The (, D) and (, N) problems in double-step digraphs with unilateral distance

Transcription

1 Electronic Journal of Grap Teory and Applications () (), Te (, D) and (, N) problems in double-step digraps wit unilateral distance C Dalfó, MA Fiol Departament de Matemàtica Aplicada IV Universitat Politècnica de Catalunya Barcelona, Catalonia {cdalfo,fiol}@maupcedu Abstract We study te (, D) and (, N) problems for double-step digraps considering te unilateral distance, wic is te minimum between te distance in te digrap and te distance in its converse digrap, te latter obtained by canging te directions of all te arcs Te first problem consists of maximizing te number of vertices N of a digrap, given te maximum degree and te unilateral diameter D, wereas te second one (someow dual of te first) consists of minimizing te unilateral diameter given te maximum degree and te number of vertices We solve te first problem for every value of te unilateral diameter and te second one for infinitely many values of te number of vertices Moreover, we compute te mean unilateral distance of te digraps in te families considered Keywords: (, D) problem, (, N) problem, unilateral distance, double-step digraps Matematics Subject Classification: C, C Introduction Te (, D) and (, N) problems ave been extensively studied for graps and digraps Miller and Sirán [9] wrote a compreensive survey about tese problems In particular, for te so-called double-step graps considering te standard diameter, te first problem was solved by Yebra, Fiol, Morillo and Alegre [], wereas Bermond, Iliades and Peyrat [], and also Beivide, Herrada, Received: September, Revised: January, Accepted: January

2 i+b i+b i+a+b i i+a i+a Figure Steps of G(N; a, b) (left), and te double-step digrap G(;, ) (rigt) Balcázar and Arruabarrena [] solved te (, N) problem In te case of te double-step digraps also wit te standard diameter, Morillo, Fiol and Fàbrega [] solved te (, D) problem and provided some infinite families of digraps wic solve te (, N) problem for teir corresponding numbers of vertices Esqué, Aguiló and Fiol [], and Aguiló and Fiol [] contributed to te second problem wit more general results Double-step digraps were proposed and studied as models for te so-called local area networks, in wic several computers placed at sort distances excange data at very ig speed, as explained in Fiol, Yebra, Alegre and Valero [] In particular, te delay in te transmission of a message between two nodes is closely related to te minimum number of necessary steps to get teir target, tat is, te distance between nodes Wat allows us to go from te combinatorial formulation (te network structure) to te algebraic one is te representation of eac digrap as an L-saped or X-saped tile (a planar region), wic periodically tessellates te plane by translations (-dimensional integer lattice) Wit tese geometric forms we can study more easily te properties related to te distance in a digrap, suc as its diameter and its mean distance In tis paper we study te (, D ) and te (, N) problems for double-step digraps, were te asterisks indicate tat we consider te unilateral distance instead of te standard distance Te former is te minimum between te distance in te digrap and te distance in its converse digrap, obtained by canging te directions of all te arcs Te plan of te paper is as follows In te next section we recall te definitions of double-step digrap, unilateral distance, and unilateral diameter Moreover, we sow tat if we take te vertices at minimum distance from a given vertex, we obtain an L-saped or X-saped tile, depending on weter we consider te standard or te unilateral diameter, respectively In Section, we give te unilateral diameter of double-step digraps wen one step is equal to In Section, we solve te (, D ) problem for every value of te unilateral diameter and also te (, N) problem for infinitely many values of te number of vertices Finally, in te last section, we derive formulas for computing te mean unilateral distance of te double-step digraps considered Preliminaries We recall te basic definitions and concepts concerning double-step digraps and unilateral distance, togeter wit teir geometric representations

