Deadlock-Free Fully Adaptive Routing in Irregular Networks without Virtual Channels

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1 Dedlock-Free Fully Adptive Routing in Irregulr Networks without Virtul Chnnels Dong Xing School of Softwre Tsinghu University Beijing , Chin Zhigng Yu Dept. of Computer Science nd Tech. Tsinghu University Beijing , Chin Jie Wu Dept. of Computer nd Info. Sciences Temple University Phildelphi, PA Astrct This pper proposes new flow control scheme in VCT-switched irregulr networks. Bsed on the new scheme, novel dedlock-free fully dptive routing lgorithm is introduced. The lgorithm does not need ny virtul chnnel. It requires tht ech input port of switch holds t lest two 1- pcket-sized uffers. The flow control scheme is proposed sed on seline routing scheme, where the downstrem nodes check the numer of sfe uffers t the upstrem nodes. The proposed fully dptive routing lgorithm is on the sis of different seline routing schemes: up*/down*, nd multiple spnning tree sed routing schemes. Extensive simultion results vlidte the effectiveness of the proposed method s compred to well-known existing pproches. Keywords Flow control, irregulr networks, dptive routing, multiple spnning trees. I. INTRODUCTION Networks of worksttions (NOWs) or clusters re recognized s good lterntives for prllel computing due to their competitive cost/performnce rtio nd wire flexiility [19]. Prcticl networks such s Autonet [25], Myrinet [1], nd Server-Net [9] re exmples of high-performnce with irregulr interconnects. The regulr networks, such s meshes/tori/hypercues, re unrelistic due to vritions in module sizes nd shpes, which re not suitle for clusters or NOWs. We consider virtul cut-through (VCT) switched NOWs ecuse we think VCT is more populr for NOWs or clusters. It is essentil to propose n effective dedlock-free dptive routing lgorithm in irregulr networks [33]. Wu nd Sheng [29] proposed dedlock-free routing scheme for irregulr networks using prefix routing. Three different multicst lgorithms were proposed for wormhole-switched irregulr networks in [11], with their respective node orders to reduce contention. Methods in Bolotin, et l. in [2] nd Meji, et l. in [17] minimize the size of the routing tle t ech router for NoCs with irregulr topologies. Flich, et l. in [8] proposed routing scheme LBDR for different vritions of the 2D meshes, which voids ny routing tles. It is found tht the LBDR still cnnot support some topologies. Rodrigo, et l. [24] proposed new mechnism, clled ulbdr, tht dpts to ny irregulr topologies derived from 2D meshes. The method in [24] still does not need ny routing tles. Cno, et l. in [3] proposed new routing methodology nd router implementtion for complex SoCs. This method voided routing tle look-up y mpping the irregulr network into 2D mesh with constnt or reduced logic cost, regrdless of the system size. A generl methodology for the design of dptive routing lgorithms for networks with irregulr topology hs een proposed in [26] y extending the Duto s protocol to the irregulr networks. Puente, et l. in [21] proposed pseudo- Hmiltonin-cycle-sed dptive routing mechnism for irregulr NOWs, which leds to pprent improvement over the clssicl up*/down* routing lgorithm, which comes t n cceptle extr cost. Puente, et l. in [22] proposed n dptive router rchitecture for irregulr NOWs, which improves network cpilities y llocting more resources to the fstest nd most-used virtul network. However, there my still exist trffic congestion sed on the method in [21], [22], ecuse single spnning tree (ST) ws used. Turn models were proposed for high-performnce routing in irregulr networks [10], [13], [18], [27]. These methods usully set the minimum numer of prohiited turns. A zoneordered lel-sed routing scheme ws proposed for irregulr networks with multiple spnning trees (s) in [18]. Any pcket is delivered long the ordered zones in [18]. There my e difficulty in finding the est zones nd zone order, especilly when the numer of STs is lrger. The Min contriutions of this pper include: (1) the generl frmework for fully dptive routing lgorithm in VCT-switched irregulr networks is presented, sed on simple flow control scheme; (2) dedlock-free fully dptive routing lgorithms re presented sed on the flow control scheme for up*/down*, nd routing schemes. In the rest of the pper, Section II presents the preliminries. The new flow control scheme for irregulr networks is in Section III. The dedlock-free dptive routing lgorithms re presented in Section IV. Simultion results re presented in Section V, nd Section VI concludes the pper. II. PRELIMINARIES Some ckground is provided first. The technique to select roots for the s, ST ssignment, nd constrined turn selection proposed in [31] re presented fter tht. A. Bckground The up*/down* routing scheme in irregulr networks ws proposed for Autonet networks in [25] y selecting single root nd then estlishing n ST. Its generl strtegy ws sed on selecting routes in n ST, where the pckets go up the ST on leving the source nd then, come ck down towrd the destintion.

