Performance Comparison between Network Coding in Space and Routing in Space

Transcription

1 Performnce omprison etween Network oding in Spce nd Routing in Spce Yunqing Ye, Xin Hung, Ting Wen, Jiqing Hung nd lfred Uwitonze eprtment of lectronics nd Informtion ngineering, Huzhong University of Science & Technology (HUST), Wuhn, , P.R. hin mil: rxiv: v1 [cs.it] 14 Jul 2014 strct Network coding in geometric spce, new reserch direction lso known s S pce In f ormtion low, is promising reserch field which shows the superiority of network coding in spce over routing in spce. Present litertures proved tht given six terminl nodes, network coding in spce is strictly superior to routing in spce in terms of single-source multicst in regulr (5+1) model, in which five terminl nodes forms regulr pentgon centered t terminl node. In order to compre the performnce etween network coding in spce nd routing in spce, this pper quntittively studies two clsses of network coding in spce nd optiml routing in spce when ny terminl node moves ritrrily in two-dimensionl ucliden spce, nd cost dvntge is used s the metric. urthermore, the upperound of cost dvntge is figured out s well s the region where network coding in spce is superior to routing in spce. Severl properties of S pce In f ormtion low re lso presented. I. Introduction Network In f ormtion low (NI) [1], studying network coding in grphs, ws proposed in NI cn improve the throughput of network nd reduce the complexity of computing the optiml trnsmission scheme [2]. The rtio of mximum throughput of network coding over tht of routing is known s coding dvntge [3]. S pce In f ormtion low (SI) [2], studying network coding in spce, ws proposed y Li et l. The spce here refers to the geometric spce. In this pper, we focus on twodimensionl ucliden geometric spce. In the SI model, informtion flows re free to propgte long ny trjectories in the spce nd my e encoded wherever they meet. The purpose is to minimize nturl network volume, which cn support end-to-end unicst nd multicst communiction demnds mong terminls in the spce, nd network volume represents the cost of constructing network. Tking the unique encoding ility of informtion flows into ccount, SI models the fundmentl prolem of informtion network design, which deserves more reserch ttention. The rtio of the minimum routing cost nd minimum network coding cost in terms of required throughput is known s cost dvntge () [3]. is used s the metric in the study of SI. urthermore, cost dvntge nd coding dvntge re dul. Yin et l. [4] studied the properties of SI, such s onvexity property nd onvex Hull property. The literture [4] proved tht if the numer of given terminl nodes is three, is lwys equl to one. However, the cses where the numer of the given terminl nodes is greter thn three hve not een discussed. Xihou et l. [5] proposed unified geometric frmework in spce to investigte the Li-Li conjecture on multiple unicst network coding in undirected grphs. Hung et l. [6] proposed two-phse heuristic lgorithm for pproching the optiml SI nd constructed the Pentgrm model. urthermore, the literture [6] proved tht the vlue of is in the Pentgrm network. Hung et l. [7] studied the regulr (n+1) model in which n terminl nodes formed regulr polygon centered t nother terminl node, nd proved tht only when n=5, network coding in spce cn e superior to routing in spce. The pentgrm network is equivlent to the regulr (5+1) model. However, the cses where ny given terminl node is llowed to move ritrrily, lso clled the irregulr (5+1) model, hve not een discussed. Zhng et l. [8] discussed the region where 1 in the irregulr (5+1) model, when