3 y w G(;,) Figure A generic L-saped tile (left), and te L-saped tile of G(;, ) (rigt) Double-step digraps A double-step digrap G(N; a, b) as set of vertices Z N = Z/NZ and arcs from every vertex i to vertices i+a mod N and i+b mod N, for i N, were a, b are some integers called steps suc tat a < b N Because of te automorpisms i i + α for α N, te double-step digraps are vertex-transitive Moreover, tey are strongly connected if and only if gcd(n, a, b) = In Fig we sow te steps of a double-step digrap G(N; a, b) and, as an example, G(;, ) It is known tat te maximum order N of a double-step digrap wit diameter k is upper bounded by te Moore-like bound N M DSD (, k) = ( ) k+, were te equality would old if all te numbers ma + nb were different modulo N, wit m, n and m + n k In fact, tis bound cannot be attained for k > For more details, see Esqué, Aguiló and Fiol [], Aguiló and Fiol [], and Fiol, Yebra, Alegre and Valero [] Given a double-step digrap G(N; a, b) tat is assumed to be strongly connected, consider te plane divided into unitary squares, centered in te integral coordinate points forming a lattice From a square or lattice point labeled wit zero, we add a mod N wen we move orizontally to te rigt to te next square, and b mod N wen we move vertically upwards Ten, te plane is covered by integers modulo N, tat is, te elements of te cyclic group Z N, as sown in Fig (rigt) wit te example of te digrap G(;, ) Note tat, in tis way, eac vertex of a digrap G(N; a, b) is related to a lattice point Moreover, tose at minimum distance from form a - dimensional lattice wic univocally caracterizes te digrap In turn, suc a lattice is determined by a basis of two (integer) vectors u = (l, y) and v = ( w, +y), satisfying N = l(+y) wy and gcd(l,, w, y) = (see Figure ) Now, if we coose any N vertices (squares) wit different labels modulo N, we get a tile wic periodically tessellates te plane and, in te cases we consider, te dimensions of suc a tile are closely related to te entries of te vectors u, v Let us see some useful cases: If all te vertices different modulo N are cosen to be at minimum (standard) distance from vertex (tis is done by following a simple algoritm tat considers te successive diagonals, as is sown again in Fig (rigt)), Brawer and Sokley [] proved tat te tiles are always L-saped tiles caracterized by its dimensions (l, + y, w, y), wit l, + y, w l, y, see again Fig (left), and te corresponding L-basis is u = (l, y), v =

4 s y= +y v -w u -y Figure Te vectors of an L-basis (left), and te dimensions of an L-saped form wit area N = l + s (rigt) ( w, + y), as sown in Fig If one of te steps, say a, equals, we can coose again an L-saped tile wit dimensions: l = b, being te quotient obtained dividing N by l, w = l s wit s being te remainder of suc a division, and y = Ten, N = l + s wit s < l (see Fig ) If all te vertices wic are distinct modulo N are cosen at minimum unilateral distance (see its definition in Subsection ) from vertex (following again a simple algoritm) we get a kind of X-saped tile, as sown in Fig Moreover, in te above case (a = ), we can rebuild te obtained tile as an L-saped tile placing te inferior alf (witout te ) onto te superior one (In oter words, te vertices at minimum unilateral distance from and to vertex are taken from two different points of te plane representing it) See again Fig Conversely, from a basis u = (l, y) and v = ( w, + y) (or distribution of zeros ) of te lattice wit gcd(l,, w, y) =, te steps a and b of te corresponding digrap of N = l(+y) wy vertices (or a tile wit te same area) can be obtained as te solutions of te following system of equations: la yb = αn ( mod N), wa + ( + y)b = βn ( mod N), wic gives a = α( + y) + βy, b = αw + βl, () were α and β must be cosen so tat te condition gcd(n, a, b) = is satisfied (See Fiol, Yebra, Alegre, and Valero [] for more details) For instance, if we apply tis metod to te example

