is shown in Fig. (b), where he sphere in consideraion is an S wih =, whose lef-cener (x, y ) is labelled by C. The verex A is labelled by A.) The verex A is conained in one of he perfecly packed spheres, which we shall call sphere B. The relaive posiion of verex A in sphere B can only be one of he following wo possibiliies: Possibiliy : he verex A is he righ-mos verex in he boom row of he sphere B. (See Fig. (a).) Possibiliy : he verex A is in he down-lef diagonal of he border of he sphere B, bu i is no he lef-mos verex of he sphere B. (See Fig. (b), (c) and (d).) Possibiliy, however, can be easily found o be impossible, since oherwise he neighboring verex o he righ of verex A and he verex below i canno boh be conained in non-overlapping spheres. (See he wo verices in he dashed circle in Fig. (a).) So only possibiliy is rue. In he following proof we use he example of = for illusraion, and assume ha he relaive posiion of he sphere B is as shown in Fig. (b). We commen ha when akes oher values or when he sphere B akes oher relaive posiions, he following argumen sill holds, which will be easy o see.

8 (a) (b) A C Fig.. A sphere in a orus. (a) (b) A A C C (c) (d) A A C C Fig.. Relaive posiions of spheres and verices. Le he sphere lef-cenered a (x, y ) be he sphere denoed by L in Fig., and le sphere B be he sphere now denoed by R in Fig.. We immediaely see ha he verex denoed by E mus be he righ-mos verex of a sphere, so he sphere conaining he verex E mus be he sphere denoed by L. Then we immediaely see ha he verex denoed by F mus be he righ-mos verex in he boom row of a sphere, so he sphere conaining he verex F mus be he sphere denoed by R. Wih he same mehod we can fix he posiions of a series of spheres L, L, L, L, and a series of spheres R, R, R, R,. Since he orus is finie, we will ge a series of spheres L, L, L, L,, L n such ha he relaive posiion of L n o L is he same as he relaive posiion of L o L (see Fig. for an illusraion) so such a series of spheres form a cycle in he orus. Since he spheres are perfecly packed in he orus, no wo spheres in his cycle overlap. Similarly, he spheres R, R,, R n also form a cycle in he orus. (Noe ha we do no make any assumpion abou wheher hese wo cycles overlap or no.)

9 R R L H R D J L G F R D I L E A R n D L C D n L n Fig.. The packing of spheres in a orus. If hose wo cycles conain all he spheres in he orus, hen we are already very close o he end of his proof. If hose wo cycles do no conain all he spheres in he orus, hen here mus be some spheres ouside he wo cycles ha are direcly aached o he down-lef side of he cycle formed by L, L,, L n. (Consider he very regular way he cycle is formed, and he resuling shape of he cycle which is invarian o horizonal and verical shifs.) Le D be a sphere direcly aached o he cycle formed by L, L,, L n, as shown in Fig.. (Noe ha we do no care abou he exac posiion of D, as long as i is direcly aached o he down-lef side of he cycle.) Then he verex I immediaely deermines ha he sphere conaining i mus be D ; similarly he verex J deermines he posiion of he sphere D ; and so on So we will ge a series of spheres D, D, D,, D n which will again form a cycle. (I is easy o see ha his cycle does no overlap he previous wo cycles.) Wih he same mehod as above, we will find more and more cycles, unil hey ogeher conain all he spheres in he orus. We can easily see ha in each of he cycles here, if here is a sphere lef-cenered a a verex (x, y), hen here are wo spheres respecively lef-cenered a ((x ) mod l, (y ) mod l ) and ((x + ) mod l, (y + ) mod l ). When oher insances of Case are rue (see he definiion of Case in previous ex), i can be shown in he same way ha whenever here is a sphere lef-cenered a a verex (x, y), here are wo spheres respecively lef-cenered a ((x ) mod l, (y + ) mod l ) and ((x + ) mod l, (y ) mod l ). By summarizing he above conclusions, we see ha his lemma is proved. Definiion.: Le be an even posiive ineger, le a be eiher + or, and le G be an l l orus. Le (x, y) be an arbirary verex in G. We define he cycle conaining (x, y) (corresponding o he parameer a) o be he se of spheres S ha are respecively lef-cenered a he verices (x, y), ((x+ ) mod l, (y+a ) mod l ), ((x+ ) mod l, (y+a ) mod l ), ((x + ) mod l, (y + a ) mod l ),