2 () Fig. 1. An irregulr network with single ST: () the irregulr network with eight nodes, nd () the ST for up*/down* routing. Let the r e the root. The distnce dist(u, r) from u to r is greter thn tht from v to r. The link (u, v) is up. Otherwise, the link is down. As the dist(u, r) = dist(v, r), the direction of link (u, v) is set from the higher ID to the lower. The routing pth from source to destintion is estlished in such fshion tht zero or more up links (towrds the root) should e trversed efore zero or more down links cn e trversed (wy from the root) in order to rech the destintion. Fig. 1() presents n irregulr network with eight nodes, nd the ST for the up*/down* routing scheme when the node 1 is selected s the root is shown in Fig. 1(). The drwcks re tht the selected pths my not e the shortest pths nd tht links ner to the root cn e congested nd ecome ottlenecks. It ws shown in [20] tht dedlock-free routing without using virtul chnnels cn e successfully used in cycles of ny size. A restricted injection mechnism is pplied to ny pcket to void dedlocks tht is trying to enter the cycle which is clled Bule Flow Control(BFC). If under no circumstnces the storge spces for pckets in cycle re llowed to ecome full, the pckets trversing long this cycle will lwys e le to dvnce. B. Root Selection for Multiple Spnning Trees The method in [31] selects multiple roots for dptive dedlock-free routing in irregulr networks. It voids congestion t the root of the ST for the trditionl up*/down* routing. We use different metrics for the first root nd the rest roots. This is resonle ecuse selection of the first root is to minimize the verge distnce etween ny pir of nodes. We hve, F R = min r { u,v 3 8 () dist r (u, v)}, (1) where dist r (u, v) is the distnce etween u nd v ccording to the up*/down* routing scheme [25] in the ST with the root r. The lrger the size of cycle, the more influentil it is on the performnce. The remining roots re selected to minimize the numer of constrined turns for removing ll cyclic input port dependencies nd minimize the verge distnce for ll pirs of nodes sed on the metric s presented in Eqution (2), SR = u,v dist(u, v), (2) CT where CT is the numer of incresing constrined turns to remove ll cyclic dependencies in the input port dependency grph. The method in [31] selects the node s the root tht produces the mximum metric s presented in Eqution (2). In Eqution (2), u,v dist(u, v) stnds for reduction for distnce etween ll pirs of nodes. In ll cses, the distnce etween pir of nodes does not increse fter the second root is inserted nd the second ST is estlished. C. Spnning Tree Assignment A simple wy to select the ST for pir of nodes is: determine the ST ccording to the length of the miniml pth for them in the corresponding ST. The ST with shorter pth is selected for the pcket. However, lod-lncing should lso e considered when selecting the ST. Tht is, the numer of pckets with the sme source should e evenly ssigned to ll STs. We cll n ST ccording to its root. Consider delivering pcket from node 5 to node 7 s shown in Fig. 2. The miniml pth selected to deliver the pcket in the ST 1 y the originl up*/down* routing scheme is 5-1-7, where the pcket is delivered long n up link (5,1) first, nd down link (1,7) fter tht. The length of the pth is two. If the pcket is delivered vi the ST 8, the miniml pth selected y the up*/down* routing scheme is Four hops re necessry. Therefore, it is etter to deliver the pcket from node 5 to node 7 long the pths in the ST 1. If pcket from node 2 to node 6 is delivered in the ST 1, the miniml pth cn e The miniml pth to deliver the pcket is when it is delivered cross the ST 8. Tht is, the pth lengths in oth STs re equl. It is ok to deliver the pcket in oth STs. D. Constrined Turn Selection for Dedlock Avoidnce Fig. 2() presents the input port dependency grph of the irregulr network with input ports s shown in Fig. 2(). The vertices of the grph re input ports. There is dependency from u to v if pcket request uffer t v when it occupies uffer t u. There my exist some potentil cyclic dependencies. We propose new technique in [31] to constrin smll numer of turns insted of introducing some prohiited turns. A scheme like the ule flow control [20] is dopted in irregulr NOWs for dedlock voidnce. Tht is, the constrined turns re llowed when