5 r r (a) r r (b) Figure From X-saped forms to L-saped forms (for te case s = l ): (a) l + even, and (b) l + odd Te squares wit an asterisk represent te fartest vertices from of G(;, ) (see again Figs and ), te dimensions of te L-saped tile are l =, =, w = y = Ten, a = α+β and b = α+β A good coice is α = and β =, wic gives te steps a = and b = More generally, for an L-saped tile wit dimensions l, + y, w = l s and y = (see Fig ), we get a = and b = l, as expected Observe tat te values of te steps a and b are not unique Indeed, given a feasible pair a, b we obtain an isomorpic digrap wit te steps γa, γb for any integer γ relatively prime to N Unilateral distance Given a digrap G = (V, A), te unilateral distance between two vertices u, v V is defined as dist G(u, v) = min{dist G (u, v), dist G (v, u)} = min{dist G (u, v), dist G (u, v)}, were dist G is te standard distance in digrap G and dist G is te distance in its converse digrap G, tat is, te digrap obtained by canging te directions of all te arcs of G From tis concept, we can define te unilateral eccentricity ecc from vertex u, te unilateral radius r of G, and te unilateral diameter D of G as follows: ecc (u) = max v V {dist G(u, v)}, r = min u V {ecc (u)}, and D = max u V {ecc (u)} As an example, if we ave G = C N, te directed cycle on N vertices, ten r = D = N/ Note tat, obviously, tese parameters ave as lower bounds te ones corresponding to te underlying grap, obtained from digrap G by canging te arcs for edges witout direction Some constructions of general digraps wit large number of vertices given te maximum degree and unilateral diameter (tat is, te (, D ) problem for digraps) were proposed by Gómez, Canale and Muñoz [, ] Te unilateral diameter of double-step digraps wit steps a = and b = l In tis section we study te unilateral diameter of te double-step digraps wit a = aving small b Altoug we ave not been able to prove tat te optimal results can be obtained always

6 s +-s- +s+ s+ s+ +s+ s s+ s+ +-s- s+ s+ Figure Tessellations for s + < l and s + l by taking suc values of te steps, computational experiments seem to support tis claim In fact, as we see in te next section, tis approac allows us to solve te (, D ) problem for every value of D, and also to solve te (, N) problem for infinitely many values of N As we ave already seen, a double-step digrap G(N;, b) wit N = l + s and s < l, can be described by an L-saped form wit dimensions l = b, = N/l, y =, and w = l s See again Fig (rigt) In tis context we ave te following result for te unilateral diameter D Proposition For N = l+s, were < l N and s l, a double-step digrap G(N; a, b) wit a = and b = l as unilateral diameter l + + s if s l, D = () l + if s = l Proof To determine te vertices at minimum unilateral distance from, it is useful to divide te L-saped form into two parts: Te left one is an L-tile wit left bottom corner (, ) and rigt top (missing) corner (s, ) (bot points correspond to vertex ) Te rigt part is a rectangle R wit left bottom corner i = (s +, ) and rigt top corner j = (l, ) (wit basis l s and eigt ) See Fig for te cases s + < l and s + l and compare te two pats to go to equivalent vertices In bot cases, notice tat, if + s + (a condition tat is always fulfilled since s < l N ), te unilateral distances from to i and to j are bot s+ As a consequence, te fartest vertices from are in (one or two) NW-SE diagonals of R or, equivalently, te same diagonals of te rectangle R wit basis l and eigt sown in Fig, in grey dased lines, wit

7 s= s= s= s=- = = = =+ =+ =+ = Figure Some cases of Proposition values l = = { l + s + if s l, l if s = l { if s l, + if s = l Consequently, te unilateral diameter D turns out to be l D + =, wence we obtain te claimed result As an example, in Fig, we sow some of te possible L-saped forms for N =, were te minimum unilateral diameter is, corresponding to te L-saped form wit l =, = and s =, tat is, te double-step digrap G(;, ); or l =, = and s = giving G(;, ) In Table, tere are te unilateral diameters corresponding to all te possible values of l for N = and N = 9 (te values in grey do not satisfy < l N ) All tese values are represented in Fig If tere is not restriction for te value of l, ten te minimum value in Eq () give us an upper bound for te unilateral diameter D Te first case in wic suc a bound is not attained corresponds to N =, were te unilateral diameter D is and te upper bound given by Eq () is Te (, D ) and (, N) problems for double-step digraps wit unilateral diameter In tis section we completely solve te (, D ) problem for double-step digraps wit unilateral diameter and give infinite families of suc digraps wic solve te (, N) problem Let us begin wit te former