10 The proof of he following lemma is omied due o is simpliciy. Lemma : Le be an even posiive ineger, le a be eiher + or, and le G be an l l orus. For any verex (x, y) in G, he cycle conaining i (corresponding o he parameer a) consiss of lcm(l,l, ) disinc spheres S. The following heorem shows he necessary and sufficien condiion for ori ha can be perfecly -inerleaved. Theorem : Le G be an l l orus where l and l. If is odd, hen G can be perfecly -inerleaved if and only if boh l and l are muliples of +. If is even, hen G can be perfecly -inerleaved if and only if boh l and l are muliples of. Proof: We consider he following hree cases one by one: Case : =. Case : is even bu. Case : is odd. Case : =. In his case, -inerleaving is equivalen o verex coloring, so he -inerleaving number of G equals G s chromaic number χ(g). Le R and R be wo rings which respecively have l and l verices. Then G is he Caresian produc of hose wo rings, namely, G = R R. I is well known [] ha for any wo graphs H and H, χ(h H ) = max{χ(h ), χ(h )}. Since l = (respecively, l = ), we ge ha χ(r ) (respecively, χ(r ) ); and χ(r ) = (respecively, χ(r ) = ) if and only if l (respecively, l ) is a muliple of. So χ(g) = if and only if boh l and l are muliples of. Since S =, we ge he conclusion in his lemma. Case : is even bu. Firsly, we prove one direcion. Assume G can be perfecly -inerleaved. We will show ha boh l and l are muliples of. Le i be an ineger used by a perfec -inerleaving on G. Then by Theorem, he spheres S lef-cenered a he verices labelled by i form a perfec sphere packing in G. By Lemma, here exiss an ineger a {+, } such ha for any cycle conaining a verex labelled by i (corresponding o he parameer a), he spheres S in he cycle are all lef-cenered a verices labelled by i and herefore hey do no overlap. By Lemma, he cycle conaining a verex labelled by i consiss of lcm(l,l, ) disinc spheres S. So such a cycle consiss of lcm(l,l, ) S = lcm(l,l, ) = lcm(l, l, ) verices. Le (x, y ) and (x, y ) be any wo verices labelled by i. We can see ha for he cycle conaining (x, y ) and he cycle conaining (x, y ), hey eiher do no overlap, or hey are he same cycle. Therefore, he verices in G can be pariioned ino several such cycles so l l is a muliple of lcm(l, l, ). Since lcm(l, l, ) is a muliple of l, l mus be a muliple of. Similarly, l mus be a muliple of, oo. So if G can be perfecly -inerleaved, hen boh l and l are muliples of. Now we prove he oher direcion. Assume boh l and l are muliples of. Le W be such a se of verices in G: W = {(x, y) x mod, y mod, x + y mod }. I is easy o verify ha he Lee disance beween any wo verices in W is a leas. Now for i =,,, and for j =,,,, define W i,j o be W i,j = {((x + i) mod l, (y + j) mod l ) (x, y) W }. Clearly hose = S ses W,, W,,, W, is a pariion of he verices in G. For each W i,j, we label he verices in i wih one disinc ineger. Clearly such an inerleaving is a perfec -inerleaving. So if boh l and l are muliples of, hen G can be perfecly -inerleaved. Case : is odd. Firsly, we prove one direcion. Assume boh l and l are muliples of +. Golomb and Welch have shown in [] ha an + + orus can be perfecly packed by he spheres S for odd. Therefore, G can also be perfecly packed by S because a orus has a oroidal opology and G can be folded ino an + + orus. Le C be a se of verices in G such ha he spheres S cenered a he verices in C form a perfec sphere packing. Then he Lee disance beween any wo verices in C is a leas. We call a se of verices D a cose of C when he following condiion is saisfied: here exis inegers a and b such ha a verex (x, y) C if and only if ((x + a) mod l, (y + b) mod l ) D. C has S differen coses in oal (including C iself), and hose coses pariion he verices of G. For each cose, we label is verices wih one disinc ineger, and we ge a perfec -inerleaving. So if boh l and l are muliples of +, hen G can be perfecly -inerleaved.