only one terminl node is plced on the vertex of the regulr pentgon is llowed to move long the circumcircle. Wen et l. [9] discussed the region where 1 in the irregulr (5+1) model, when only one terminl node tht is plced on the vertex of the regulr pentgon is llowed to move in spce ritrrily. ut the cse where the center terminl node is llowed to move ritrrily s well s the properties ssocited with this cse hve not een discussed. This pper studies two clsses of irregulr (5+1) Model nd compres their differences nd similrities in order to study the performnce of network coding in spce nd the properties of SI. The contriution of this pper is tht we quntittively compre the performnce etween network coding in spce nd routing in spce through studying two clsses of irregulr (5+1) Model, nd we otin some properties of SI, the upper ound of nd the region where 1. The orgniztions of this pper re s follows. Model nd definitions re descried in Section II. The performnce of network coding in spce nd routing in spce re studied in Section III nd Section IV, respectively. The numericl nlysis nd results re presented in Section V. Some properties of network coding in spce re discussed in Section VI. Lstly, the conclusions re given in Section VII. II. Model nd efinitions The purpose is to find the min-cost of multicst network coding in spce. The cost is defined s e( e f e ) [2] where

2 e is the length of link e [2] nd f e is the flow rte of link e. efinition 1: (ost dvntge () [3]) is defined s the rtio of the minimum routing cost nd minimum network coding cost in terms of required throughput. is used s the metric to quntittively compre the performnce etween network coding in spce nd routing in spce. efinition 2: (Regulr (5+1) model [6]) Given (5+1) terminl nodes in two-dimensionl ucliden spce, five terminl nodes re the vertices of regulr polygon, whose circumcenter is the terminl node (See ig.1). The center terminl node is considered s the source terminl node, nd the remining five terminl nodes re sink terminl nodes. ig. 1: Regulr (5+1) model r ' ig. 2: Irregulr (5+1) model: () Node lss I; () Node lss II efinition 3: (Irregulr (5+1) Model) ne of the terminl nodes in the regulr (5+1) model devites from its originl position. There re minly two clssifictions: Node lss I nd Node lss II. The terminl node denotes the single source terminl node nd the terminl nodes denote five sink terminl nodes (See ig.2). efinition 4: (Node lss I) The terminl node t the circumcenter is llowed to move ritrrily inside the circle. In this Node lss, the terminl node is the center terminl node tht is llowed to move ritrrily, while node is the center of the circumcircle tht is fixed. r is depicted s the distnce etween the center terminl node nd node, ndθis denoted s where is the midpoint of the line (See ig.2 ()). efinition 5: (Node lss II) ne of the five terminl nodes on the circumcircle is llowed to move ritrrily. In this Node lss, node is t the plce of one sink terminl node in regulr model. r is depicted s the distnce etween the terminl nodes nd, ndαis denoted s (See ig.2 ()). III. Performnce of Network oding in Spce. ost of Network oding in spce for Node lss I The construction of network coding in spce is depicted in ig.3 (), nd the hollow nodes re rely nodes. The shded region where network coding in spce works when 0 θ 36 o nd r 0is shown in ig.3 (). nly the cse thtθrnges from 0 to 36 o clockwise is necessrily the one to e discussed out ecuse of the symmetry nd = 72 o. should e smller thn 120 o ccording to the Lune property [10], otherwise SI cn not help. s shown in ig.3 (), 120 o when the terminl node moves into