the uffer requirement t the input port is stisfied. The following metric is used to select constrined turns. m(v i ) = size(c), (3) v i C where the input port i of node v is contined in the cycle C. Our method ssigns enefit metric size(c), the size of the cycle C, for ech input port when the node nd its input port re contined in the cycle. The size of cycle is the numer of links to estlish the cycle. The reson why we ssign ech input port the size of the cycle is tht cycle of igger size cn e more influentil to the performnce. Our method selects the input port with the most metric. The enefit metric for ech node nd its input port is updted fter constrined turn node hs een selected. Tht is, the enefit metric of ny node nd its input port, tht re contined in the

3 1 d c c 1 d c c c c 2 c 3 5 2c c d 4c 1 c () : constrined turn in spnning tree with root 8 : the spnning tree with root () : constrined turn in spnning tree with root 1 : the spnning tree with root 1 Fig. 2. Cyclic chnnel dependencies in n irregulr network with STs: () the irregulr network with input ports, nd () the cyclic dependencies. flow-control-irregulr-network() Inputs: coordintes of the current node C, coordintes of the destintion D, free uffers: (f 1, f 2,..., f n), sfe pckets: (s 1, s 2,..., s n ) Output: selected output chnnel. 1) S := ; ch := null; 2) if C = D, ch := internl; 3) for ech next node i of C in the shortest pth to d ccording to the seline routing function R, do if (flow-control(i)) S S {i} ; 4) if S, ch := select(s). Fig. 3. The generl frmework of fully dptive routing lgorithm in irregulr networks sed on the new flow control scheme. cycle, is reduced y the size of the cycle. The ove process continues until ll cycles hve een removed. As shown in Fig. 2, the input port of 4 c, 3, 5, nd 7 re selected to remove ll six cyclic input port dependencies. III. THE NEW FLOW CONTROL SCHEME FOR IRREGULAR NETWORKS Recently, Luo nd Xing [15] proposed fully dptive routing lgorithm for tori without ny virtul chnnels sed on new flow control scheme. Tht work is the initil motivtion of this pper. In this section, we propose the generl frmework of new fully dptive routing lgorithm for irregulr networks sed on the flow control s shown in Fig. 3. The seline routing function R is dedlock-free, where it cn e the up*/down* routing scheme, -sed routing [31], nd ny other refined dedlock-free routing schemes. The input uffers of flow-control-irregulr-network() re orgnized s dynmiclly llocted multi-queues. Two queues re needed to void dedlocks. By this cse, the performnce would e even etter. Two clsses of pckets, sfe nd unsfe pckets, re defined in flow-control-irregulr-network(). Bsed on routing lgoflow-control(i) Inputs: the numer of free uffers: f i, nd the numer of sfe pckets: s i. Output: whether the pcket cn route to the next node. 1) If (f i > 1) return true; exit. 2) If (f i = 1 nd s > 0), return true; exit. 3) If (f i = 1 nd s i = 0), nd the next hop conforms to the seline routing scheme R, return true; exit. 4) Return flse. Fig. 4. The generl flow control function using R s the seline routing scheme. rithm R s the seline routing scheme in n irregulr network, pcket is sfe to the downstrem node if it is delivered long hop provided y the dedlock-free seline routing scheme R; otherwise, it is unsfe. In the rest of this pper, we sy pcket is sfe or unsfe mens tht it is sfe or unsfe to the current node if not specificlly defined. Fig. 3 presents the generl frmework of the fully dptive routing lgorithm for irregulr networks sed on the new flow control scheme nd seline routing scheme R. Here, f i, nd s i stnd for the numer of free uffers nd sfe pckets in the input port of the ith neighor directly connected to the current node C, respectively. The input of this lgorithm including coordintes of the current node nd the destintion, the numer of free uffer numers nd specil pcket numers of ll neighoring input ports. The ville chnnel set nd the selected output chnnel re initilized s nd null. If the current node is equl to the destintion, the internl chnnel is selected. Otherwise, for ech upstrem node v in the minimum pths from the current node to the destintion checks whether the pcket would e forwrded through the function flow-control() s presented in Fig. 4. The function flow-control(i) returns 1 when the pcket is sfe to the ith neighor nd free uffer is ville. Otherwise, it returns 0 when the input port is unsfe to the pcket. Function flow-control(i) voids filling ny input port with only unsfe