11 Now we prove he oher direcion. Assume G can be perfecly -inerleaved. Le i be an ineger used by a perfec - inerleaving on G. Then by Theorem, he spheres S cenered a he verices labelled by i form a perfec sphere packing in G. Golomb and Welch presened in [] a way o perfecly pack spheres S in a orus when is odd, which can be described as eiher of he following wo condiions is rue: () whenever here is a sphere S cenered a a verex (x, y), here are wo spheres respecively cenered a ((x + + ) mod l, (y + ) mod l ) and ((x ) mod l, (y + + ) mod l ); () whenever here is a sphere S cenered a a verex (x, y), here are wo spheres respecively cenered a ((x + ) mod l, (y + + ) mod l ) and ((x + ) mod l, (y + ) mod l ). I is well known ha ha way of packing is in fac he only way o perfecly pack S for odd, whose feasibiliy requires boh l and l o be muliples of + l and l are muliples of +.. So if G can be perfecly -inerleaved, hen boh Below we presen he complee se of perfec sphere packing consrucions. Bu firsly le s explain a few conceps. Le G be an l l orus ha is perfecly packed by spheres S here are l l S such spheres. Define e as e = l l S, and le s say hose spheres are cenered (or lef-cenered) a he verices (x, y ), (x, y ),, (x e, y e ). By verically (respecively, horizonally) shifing he spheres in G, we mean o selec some ineger s, and ge a new se of perfecly packed spheres ha are cenered (or lef-cenered) a (x + s mod l, y ), (x + s mod l, y ),, (x e + s mod l, y e ) (respecively, a (x, y + s mod l ), (x, y + s mod l ),, (x e, y e + s mod l )). By verically reversing he spheres in G, we mean o ge a new se of perfecly packed spheres ha are cenered (or lef-cenered) a ( x mod l, y ), ( x mod l, y ),, ( x e mod l, y e ). Afer such a shif or reverse operaion, echnically speaking, he way he spheres are perfecly packed in G are changed however, he paern of he sphere packing essenially remains he same. Consrucion.: The complee se of perfec sphere packing consrucions Inpu: A posiive ineger. An l l orus G, where () boh l and l are muliples of if is even and, () l is even if =, and () boh l and l are muliples of + if is odd. Oupu: A perfec packing of he spheres S in G. Consrucion:. If is even and, hen do he following: Le A, A,, A l gcd(, l ) be gcd( l, l ) inegers, where A i can be any ineger in he se {,,, } for i =,,, gcd( l, l ). Find he gcd( l, l ) cycles in G (corresponding o he parameer ) respecively conaining he verex (, ), ( i= A i, i= ( + A i)), ( i= A i, i= ( + A i)),, ( gcd( l, l ) i= A i, gcd( l, l ) i= ( + A i )). The spheres S in hose gcd( l, l ) cycles form a perfec sphere packing in he orus.. If =, he do he following: The l l orus G has l rows, each of which can be seen as a ring of l verices. When =, he sphere S simply consiss of wo horizonally adjacen verices. Spli each row of G ino l ll spheres in any way. The resuling spheres form a perfec sphere packing in he orus.. If is odd, hen do he following: Find such a se of ll S spheres S : each of he spheres is cenered a a verex (i(m + ) + j ( m) mod l, i m + j(m + ) mod l ) for some inegers i and j. Those spheres form a perfec sphere packing in he orus.. Horizonally shif, verically shif, and/or verically reverse he spheres in G in any way. Theorem : Consrucion. is he complee se of perfec sphere packing consrucions. Proof: We consider he following hree cases. For each case, we need o prove wo hings: firsly, he Inpu par of Consrucion. ses he necessary and sufficien condiion for a orus o have perfec sphere packing; secondly, he Consrucion