rc ĀG where G=120 o. r ' ' ig. 3: Network coding model for Node lss I: () construction of network coding in spce; () region where network coding in spce works our source flows in spce lterntely trnsmit messges nd while one coding flow trnsmits the encoded messge, s shown in ig.3 (). onsequently, five sink terminl nodes receive two its of different messges simultneously, nd f e = 1 2 under the ssumption tht the mximum flow of ech sink is unit. Thus the cost of network coding in spce cn e clculted s follows: L N I = 1 2 e( e )= 1 2 r 2 4 sin 66 o r cosθ+4 sin 2 66 o r 2 4 sin 66 o r cos(θ+72 o )+4 sin 2 66 o r 2 4 sin 66 o r cos(θ+144 o )+4 sin 2 66 o r 2 4 sin 66 o r cos(144 o θ)+4 sin 2 66 o r 2 4 sin 66 o r cos(72 o θ)+4 sin 2 66 o. ost of Network oding in spce for Node lss II The construction of network coding in spce is depicted in ig.4 (). The shded region where network coding in spce works when 0 α 48 nd r 0 is shown in ig.4 ()., nd should e smller thn 120 o ccording to the Lune property [10], otherwise, SI cn not help. rom ig.4 (), 120 o when the terminl node moves cross the dshed line where = 48 o ; 120 o when the terminl node moves cross the dshed line where = 120 o ; nd 120 o when the terminl node moves inside rc ŌR where R= 120 o. ' ig. 4: Network coding model for Node lss II: () construction of network coding in spce; () region where network coding in spce works The cost of network coding in spce cn e clculted s follows [9]: G '' R ' '

3 L N II = 3 cos r 2 2r cos (132 α) r 2 cos (132 +α) IV. Performnce of Routing in Spce. ost of Routing in Spce for Node lss I The cost of routing in spce for Node lss I cn e otined y the exct lgorithms [11] of ucliden Steiner Minimum Tree (SMT). The min steps re s follows. irst, generte ll the constructions of ull Steiner Tree (ST). Second, enumerte ll the possile Steiner Trees which re the conctentions of STs. Third, clculte the cost of the Steiner Trees nd choose the minimum one s SMT. 1) Genertion of STs : Generte ll the possile STs of the six terminl nodes in the irregulr (5+1) model. 2) onctentions of STs: We minly consider the conctention of one ST with three terminl nodes nd nother ST with four terminl nodes, nd the resons re similr to [8] [9]. Moreover, the intersection of STs must e the center terminl node rther thn ny of the terminl nodes on the circumcircle. ll the cses where the intersection of STs is the terminl node on the circumcircle cn e represented y the three cses shown in ig.5. ccording S1 S2 ig. 5: Three cses tht should e pruned: () +; () +; (c) + to [10], if two STs shre node z, then the two edges meet t z nd mke t lest 120 o with ech other. However, when two STs shre the terminl node (See ig.5 ()), S 1 S 2 < = 108 o < 120 o. Hence, ig.5 () should e pruned. In ddition, similr proof cn e pplied to ig.5 () nd ig.5 (c). The conctentions of STs cn e divided into five cses, s shown in ig.6, ig.7 nd ig.8. The sucses shown in ig.7 () nd ig.7 (c) re the degenertions of ig.7 (), nd the sucses shown in ig.8 () nd ig.8 (c) re the degenertions of ig.8 (). ig. 6: irst three cses: () +; () +; (c) + (c) () () ig. 7: The fourth cse + hs three sucses: () node is nondegenerte; () node degenertes nd is elow the line ; (c) node degenertes nd is ove the line () () ig. 8: The fifth cse + hs three sucses: () node is nondegenerte; () node degenertes nd is on the right of the line ; (c) node degenertes nd is on the left of the line In the fourth cse, the terminl node is nondegenerte s shown in ig.7 (), when it moves inside the shded region shown in ig.9. s shown in ig.11, nd M re two equilterl tringles constructed to find the ST, nd