4 flow-control(i) Inputs: the numer of free uffers: f i, nd the numer of sfe pckets: s i. Output: whether the pcket cn route to the next node. 1) If (f i > 1) return true; exit. 2) If (f i = 1 nd s > 0), return true no mtter whether the next hop conforms to up*/down* routing; exit. 3) If (f i = 1 nd s i = 0), nd the next hop conforms to the up*/down* routing scheme, return true; exit. 4) Return flse. Fig. 5. Flow control function using up*/down* routing s the seline routing scheme. -sed-route(v,d) 1) Deliver the pcket from v to d if v is in the miniml pths from v to d in the ssigned ST in the following wy until it reches the destintion d, ) If the link is constrined turn, it cn e delivered if the input port of the next hop contins two empty uffers. ) Otherwise, it cn e delivered if the input port of the next hop contins t lest one empty uffer. c) Consume the pcket if the locl node hs een the destintion d. Fig. 6. -sed dedlock-free routing. pckets. It checks how mny free uffers (f) nd sfe pckets (s) in the input uffer of the ith input port.our method selects n output chnnel from S if it is not. Otherwise, the pcket is put in the witing list nd locked. The key point of the lgorithm flow-control-irregulrnetwork() is the flow-control(i) function. It voids filling ny input port with only unsfe pckets. The input uffers re orgnized s dynmiclly llocted multi-queues. There re two queues t ech input port: n sfe pcket queue nd unsfe pcket queue. All sfe pckets to node in n input port re linked s sfe pcket queue. The rest re linked s n unsfe pcket queue. So the sfe pckets would not e locked y unsfe pckets. In ll cses, no cyclic dependencies will e introduced y the dptive hops, therefore, the proposed fully dptive routing lgorithm is dedlock-free. We shll not provide more detiled proof on dedlock freedom of the proposed lgorithm in this pper. Fig. 5 presents the flow control function for the seline routing scheme up*/down* routing. Wht is ment y the next hop conforms to the up*/down* routing is tht one of the following three conditions must e met when the ith neighor of C is in the shortest pth from C to d: (1) the pcket reches the current node C vi n up link nd expects to rech the ith neighor vi n up link; (2) the pcket reches C y n up link nd the next node vi down link; (3) the pcket reches C y down link nd the next node vi down link. The sed routing scheme [31] cn e used s the seline routing scheme. The preliminry section presents more detils of the lgorithm. The -sed routing scheme provides dedlock-free routing ecuse ny possile cycle contins t lest one constrined turn, which introduces no dedlock configurtion. The following rules determine whether flow-control-(i) Inputs: the numer of free uffers: f i, nd the numer of sfe pckets: s i. Output: whether the pcket cn e delivered to the next node. 1) If (f i > 1) return true; exit. 2) If (f i = 1 nd s i > 0), no mtter whether the next hop conforms to the seline routing scheme R, return true; exit. 3) If (f i = 1 nd s i = 0), the next hop conforms to the seline routing scheme nd the input port of the next node i is not constrined turn, return true; exit. 4) Return flse. Fig. 7. The flow control function using sed routing s the seline routing scheme. pcket cn e delivered to the next node when one of the sed routing schemes is used s the seline routing scheme. Rememer tht pcket is ssigned to single ST. As shown in Fig. 7, ny pcket is delivered in the sme ST sed on up*/down* routing except the dptive hops. f > 1, the pcket could e delivered ecuse there is more thn one free uffers in the next node. f = 1 nd s > 0 no mtter whether input port of the next node is constrined turn, the pcket could e delivered ecuse there is t lest one sfe pcket in the next node which cn lwys free the occupied uffer. f = 1 nd s = 0 nd the next node is not constrined turn, the pcket could e delivered if the hop conforms to the seline routing scheme; otherwise, keep the pcket in the witing list. f = 0, keep the pcket in the witing list. The -sed dedlock-free dptive routing is presented in Fig. 6. It is cler tht the pcket cn e delivered to the neighor in shortest pth if the input