12 par of Consrucion. generaes perfec sphere packing correcly, and every perfec sphere packing ha exiss is a possible oupu of i. Case : is even and. In his case, since a sphere S occupies rows and columns, for he l l orus G o have perfec sphere packing, i mus be ha l and l. We can show ha l in he following way assume l = and spheres S are perfecly packed in G; say a sphere S is lef-cenered a (x, y) in G; hen he wo verices, (x ( ) mod l, y mod l ) and (x + ( ) mod l, y mod l ), canno boh be conained in spheres (see he proof of Theorem for a very similar argumen), and ha conradics he saemen ha spheres are perfecly packed in G. Therefore, if G can be perfecly packed by spheres, l and l. Then, from Theorem and Theorem, we see ha G can be perfecly packed by spheres if and only if boh l and l are muliples of. So he Inpu par of Consrucion. correcly ses of he necessary and sufficien condiion for a orus o have perfec sphere packing. Lemma and is proof have shown ha when spheres are perfecly packed in a orus, hose spheres can be pariioned ino cycles. By observing he shape of he border of a cycle, we see ha wo adjacen cycles can freely slide along each oher s border and here are possible relaive posiions beween wo adjacen cycles. In Consrucion., he possible relaive posiions are deermined by A i, a variable ha can ake possible values. Now i is easy o see ha Sep of Consrucion. provides a perfec sphere packing (which akes one of many possible forms, depending on he value of he A i s), and is Sep changes he posiions of he spheres o furhermore cover all he possible cases of perfec sphere packing. () Case : =. We skip he proof for his case due o is simpliciy. () Case : is odd. In his case, Consrucion. re-produces he sphere-packing mehod presened in [], which is commonly known as he unique way o pack spheres for odd (see he final paragraph of he proof of Theorem for a more deailed inroducion). Now we presen perfec -inerleaving consrucions ha are based on perfec sphere packing. Consrucion.: Perfec -inerleaving consrucions Inpu: A posiive ineger. An l l orus G, where boh l and l are muliples of if is even, and boh l and l are muliples of + if is odd. Oupu: A perfec -inerleaving on G. Consrucion: () If, hen do he following: Use Consrucion. o ge a perfec sphere packing in G. Label each of hose spheres wih S disinc inegers, in such a way ha all he spheres have he same inerleaving paern, and every ineger is used exacly once in each sphere. () If =, hen do he following: For every verex (i, j) of G, if i + j is even, label i wih he ineger, oherwise label i wih he ineger. The following example illusraes how o use Consrucion. o obain perfec sphere packing, and how o use Consrucion. o obain perfec -inerleaving. Example.: Le =, and le G be an orus. Firsly, we use Consrucion. o find a perfec sphere packing in G. Since is even, he Sep of Consrucion. is execued. We choose A, A,, A gcd( l, l ) o be A =, A =. (Noe ha here gcd( l, l ) =.) Then he gcd( l, l ) = cycles in G are as shown in Fig. (a), which are hree ses of spheres S respecively of hree differen background shades. The spheres in hose cycles form a perfec packing in G. Nex, we use Consrucion. o perfecly -inerleave G. Le he perfec sphere packing remain as i is; and label all he spheres wih he same inerleaving paern, using S = disinc inegers. The resuling perfec -inerleaving on G is shown in Fig. (b).