the old-solid lines construct the ST, in which node S is Steiner node. We construct the rc N tht stisfies N=120 o. s every Steiner node of Steiner Tree hs exctly three lines meeting t 120 o [10], node S is on the rc N when the terminl node moves outside the rc N s shown in ig.11 (). However, when the terminl node moves on the rc N s shown in ig.11 (), =120 o nd the terminl node degenertes into Steiner node. urthermore, when the terminl node moves inside the rc N, >120 o, nd the terminl node lso degenertes. I ig. 9: Region tht the terminl node does not degenerte 120 l (c) ig. 10: the Steiner node S should lie in The terminl node degenertes s shown in ig.7 (), when it moves elow the line nd is outside the shded region shown in ig.9. s the terminl node is elow the line, terminl nodes,,, form convex quilterl, nd ccording to [9] [10], ig.7 () is otined. In ddition, the terminl node degenertes s shown in ig.7 (c), when it moves ove the line nd is outside the shded region shown in ig.9. When the S (c)

4 120 M S N ig. 11: Two cses with supplementry lines: () node lies outside rc N; () node is on the rc N terminl node is ove the line nd forms concve quilterl, the Steiner node should lie in tringle rther thn in tringle. s shown in ig.10, <. Let = β, then = 108 o β. Suppose 120 o, otherwise no Steiner node could possily exist in [10]. Let = x, = = l, then = x 2 + l 2 2xl cos(108 o β), = x 2 + l 2 2xl cosβ. (1) If the Steiner node S lies inside, L 1 denotes the cost of the ST nd it is given y: L 1 = x 2 + l 2 2xl cos(168 o β)+ x 2 + l 2 2xl cosβ, (2) else if the Steiner node S lies inside, L 2 denotes the cost of the ST nd it is given y: L 2 = x 2 + l 2 2xl cos(β+60 o )+ x 2 + l 2 2xl cos(108 o β). Let (β)=l 1 L 2 (0 β 54 o ). lcultions show tht (β) 0, (β) mx =(54 o )=0. Hence, L 1 L 2, nd the Steiner node should lie in tringle nd ig.7 (c) is otined. The fifth cse hs three sucses shown in ig.8, similrly to the fourth cse. 3) omputtions of SMT: omputtions of SMT re divided into five cses, nd SMT is the one tht hs the minimum cost. (1) The first cse: + (See ig.12) r ' G ig. 12: etiled clcultion of the first cse +, G nd H re three equilterl tringles, nd node is the circumcenter, s shown in ig.12. In, =36 o +θ=α, = r nd =1. ccording to the lw of cosines, 2 = 1+r 2 2r cosα, cos( )=(1 r cosα)/. In ddition, we cn otin sin( )=r sinα/, tn( )=r sinα/(1 r cosα) y using the lw of sines. M S H 120 N In G, G = 108 o, G = 2 sin 66 o = β. Similrly we cn otin tn( G ) = (β sin 108 o )/(1 β cos 108 o ), cos( G )=(1 β cos 108 o )/G, sin( G )= (β sin 108 o )/G. In G, 2 = 1+r 2 2r cosα, G 2 = 1+β 2 2β cos 108 o, cos( G) = cos(60 o + + G) = 1/(K G)[0.5 rsin(α+30 o ) β sin 138 o +βr sin(138 o +α)]. y using the lw of cosines, cos G = ( 2 + G 2 G 2 )/(2 G). s result, G 2 = 1+r 2 + 2r sin(θ+ 6 o )+4r sin(θ 6 o ) sin 66 o + 4(sin 2 66 o ) 4 sin 66 o cos 168 o. In H, H = 36 o, = r, H = β = 2 sin 66 o. ccording to the lw of cosines, H 2 = r 2 4r sin 66 o cos(72 o θ)+4 sin 2 66 o. Hence, L I 1 = G+H. L I 1 = [r 2 4 sin 66 o rcos(72 o θ)+4 sin 2 66 o ] 1 2+ [1+r 2 + 2r sin(θ+6 o )+4 sin 66 o r sin(θ 6 o )+4 sin 2 66 o (2) The second cse: + (See ig.6 ()) L I 2 = [r 2 4 sin 66 o r cosθ+4 sin 2 66 o ] 2+[1+r 1 2 2r cos(168 o θ)+4 sin 66 o r sin(66 o θ)+4 sin 2 66 o (3) The third cse: + (See ig.6 (c)) L I 3 = [r 2 4 sin 66 o r cos(θ+72 o )+4 sin 2 66 o ] [1+ r 2 2r sin(θ 6 o ) 4 sin 66 o r sin(θ+6 o )+4 sin 2 66 o (4) The fourth cse: + (See ig.7) When the terminl node is nondegenerte, then <120 o, i.e. r cos(36 o θ)<cos 72 o nd cos( )= 1+2r 2 2r cos(36 o +θ)+4 sin 2 66 o 4r sin 66 o cos(72 o θ) 16 sin 2 36 o sin 2 84 r o > 1 2 4r sin 66 o cos(72 o θ)+4 sin 2 66 o 2, 8 1+r 2 2r cos(36 o +θ) L I 4 1 = [1+ r 2 2r cos(36 o +θ)] 2+ 1 [r 2 4 sin 66 o r cos(72 o θ)+4 sin 2 66 o ] [r 2 4 sin 66 o r cos(θ+144 o )+4 sin 2 66 o ] 1 2 When the terminl node degenertes nd is elow the line, then 120 o, i.e. r cos(36 o θ)<cos 72 o nd cos( )= 1+2r 2 2r cos(36 o +θ)+4 sin 2 66 o 4r sin 66 o cos(72 o θ) 16 sin 2 36 o sin 2 84 r o 1 2 4r sin 66 o cos(72 o θ)+4 sin 2 66 o 2, 8 1+r 2 2r cos(36 o +θ) L I 4 2 = [r 2 4 sin 66 o r cos(144 o +θ)+4 sin 2 66 o ] 1 2+ [1+ r 2 2r sin(66 o +θ) 4 sin 66 o r sin(102 o θ)+4 sin 2 66 o When the terminl node degenertes nd is ove the line, i.e. r cos(36 o θ) cos 72 o, L I 4 3 = [r 2 4 sin 66 o r cos(θ+144 o )+4sin 2 66 o ] 1 2+ [1+r 2 2r sin(66 o +θ) 4 sin 66 o r sin(102 o θ)+ (5) The fifth cse: + (See ig.8) When the terminl node is nondegenerte: L I 5 1 = [1+r 2 2r cos(36 o θ)] 1 2+ [r 2 4 sin 66 o r cos(θ+ 72 o )+4 sin 2 66 o ] 1 2+ [r 2 4 sin 66 o r cos(144 o θ)+4 sin 2 66 o ] 1 2 When the terminl node degenertes nd is on the right of the line : L I 5 2 = [1+r 2 2r cos(36 o θ)] 1 2+ [r 2 4 sin 66 o r cos(θ+ 72 o )+4 sin 2 66 o ] 1 2+ [r 2 4 sin 66 o r cos(144 o θ)+4 sin 2 66 o ] 1 2 When the terminl node degenertes nd is on the left of the line : L I 5 3 = [r 2 4 sin 66 o r cos(144 o θ)+4 sin 2 66 o ] 1 2+ [1+ r 2 2r sin(66 o θ) 4 sin 66 o r sin(102 o +θ)+4 sin 2 66 o

5 . ost of Routing in Spce for Node lss II Methods re similr with Section IV-. etils refer to [9].. Node lss I V. Numericl nlysis nd Results The functionl reltion of nd (x, y) in three-dimensionl is shown in ig.13 (), where only 1 is figured out nd crtesin coordintes (x, y) re otined from polr coordintes (r, θ). urthermore, ig.13 () shows tht chieves its mximum vlue of when r=0. () () ig. 13: ost dvntge 1 (3-): () Node lss I; () Node lss II The two-dimensionl region where 1 is shown in ig.14 (), nd it is otined y projecting ig.13 () to XY plne. When projected to XY plne, ig.13 () turns out to e sector whose ngle is 36 o nd its rdius is etween 0.20 nd Tking the symmetry of the Node lss I model into considertion, the finl projection in two-dimensionl is shown in ig.14 (). The performnce of network coding in spce is superior to routing in spce in the shded region, nd the mximum vlue of is chieved when the terminl node is t the center of the circumcircle (i.e. r=0). ig. 14: Region where cost dvntge 1 (2-): () Node lss I; () Node lss II urthermore, it cn e proved tht the region where network coding in spce outperforms routing in spce is circle, nd the mximum vlue of cn e chieved when the terminl node is t the centre of the circumcircle. irst, trnsform the prolem of 1 to the prolem of L N I min{l I i } (i= 1, 2,..., 5), nd mke sure there is no discontinuity point in the region where 1. Here L I 4 nd L I 5 represents the minimum vlue of L I 4 j ( j=1, 2, 3), nd L I 5 represents the minimum vlue of L I 5 k (k=1, 2, 3). Second, whenθ is fixed, study the monotonicity of function f (r,θ)=l N I min{l I i } in order to confirm tht the region where 1 is not n nnulus or something similr. Third, when r is fixed, study the monotonicity of function f (r,θ)=l N I min{l I i } in order to confirm tht the region where 1 is circle. SMT should e one of the five cses. urthermore, the prolem of otining the region where 1 is equivlent to the the prolem of otining the region where L N I min{l I i }. L N I nd L I 1 to L I 5 re ll continuous functions, ecuse they re the liner comintions of some sic functions. s result, function Y i = L I i L N I (i=1,2,...,5) is lso continuous function. In ddition, f (r,θ) 0 is threedimensionl curved surfce