port contins more thn one free uffer (f > 1). The pcket cn e delivered to the free uffer of the neighor if it contins one free uffer nd one sfe uffer no mtter whether the input port of the next node is not constrined turn ecuse the sfe uffer cn lwys e freed. The pcket cn e delivered to the next node if the hop conforms to the seline routing scheme, there exists free uffer nd no sfe uffer in the input port of the next node, nd the input port of the next node is not constrined turn. In ll other cses, the pcket cnnot e delivered to the next node. Wht is ment y conform to the sed routing scheme? Two consecutive hops, if none of them is n dptive hop provided y the sed routing, they must follow one of the following rules: (1) the pcket reches the current node C vi n up link nd expects to rech the next node vi n up link; (2) the pcket reches C y n up link nd the next node vi down link; (3) the pcket reches C y down link nd the next node vi down link. IV. DEADLOCK-FREE ADAPTIVE ROUTING BASED ON THE NEW FLOW CONTROL SCHEME We present new dptive routing scheme with unfixed STs. We then introduce the fully dptive routing lgorithms

5 dptive-route-up-down(c, d) Inputs: the current node C, nd the destintion d, directions for ll links in the ST, nd sttes of uffers t the neighors. Output: the next chnnel. 1) For ech neighor i of C in shortest pth to d, do steps 2), 3), 4), nd 5); 2) if the pcket reches C long up link nd the link to the next node i is still up link, nd oth hops re provided y the seline routing scheme, S S {i} when flowcontrol(i) is true; 3) if the pcket reches C long up link nd the link to the next node i is down link, nd oth hops re provided y the seline routing scheme, S S {i} when flowcontrol(i) is true; 4) if the pcket reches C long down link nd the link to the next node i is still down link, nd oth hops re provided y the seline routing scheme, S S {i} when flow-control(i) is true; 5) for ll other neighor i of C in shortest pth to d, i is not in S, if flow-control(i) is true nd i is not in S, S S {i}; 6) if S, ch := select(s). Fig. 8. Adptive routing using up*/down* routing s the seline routing scheme. with different seline routing schemes without ny virtul chnnel: (1) up*/down* routing, (2) sed routing, nd (3) unfixed sed routing. A. Fully Adptive Routing Using Up*/Down* Routing s the Bseline Routing Scheme We first propose the fully dptive routing lgorithm y using the up*/down* routing scheme s the seline routing scheme. Fig. 8 presents the detiled routing lgorithm. The inputs of the lgorithm includes the coordintes of the current node, nd the destintion, directions of the ll links of the ST, the root of the ST, nd the uffer sttes t the input ports of the current node C s neighors. Steps 2), 3), nd 4) present ville inputs port of the neighors s the next hop which conform to the seline routing scheme, up*/down* routing. The routing lgorithm in Fig. 8 shows three different cses when the next hop conforms to the seline routing scheme up*/down* routing. The next node must e in the shortest pths from C to d. The three cses include: (1) the pcket reches the current node long up link nd the link to the next node i is still up link, where oth hops re provided y the seline routing scheme; (2) the pcket reches C long up link nd the link to the next node i is down link, where oth hops re provided y the seline routing scheme; nd (3) the pcket reches C long down link nd the link to the next node i is still down link, where oth hops re provided y the seline routing scheme. After ll next hops long the shortest pths hve een identified, our method checks the next hops tht do not conform to the seline routing scheme. However, one of the uffer conditions presented y flow-control(i) in Fig. 4 must e stisfied. The selection function finlly selects one of the next hops in S to deliver the pcket. The possiility to estlish cyclic dependency y the proposed fully dptive routing lgorithm is tht one or more dptive hops. However, ny dependencies estlished y the dptive-route-(c,d) Inputs: the current node C, nd the destintion d; directions of ll links in ; ssignment of the pckets to ll STs. Output: the next chnnel. 