8 Te (, D ) and (, N) problems in double-step digraps C Dalfó, MA Fiol 9 9 = = = = = Figure Some of te possible L-saped forms for N = Te numbers indicate te distance from to eac vertex Eac vertex and its fartest vertices are in bold D l D D l D N= N=9 Figure Te minimum unilateral diameter D wit respect to l for N = and N = 9 Te (, D ) problem In our context, te (, D ) problem consists of finding te double-step digrap G(N; a, b) wit maximum number of vertices given a unilateral diameter D, tat is, to find te steps tat maximize te number of vertices for suc a unilateral diameter To get a Moore-like bound (see Miller and Sirán [9]), notice tat at distance k =,,, D from vertex tere are at most (k + ) vertices (k + of tem going forward and te oter k + going backwards) Ten, tis gives N M(, D ) = ( D + ) = (D ) + D + () Moreover, if te maximum is attained, we get an optimal X-saped tile wic tessellates te plane, as sown in Fig 9 (rigt) for te case N = In te following result we sow tat tis Moore-like bound can be always attained Proposition For eac integer value k, te double-step digrap G(N;, b), wit N = M(, k) = k + k + and b = k + as unilateral diameter D = k

9 Table Te unilateral diameter D wit respect to l for N = and N = 9 (in bold tere are te values tat satisfy < l N ) N= N=9 N= N=9 N= N=9 N= N=9 l D D l D D l D D l D D Proof Tis corresponds to one of te cases of Proposition Indeed, it suffices to take l = = k + and s = k In Table tere are te computer data of te minimum unilateral diameter for eac number of vertices ( N ) wit steps a = and b = l In bold, tere are te cases corresponding to te (, D ) problem Te (, N) problem In our context, te (, N) problem consists of finding te minimum unilateral diameter D in double-step digraps given a number of vertices N, tat is, to find te steps tat minimize te unilateral diameter for suc a number of vertices We begin wit te following general upper bound for te unilateral diameter Proposition Given any number of vertices N, tere exists a double-step digrap wit unilateral diameter D satisfying D (N + ) Proof To prove te upper bound, we use again te constructions of Proposition Ten, suppose tat, in te worst case, te division of N by l, were l, gives a remainder s = l Tus, N = l + s wit = N s = N+ and tis gives a unilateral diameter l l l + + s N + D = = + l l 9

10 Figure 9 Te digrap G(;, ) (left), and its corresponding X-saped tile (rigt) Now, since we can coose te value of l, we want to minimize te function φ(l) = N+ + l, wic l is attained at l = (N + )/ Ten, te claimed upper bound is obtained by considering tat l must be an integer To solve te (, N) problem for double-step digraps wit minimum unilateral diameter we consider te case s = l of Proposition Moreover, in tis case, to keep track of te excluded vertices from te maximum M(, k) (tose corresponding to te wite squares in Fig ), we define r as te subindex of te triangular number T r = r = ( ) r+ Proposition (a) If r < ( k + 9 ), te double-step digrap G(N; a, b), wit number of vertices N = k + k + r(r + ) and steps a = and b = l = k r +, as minimum unilateral diameter D = k (b) If r < k +, te double-step digrap G(N; a, b), wit number of vertices N = k + k r and steps a = and b = l = k r +, as minimum unilateral diameter D = k Proof (a) In Proposition, take l = k + r, = k + + r, and s = l Tis corresponds to a double-step digrap wit number of vertices N = l + s = (k + r)(k + + r) + k r = k +k+ r(r+) and unilateral diameter D = (l+ )/ = k Tus, if te number r(r+) of excluded vertices from te maximum M(, k) is at most k +, te digrap as minimum unilateral diameter for tis N Tis comes from te fact tat M(, k) M(, k ) = k + Using te triangular numbers, te condition is T r k + As T r is an even number, we get T r k T r k T r = r(r + ) < k + So, r + r (k + ) < and, ence, r < ( + k + 9) (b) Using te same proposition, take l = k + r, = k + r, and s = l, wic corresponds