nd s result its projection on the XY Plne is continuous, which mens tht there exists no discontinuity point in the region where 1. rom equtions L I 1 to L I 5, we cn clculte it s follows: d(l N I ) = 0.25 r 2 4r sin 66 o cosθ+4 sin 2 66 o r 2 4r sin 66 o cos(θ+72 o )+4 sin 2 66 o r + 2 4r sin 66 o cos(θ+144 o +4 sin 2 66 o ) 0.25 r 2 4r sin 66 o cos(144 o θ)+4 sin 2 66 o r 2 4r sin 66 o cos(72 o θ)+4 sin 2 66 o, + d(l I 1 ) r+sin(θ+6 = o )+4 sin 66 o sin(θ 6 o ) 1+r 2 +2r sin(θ+6 o )+4 sin 66 o r sin(θ 6 o )+4 sin 2 66 o 4 sin 66 o cos 168 o r 2 sin 66 o cos(72 o θ) r 2 4 sin 66 o r cos(72 o θ)+4 sin 2 66 o Let y 1 = d(l N I ) d(l I 1), whenθ = θ 0 (θ 0 [0, 36 o ]), r [0,0.24]. r is restricted to [0,0.24] ccording to the projection of ig.13 (). y mtl, we find tht r [0, 0.24], y 1 0. Similr results cn e otined when we clculte the other four functions y 2 = d(l N I ) d(l I 2) to y 5 = d(l N I ) d(l I 5), which mens tht Y i = L N I L I i (i=1,2,...,5) re ll monotonous incresing. s mentioned ove, the function f (r,θ)=l N I min{l I i } is continuous, thus the function f (r,θ 0 ) = L N I min{l I i } is lso monotonous incresing whenθis fixed. The significnce of this result is tht if r 0, when r= r 0, f (r 0,θ 0 )=L N I min{l I i }=0, then r 0 is the only prmeter tht cn stisfy the eqution L N I = min{l I i }. In other words, the region where 1 is not n nnulus. In ddition, we find tht when r=0, = Let f (r 0,θ) = L N I min{l I i } (i=1,2,...,5), when r = r 0 (r 0 [0.20, 0.24]),θ [0, 36 o ]. lcultions show tht d f (r when r (0.20 r 0.24) is fixed, 0,θ) dθ 0, which mens tht if r 1 [0.20, 0.24], f (r 1, 0)=0 nd f (r 1, 36 o )=0, then f (r 1,θ)=0 stisfies ll of the possiilities whenθrnges from 0 to 36 o, which mens the region tht stisfies 1 is circle. urthermore, r 1 indeed exists nd r 1 =0.225 y mtl.. Node lss II The functionl reltion of nd (x, y) for Node lss II in three-dimensionl is shown in ig.13 (), where only 1 is depicted, nd crtesin coordintes (x, y) re otined from polr coordintes (r, α). urthermore, ig.13 () shows tht chieves its mximum vlue of when (r, α)=(1,0). The coordintes of node in the irregulr (5+1) model is (r, α)=(1,0). rom ig.13 (), the closer the terminl node

6 moves to the node, the greter the vlue of gets. chieves its mximum vlue when the terminl node coincides with the node. The two-dimensionl region where 1 for Node lss II is shown in ig.14 (). VI. Properties nd iscussion ifferent from tht of routing in spce, some properties of SI cn e descried s follows: (1) ither the center terminl node or one terminl node on the circumcircle moves ritrrily, the mximum vlue of is chieved when the irregulr (5+1) model turns ck to the regulr (5+1) model. (2) The numer of rely nodes cn e greter thn n 2 in SI while this numer cn not e greter thn n 2 in SMT. (3) given terminl node cn hve degree which cn e greter thn three while the degree cn not e greter thn three in SMT. urthermore, the center terminl node (for Node lss I) cn only move in the dshed circle shown in ig.15, whose dimeter is round The minimum distnce etween one terminl node on the circumcircle nd the center terminl node is round (i.e. if the terminl node for Node lss II moves inside the solid-line circle shown in ig.15, will e less thn one), which is nerly equl to the dimeter mentioned ove. Thus, we conjecture Position Independence Property. rom ig.15, whtever the position of ny terminl node outside the solid circle, or if ny terminl node does move nywhere outside the solid circle, the distnce etween the moving node nd the center terminl node will lwys e greter thn the minimum distnce of required to chieve 1. The