13 (a) G ,,,,, / / / # # # # # $ $ $ $ $ ,,,,, / / / # # # # # $ $ $ $ $ # # # # # : : :,,,,, $ $ $ $ $ ,,,,,..... # # # # # $ $ $ $ $ : : : % % % % % & & & & & : : : % % % % % & & & & & : : : ; ; ; ; < < < < % % % % % & & & & & % % % % % ' ' ' ' ' & & & & & ( ( ( ( ( ; ; ; ; < < < < ' ' ' ' ' ( ( ( ( ( ; ; ; ; < < < < ' ' ' ' ' ( ( ( ( ( ; ; ; ; < < < < ' ' ' ' ' ) ) ) ) ) )!!!!!! ( ( ( ( ( * * * * * " " " " " = = = ) ) ) ) ) ) * * * * * > >!!!!!! " " " " " = = = > > ) ) ) ) ) ) * * * * *!!!!!! " " " " " ) ) ) ) ) ) * * * * *!!!!!! " " " " " (b) G Fig.. Example of perfec sphere packing using Consrucion. and perfec -inerleaving using Consrucion.. We commen ha Consrucion. provides he complee se of perfec -inerleaving consrucions ha have he following propery: for any wo inegers, he wo ses of verices respecively labelled by hose wo inegers are coses of each oher in he orus. Wha is more, in [], hree -inerleaving consrucions for wo-dimensional arrays were presened, all based on laice inerleavers. Those hree consrucions can also be applied o ori because of heir periodic paerns. Our Consrucion. generalizes he resuls in [] in wo ways: firsly, i covers more consrucions based on laice inerleavers, wih he resuls of [] included as special cases; secondly, when is even, i also covers consrucions ha do no use laice inerleavers, which we can make happen by simply leing any A i and A j ake differen values. III. ACHIEVING AN INTERLEAVING DEGREE WITHIN ONE OF THE OPTIMAL In his secion, we presen a novel -inerleaving consrucion, wih which we can -inerleave any large enough orus wih a degree wihin one of he opimal. The consrucion presened here will also be used as a building block in Secion IV for opimal -inerleaving. A. Inerleaving Consrucion Definiion.: Given a posiive ineger, if is odd, hen P is defined o be a sring of inegers a, a,, a, where a = + and a i = for i < ; if is even, hen P is defined o be a sring of inegers a, a,, a, where a = and a i = for i <. (For example, if =, hen P = ; if =, hen P =, ; if =, hen P =,.) Given a posiive ineger, if is odd, hen Q is defined o be a sring of inegers b, b,, b +, where b + = + and b i = for i < + ; if is even, hen Q is defined o be a sring of inegers b, b,, b +, where b + = and b i = for i < +. Given a posiive ineger, an offse sequence is a sring of P s and Q s. (As an example, an offse sequence consising of P and Q s can be P QQ, QP Q or QQP.) The offse sequence is also naurally seen as a sring of inegers

14 Fig.. An example of -inerleaving wih he hree feaures. which is he union of he inegers in is P s and Q s. (For example, when =, if an offse sequence consising of P and Q s is P QQ, hen he offse sequence is also seen as,,,, ; when =, if an offse sequence consising of P s and Q s is P QP P Q, hen he offse sequence is also seen as,,,,,,,,,,,.) The number of inegers in an offse sequence is called is lengh. In his secion, we are paricularly ineresed in one kind of -inerleaving on an l l orus, which has he following feaures: Feaure : l = S +. (In oher words, if is odd, hen l = + + ; if is even, hen l = +.) Feaure : The degree of he -inerleaving equals l. And in every column of he orus, each of he l inegers is assigned o exacly one verex. Feaure : If he verex (a, b ) and he verex (a, b ) are labelled by he same ineger, hen for i =,,, l, he verex ((a + i) mod l, b ) and he verex ((a + i) mod l, b ) are labelled by he same ineger. Example.: Fig. shows a -inerleaving on an l l orus which has he above hree feaures. There =, l = S + = and l =. Now le s fixed an ineger i, where i, and say he se of verices labelled by i are (x, ), (x, ),, (x l, l ). Then he following sring of inegers: (x x ) mod l, (x x ) mod l,, (x x ) mod l, (x x ) mod l, equals,,,,,,,. Since when =, P = and Q =,, he above sring of inegers acually equals P P P QP Q, which is an offse sequence of lengh l. We commen ha his phenomenon is no a pure coincidence offse sequences do help us find -inerleavings ha have he above hree feaures. In fac, we can prove ha in many cases (e.g., when = or ), for