11 D N Figure Te minimum unilateral diameter D wit respect to te number of vertices N, for N (Te largest points correspond to te (, D ) problem, and te tick lines to te upper bound given in Proposition ) to a double-step digrap wit order N = l + s = (k + r)(k + r) + k r = k + k r and, as before, unilateral diameter D = (l + )/ = k Now, for te digrap to ave minimum unilateral diameter, te number of excluded squares from M(, k) must satisfy T r + T r + k + = (r + )r + r(r ) + k + = r + k + k + Ten, r k and, ence, r k or r < k + As said before, in Table tere are te computer data of te minimum unilateral diameter for eac number of vertices ( N ) wit steps a = and b = l Te (, D ) problem, wic are in bold, correspond to r = in case (a) In grey, tere are te cases corresponding to te (, N) problem solved wit Proposition As sown in Fig, te unilateral diameter D does not increase monotonically wit te number of vertices N Note tat if we fix r for any k large enoug, we get an infinite family of digraps wit minimum unilateral diameter for eac number of vertices See some examples of te cases of Proposition in Table

12 Table Some results of te (, D ) and (, N) problems solved wit Proposition Problem l + r l N = l + l D (, D ) even k + k + k + k + k (, N) even k k + k + k k (, N) even k k + k + k k (, N) even k k + k + k k (, N) odd k + k + k + k k (, N) odd k k + k + k k (, N) odd k k + k + k k (, N) odd k k + k + k 9 k Te mean unilateral distance for double-step digraps In te next result, we give te mean unilateral distance for te double-step digraps of Proposition Tere is an example of eac kind of L-saped form in Fig Proposition For N = l+s, were < l N and s l, a double-step digrap G(N; a, b) wit a = and b = l as mean unilateral distance d, were: (a) For s = : (a) If l, are [ even: ( d = l (a) If l is even, [ and is odd: d = l + (a) If l is odd, [ and is even: ( d = l (a) If l, are odd: d = l (b) For s = l : ) ( + (l ) l( l) + ( l+ ) l+ ( + (l ) l( l) + l+ ) ( + (l ) l( l) + l+ [ + ( + + (l ) l( l) + ( ) l+ (b) If l + is even: ( d = l (l )( l) + l+ l+l l+ ) (b) If l + is odd: ( d = l (l )( l) + ) l+ l+l

13 Table Minimum unilateral diameter D for eac number of vertices N, N, wit steps a = and b = l (te cases of te (, D ) problem are in bold, te cases wit s = are in wite, wit s l are in ligt grey, and wit s = l are in dark grey) N D l s N D l s N D l s

14 (c) For s l : (c) If l,, s are [ even: ( d = (s + ) s( s) + ( ) ) s+ l+s + (l s )(l s)(l + s ) + (l s ) ( ( l++s (c) If l, are[ even, and s is odd: d = (s + ) l+s + ( s( s) + s+ (l s )(l s)(l + s ) + (l s ) ( l++s (c) If l, s are[ even, and is odd: d = (s + ) l+s + ) l++s (l s + )(l s + ) + s+ ) (l s)(l s + ) ( s( s) + s+ (l s )(l s)(l + s ) + (l s ) ( l++s + s+ ) + (l s + )(l s + ) (c) If l is odd, [ and, s( are even: d = (s + ) s( s) + ( ) s+ )) l+s + (l s )(l s)(l + s ) + (l s ) ( l++s (l s + )(l s + ) (c) If l is even, [ and, ( s are odd: d = (s + ) s( s) + ( ) ) s+ l+s + (l s )(l s)(l + s ) ( ( + (l s ) l++s ) ( l++s ) (l s)(l s + ) (c) If l, s are[ odd, and is even: d = (s + ) l+s + ( s( s) + s+ (l s )(l s)(l + s ) + (l s ) ( ( l++s ) ( l++s (c) If l, are[ odd, and s is even: d = (s + ) l+s + ( s( s) + s+ (l s )(l s)(l + s ) + (l s ) ( ( l++s (c) If l,, s are [ odd: ( d = (s + ) s( s) + ( ) s+ ) l+s + s+ ) ) (l s)(l s + ) + s+ ) ) ( l++s ) (l s + )(l s + )