mximum moving distnce of the center terminl node is equivlent to the minimum distnce etween the center terminl node nd ny other terminl nodes on the circumcircle. Thus, the position of the terminl node does not influence the performnce of SI. In ddition, ny two terminl nodes cn not move too close to ech other, otherwise SI cn not help. or exmple, if the terminl node moves outside the dshed circle nd gets too close to the terminl nodes nd, then it will e too fr from the terminl nodes, nd, which will mke routing in spce superior to network coding in spce, mening tht SI cn not help ig. 15: Position Independence There re following interesting questions to nswer: When ny two terminl nodes re llowed to move ritrrily, does chieve its mximum vlue only when the irregulr (5+1) model turns ck to the regulr (5+1) model? When ny three terminl nodes re llowed to move ritrrily, does chieve its mximum vlue only when the irregulr (5+1) model turns ck to the regulr (5+1) model? Wht if ny four terminl nodes re llowed to move ritrrily? Wht if ny five terminl nodes in this regulr (5+1) model re llowed to move ritrrily, which is equivlent to ll the terminl nodes re llowed to move ritrrily? urthermore, if given six terminl nodes whose positions re ritrry, when nd how is SI superior to routing in spce? In other words, when given n terminl nodes in spce ritrrily, does SI help when n=6? oes chieve its mximum vlue only when the six terminl nodes form the regulr (5+1) model? It is known tht Yin et l [4] proposed questions whether SI cn help when n=4 or n=5. urthermore, Yin hs lredy proved tht SI does not help when n=3. VII. onclusions This work compres the performnce etween network coding in spce nd routing in spce sed on two clsses of irregulr (5+1) model, which re Node lss I nd Node lss II. urthermore, in oth irregulr models chieves its mximum vlue of only when the irregulr (5+1) model turns ck into regulr (5+1) model. The ongoing work is to nswer ove questions in order to study SI properties. cknowledgment This reserch ws supported y Ntionl Nturl Science oundtion of hin (No ). The uthors thnk Zhidong Liu, Wei Xiong nd Rui Zhng for their constructive comments. References [1] R. hlswede, N. i, S.Y.R. Li, nd R.W. Yeung. Network informtion low. I Trns. on Informtion Theory, 46(4): , [2] Z. Li,. Li, nd L.. Lu. n chieving mximum multicst throughput in undirected networks. I Trns. on Informtion Theory, 52(6): , [3] S. Mheshwr, Z. Li, nd. Li. ounding the coding dvntge of comintion network coding in undirected networks. I Trns. on Informtion Theory, 58(2): , [4] X. Yin, Y. Wng, X. Wng, X. Xue, Z. Li, Min-ost Multicst Networks in ucliden Spce. I Interntionl Symposium on Informtion Theory (ISIT), [5] T. Xihou,. Wu, J. Hung, Z. Li, Geometric rmework for Investigting the Multiple Unicst Network oding onjecture. Netcod, [6] J. Hung, X. Yin, X. Zhng, X. u nd Z. Li, n Spce Informtion low: Single Multicst. Netcod, [7] J. Hung,. Yng, K. Jin, Z. Li, Network oding in Two-dimension ucliden Spce. Journl of hongqing University of Posts nd Telecommunictions (Nturl Science dition) ct [8] X. Zhng nd J. Hung, Superiority of Network oding in Spce for Irregulr Polygons. I 14th Interntionl onference on ommuniction Technology (IT), [9] T. Wen, X. Zhng, X. Hung, J. Hung, ost dvntge of Network oding in Spce for Irregulr 5+1 Model. I 11th Interntionl onference on ependle, utonomic nd Secure omputing (IS), [10].N. Gilert nd H.. Pollk. Steiner Miniml Trees. SIM Journl on pplied Mthemtics, 16(1): 1-29, 1968 [11] P. Winter nd M. Zchrisen, xct lgorithms for Plne Steiner Tree Prolems: omputtionl Study. omintoril ptimiztion Volume 6, 2000, pp