any -inerleaving on a orus ha has he above hree feaures, afer horizonally shifing and/or verically reversing he inerleaving paern, he resuling inerleaving will have he same phenomenon as he example shown here. The following consrucion oupus a -inerleaving ha has he hree feaures. Consrucion.: Inpu: A posiive ineger. An l l orus, where l = S +. An ineger m ha equals. Two inegers p and q ha saisfy he following equaion se if is odd: and saisfy he following equaion se if is even: pm + q(m + ) = l p(m + m + ) + q(m + m + ) mod (m + m + ) p and q are non-negaive inegers, p + q >. pm + q(m + ) = l p(m m + ) + q(m + m) mod (m + ) p and q are non-negaive inegers, p + q >. () ()

15 Oupu: A -inerleaving on he l l orus ha saisfies Feaure, Feaure and Feaure. Consrucion: Le S = s, s,, s l be an arbirary offse sequence consising of p P s and q Q s. For j =,,, l and for i =,,, l, label he verex (( j k= s k + i) mod l, j mod l ) wih he ineger i. Example.: Le =, l =, l =, m =, p =, and q =. We use Consrucion. o -inerleave an l l orus. Say he offse sequence S is chosen o be P P P QP Q. Then Consrucion. oupus he -inerleaving shown in Fig.. We explain Consrucion. a lile bi. The Equaion Se () (for odd ) and he Equaion Se () (for even ) ensure ha he offse sequence S, which consiss of p P s and q Q s, exiss. Furhermore, for any ineger j ( j l ), if (a, j) and (b, (j + ) mod l ) are wo verices labelled by he same ineger, hen b a s j mod l namely, he offse sequence S indicaes he verical offses of any wo verices in adjacen columns ha are labelled by he same ineger. I is simple o verify ha he -inerleaving oupu by Consrucion. saisfies all he hree feaures Feaure, and lised earlier in his subsecion. The following lemma will be used o prove he correcness of Consrucion. and also in fuure analysis. Lemma : Le i {,,, S } be any of he inegers used by Consrucion. o inerleave he l l orus. Le {(b, ), (b, ),, (b l, l )} be he se of verices in he orus ha are labelled by i. Le m and S have he same meaning as in Consrucion. (namely, m =, and S = s, s,, s l is he offse sequence consising of p P s and q Q s uilized by Consrucion.). For any wo inegers j and j ( j j l ), we define L j j as L j j = [(j j ) mod l ] + min{(b j b j ) mod l, (b j b j ) mod l }. Then we have he following conclusions: Case : is odd, j j m mod l, and s j, s (j +) mod l, s (j +) mod l,, s (j ) mod l do no all equal. In his case, b j b j (m + ) mod l and L j j =. Case : is odd, j j m + mod l, and exacly one of s j, s (j +) mod l, s (j +) mod l,, s (j ) mod l equals +. In his case, b j b j m mod l and L j j =. Case : is even, j j mod l, and s j =. In his case, b j b j mod l and L j j =. Case : is even, j j m mod l, and s j, s (j +) mod l, s (j +) mod l,, s (j ) mod l do no all equal. In his case, b j b j m mod l and L j j =. Case : is even, j j m + mod l, and exacly one of s j, s (j+) mod l, s (j+) mod l,, s (j ) mod l equals. In his case, b j b j m mod l and L j j =. If none of he above five cases is rue, and j j mod l, hen L j j >. If none of he above five cases is rue, and j j mod l, hen L j j. Proof: Le = + if is odd, and le = if is even. The offse sequence S consiss of P s and Q s, so i has he following propery: for any k {,,, l } such ha s k =, he following m inegers s (k+) mod l, s (k+) mod l,, s (k+m ) mod l all equal, and eiher s (k+m) mod l or s (k+m+) mod l equals. Also noe ha b j b j s j + s (j+) mod l + s (j+) mod l + + s (j ) mod l mod l. Based on hose wo observaions, his lemma can be proved wih sraighforward compuaion. Theorem : Consrucion. is correc. Proof: Le (b j, j ) and (b j, j ) be any wo verices labelled by he same ineger in he l l orus ha was inerleaved by Consrucion.. The Lee disance beween hem is d((b j, j ), (b j, j )) = min{(j j ) mod l, (j j ) mod l } + min{(b j b j ) mod l, (b j b j ) mod l } = min{l j j, L j j }. From Lemma, i is clearly ha neiher L j j nor L j j is less han. Therefore d((b j, j ), (b j, j )). So Consrucion. -inerleaved he orus. And as menioned before, his -inerleaving saisfies Feaure, Feaure and Feaure.