15 s= (a) s= (a) s= (a),(a) s s=- s (b) s=- s s=- s (b) s s+ s s+ (c),(c) s+ (c) s+ s s+ s+ (c),(c),(c) s s+ s+ (c),(c) Figure Te cases of Proposition (te vertices wit an asterisk are te fartest from ) + (l s )(l s)(l + s ) + (l s ) ( l++s (l s)(l s + ) Proof We only prove case (c) because te oter proofs are very similar (c) If l, and s even: We divide te L-saped form into two parts, te left one and te rigt one In te left part, we ave an L-saped form wit widt s + and eigt (wit s ) In te rigt part, tere is an L-saped form wit widt l s and eigt (wit s = ) See case (c)

16 in Figure Ten, in te left part, te sum of te number of vertices times teir distance is ( ) (s + )s ( ( )) s + + (s + )(s + ) + (s + )(s + ) + + (s + ) + (s + ) s + (( ) ( ) ( )) s + = [( +(s + ) s + ) ] ( s) + (s + ) s + ( ) [( ) ] s + s + s + = + (s + ) s(s + ) + (s + ) s + ( ( ) ) s + = (s + ) s( s) + For te rigt part, we ave [(s + ) + (s + ) + + (l )(l s [ ( l + + s +(l s ) (l s + ) + + = [(s ) + (s ) + + (s )(l s +[ (l s)(l s [ ( l + + s +(l s ) (l s ) ( l s ) [( ) ( ) ( l s = (s )(l s )(l s) ( ) l + + s l + + s +(l s ) (l s )(l s + )(l s + ) ( ) l s + = (s )(l s )(l s) + ( ) l + + s l + + s +(l s ) (l s )(l s + )(l s + ) = (l s )(l s)(l + s ) ( (l ) ) + + s +(l s ) l + + s (l s + )(l s + ) Finally, considering bot parts, we obtain te mean unilateral distance dividing by te number of vertices N = l + s

17 Acknowledgement Tis researc was supported by te Ministry of Science and Innovation (Spain) and te European Regional Development Fund under project MTM--C-- and by te Catalan Researc Council under project 9SGR References [] F Aguiló and MA Fiol, An efficient algoritm to find optimal double loop networks, Discrete Mat (99), 9 [] R Beivide, E Herrada, JL Balcázar, and A Arruabarrena, Optimal distance networks of low degree for parallel computers, IEEE Trans Comput (99), 9 [] JC Bermond, G Iliades, and C Peyrat, An optimization problem in distributed loop computer networks, in Combinatorial Matematics: Proceedings of te Tird International Conference (New York, 9), Ann New York Acad Sci (99), [] A Brawer, and J Sokley, On a problem of Frobenius, J Reine Angew Mat (9), [] P Esqué, F Aguiló, and MA Fiol, Double commutative-step digraps wit minimum diameters, Discrete Mat (99), [] MA Fiol, JLA Yebra, I Alegre, and M Valero, A discrete optimization problem in local networks and data alignment, IEEE Trans Comput C- (9), [] J Gómez, EA Canale, and X Muñoz, On te unilateral (, D )-problem, Networks () (), [] J Gómez, EA Canale, and X Muñoz, Unilaterally connected large digraps and generalized cycles, Networks () (), [9] M Miller and J Sirán, Moore graps and beyond: A survey of te degree/diameter problem, Electron J Combin () (), DSv [] P Morillo, MA Fiol, J Fàbrega, Te diameter of directed graps associated to plane tesselations, Ars Combin A() (9), [] JLA Yebra, MA Fiol, P Morillo, and I Alegre, Te diameter of undirected graps associated to plane tesselations, Ars Combin B() (9), 9