16 B. Exisence of Offse Sequences The feasibiliy of Consrucion. depends only on one hing wheher he wo inpu parameers p and q exis or no. The following heorem shows ha when he widh of he orus, l, exceeds a hreshold, p and q are guaraneed o exis. Theorem : Le be an odd (respecively, even) posiive ineger. When l ( + )( S + ), here exiss a leas one soluion (p, q) o he equaion se () (respecively, equaion se ()), which is shown in he Inpu par of Consrucion.. Proof : Firsly, le s assume is odd. The equaion se () is as follows: pm + q(m + ) = l p(m + m + ) + q(m + m + ) mod (m + m + ) p and q are non-negaive inegers, p + q >. where m =. We inroduce a new variable z, and ransform he above equaion se equivalenly o be: ( ) ( ) ( ) which is he same as: which equals: ( p q m m + m + m + m + m + ) = p q = l z(m + m + ) p and q are non-negaive inegers; z is a posiive ineger. ( m m + m + m + m + m + ) ( l z(m + m + ) p and q are non-negaive inegers; z is a posiive ineger. p = (m + )(m + m + )z (m + m + )l q = (m + m + )l m(m + m + )z p and q are non-negaive inegers; z is a posiive ineger. There exiss a soluion for he variables p, q and z in he above equaion se if and only if he following condiions can be saisfied: which is equivalen o: (m + )(m + m + )z (m + m + )l (m + m + )l m(m + m + )z z is a posiive ineger. { (m +m+)l (m+)(m +m+) z (m +m+)l m(m +m+) z is a posiive ineger. To enable a value for z o exis ha saisfies he above condiions, i is sufficien o make (m +m+)l m(m +m+) (m +m+)l (m+)(m +m+) ha is, o make l m(m + )(m + m + ) = ( + )( S + ). Therefore when l ( + )( S + ), here exiss a leas one soluion (p, q) o he equaion se (). When is even, he conclusion can be proved in a very similar way. We skip is deails. ) Corollary : When l ( +)( S +), Consrucion. can be used o oupu a -inerleaving on an ( S +) l orus. Proof: When l ( + )( S + ), all he parameers in he Inpu par of Consrucion. exis, including p and q.

17 (a) (b) A B C D E Fig.. Examples of iling ori C. Inerleaving wih Degree wihin One of he Opimal In his subsecion, we will show how o inerleave a large enough orus wih he degree wihin one of he opimal. We define he simple erm of iling ori here. By iling several inerleaved ori verically or horizonally, we ge a larger orus, whose inerleaving is he sraighforward combinaion of he inerleaving on he smaller ori. I is bes explained wih an example. Example.: Three inerleaved [ ] ori A, B and C are shown in Fig.. The orus D is a orus, go by iling A and A B verically in he form of B [ ]. The orus E is a orus, go by iling one copy of A and wo copies of C horizonally in he form of C A C. The following consrucion -inerleaves a large enough orus wih a mos S + disinc inegers. Consrucion.: -inerleave an l l orus G, where l S ( S + ) and l ( + )( S + ), using a mos S + disinc inegers.. Le G be an ( S + ) l orus ha is -inerleaved by Consrucion., using he inegers,,, S. Le {(c, ), (c, ),, (c l, l )} be he se of verices in G labelled by he ineger.. Le G be an ( S +) l orus. Label he verices {(c, ), (c, ),, (c l, l )} in G wih he ineger S +.. For j =,,, l and for i =,,, S +, label he verex ((c j + i) mod ( S + ), j) in G wih he ineger i.. Le x and y be wo non-negaive inegers such ha l = x( S + ) + y( S + ). Tile x copies of G and y copies of G verically o ge an l l orus G. (Then G has been -inerleaved using a mos S + disinc inegers.) Example.: We use Consrucion. o -inerleave a orus G, where =. The firs sep is o use Consrucion. o -inerleave a orus G. Say he offse sequence seleced in Consrucion. is S = QQQ =,,,,,, hen G is as shown in Fig.. Then he orus G is as shown in he figure. By iling one copy of G and one copy of G verically, we ge he -inerleaved orus G. S + = disinc inegers are used o inerleave G. Theorem : Consrucion. is correc. Proof: I is a known fac ha for any wo relaively prime posiive inegers A and B, any ineger C no less han (A )(B ) can be expressed as C = xa + yb where x and y are non-negaive inegers. Therefore in Consrucion., since l S ( S + ), l indeed can be expressed as l = x( S + ) + y( S + ), as shown in he las sep of Consrucion.. So he consrucion can be execued from beginning o end successfully. Now we prove ha he consrucion does -inerleave G ha is, for any wo verices (a, b ) and (a, b ) labelled by he same ineger i in G, he Lee disance beween hem is a leas. We consider hree cases.

18 G G G Fig.. Examples of Consrucion.. Case : b = b, which means ha (a, b ) and (a, b ) are in he same column of G. We see every column of G as a ring of lengh l (because i is oroidal). Then, observe he inegers labelling a column of G, and we can see ha on he column, he inegers following an ineger S + and before he nex ineger S + mus be,,, S,,,, S,,,,, S, where he paern,,, S appears a leas once. Therefore since (a, b ) and (a, b ) are labelled by he same ineger, he Lee disance beween hem mus be a leas S + >. Case : b b, and i S +. In his case, le s firs observe wo conclusions: The inerleaving on G is -inerleaving. (See Consrucion. for he definiion of G.) This can be proved as follows: any wo verices labelled by he same ineger in G can be expressed as ((c j + i ) mod ( S + ), j ) and ((c j + i ) mod ( S + ), j ) (see he Sep and Sep of Consrucion.); hen, d G (((c j + i ) mod ( S + ), j ), ((c j + i ) mod ( S + ), j )) = d G ((c j, j ), (c j, j )) d G ((c j, j ), (c j, j )). Le (α, j) and (β, j) be wo verices respecively in G and G boh of which are labelled by he same ineger. Then i is simple o see ha β = α or β = α+. Since G has S + rows and G has S + rows, we have d G ((β, j), (, j)) d G ((α, j), (, j)) and d G ((β, j), ( S +, j)) d G ((α, j), ( S, j)). Tha is, if u and v are wo verices respecively in G and G boh of which are in he j-h column and labelled by he same ineger, he verical disance from v o he wo borders of G is no less han he verical disance from u o he wo borders of G. According o Consrucion., G is go by verically iling x copies of G and y copies of G. Le s call each of hose x + y ori a componen orus of G. Now, if (a, b ) and (a, b ) are in he same componen orus of G, we know he Lee disance beween hem in G is no less han he Lee disance beween hem in ha componen orus, which is a leas because ha componen orus is -inerleaved. If (a, b ) and (a, b ) are no in he same componen orus of G, we do he following. We firsly consruc a orus G which is go by verically iling x+y copies of G. I is simple o see ha G is -inerleaved. We call each of he x + y copies of G in G a componen orus of G. Le s say (a, b ) and (a, b ) are respecively in he k -h and k -h componen orus of G. Le (c, b ) and (c, b ) be he wo verices labelled by he ineger i ha are respecively in he k -h and k -h componen orus of G. Observe he shores pah beween (a, b ) and (a, b ) in G, and we see ha i can be spli ino such hree inervals: from (a, b ) o a border of he k -h componen orus, from he border of he k -h componen orus o he border of he k -h componen orus, and from he border of he k -h componen orus o (a, b ). There is a corresponding (no necessarily shores) pah connecing (c, b ) and (c, b ) in G, which can be spli ino such hree inervals similarly. And each of he hree inervals of he firs pah is a leas as long as he corresponding inerval of he second pah. G is -inerleaved, so he second pah s lengh is a leas. So he Lee disance beween (a, b ) and (a, b ) in G is a leas. Case : b b, and i = S +. In his case, i is simple o see ha he wo verices in G, (a + mod l, b ) and (a + mod l, b ), are boh labelled by he ineger. Based on he conclusion of Case, d G ((a + mod l, b ), (a + mod l, b )). So d G ((a, b ), (a, b )) = d G ((a + mod l, b ), (a + mod l, b )). So Consrucion. correcly -inerleaved G.