1) For ech neighor i of C in shortest pth to d, do steps 2), 3), 4), nd 5); 2) if the pcket reches C long up link vi the seline routing function provided hop nd the link to the next node i is still up link in the jth ST, if flow-control-(i) is true, S S {i}; 3) if the pcket reches C long up link nd the link to the next node i is down link in the jth ST, nd oth links re provided y the seline routing function, if flow-control- (i) is true, S S {i}; 4) if the pcket reches C long down link nd the link to the next node i is still down link in the jth ST, nd oth links re provided y the seline routing function, if flow-control-(i) is true, S S {i}; 5) for ll other neighor i of C in shortest pth to d, if flowcontrol-(i) s presented in Fig. 7 returns true, nd i is not in S, S S {i}; 6) if S, ch := select(s). Fig. 9. Adptive routing using sed routing s the seline routing scheme. dptive hops cn e eliminted ecuse pcket cn dvnce y hop not provided y the seline routing scheme when the input port of the next node is sfe to the pcket. B. Fully Adptive Routing Using Multiple Spnning Tree Bsed Routing s the Bseline Routing Scheme Fig. 9 presents the fully dptive routing lgorithm sed on the new flow control scheme when the sed routing scheme is used s the seline scheme. The sed routing scheme is very simple, which is presented in Fig. 6. Initilly, the s re estlished nd the pckets re ssigned to the STs for lod-lncing considertion. A pcket is delivered long the fixed ST. However, some of the hops, tht do not conform to the seline routing scheme, do not follow this. Similr conditions s presented in Fig. 8 must e met when oth consecutive hops in the sme ST re provided y the seline routing scheme. The three cses include: (1) the pcket reches the current node long n up link nd the link to the next node i is still n up link; (2) the pcket reches C long n up link nd the link to the next node i is down link; nd (3) the pcket reches C long down link nd the link to the next node i is still down link. Our method then checks ll other next nodes in the shortest pths from the current node to the destintion whether one of the uffer conditions s presented in Fig. 7 cn e met. If so, the next node cn e provided y the fully dptive routing lgorithm. In ny cse, the pcket wits for one of the uffers provided y the seline routing scheme in the fixed ST when trffic congestion occurs. C. Dedlock Freedom Proof The fully dptive routing lgorithms re dedlock-free ecuse: (1) there re no cyclic dependencies in the sme ST ecuse the pckets follow the dedlock-free seline

6 Ltency(cycles) Trffic (flits/cycle/switch) ltency(cycles Trffic (flits/cycle/switch) Ltency(cycles) Trffic (flits/cycle/switch) A 0.3 A A 0.2 A Injection Rte(flits/cycle/switch) Injection Rte(flits/cycle/switch) Fig. 10. Performnce comprison with previous methods (,, A, ) in n irregulr network (32,64) A 0.12 A 100 A 0.08 A Injection Rte(flits/cycle/switch) Injection Rte(flits/cycle/switch) Fig. 11. Performnce comprison with previous methods (,, A, ) in n irregulr network (64,128) A 120 A A A Injection Rte(flits/cycle/switch) Injection Rte(flits/cycle/switch) Fig. 12. Performnce comprison with previous method (,, A, ) in n irregulr network (128,256). routing such s: up*/down* routing or sed routing, whose proofs cn e found in [25] nd [31], respectively; (2) oviously, the dptive hops provided y the new flow control scheme cnnot introduce ny cyclic dependencies [15]; (3) no cyclic dependencies cn e estlished cross different STs, ecuse ny cycle contins t lest one constrined turn. In ll, we cn conclude tht the ove fully dptive routing lgorithms re dedlock-free. V. SIMULATION RESULTS The proposed routing lgorithms nd flow control scheme for VCT-switched irregulr networks hve een implemented in cycle-ccurte simultor y C++ [31]. Different seline routing schemes re used including up*/down* routing (), multiple spnning tree sed routing (). The implemented fully dptive routing lgorithms sed on the proposed new flow control scheme re A nd A. The previous methods, the originl up*/down* routing scheme [25] (), the dptive routing scheme sed on Duto s protocol [26] (), nd the sed routing scheme () [31], re lso implemented. Links re rndomly dded to the network, while the degree of ech node is no more thn four [32]. Two metrics re used to evlute the performnce of method: the ltency to deliver pcket (cycle) nd the ccepted trffic (flit/cycle/switch). The size of pcket is set to 16 flits for ll simultion results. The topology of the irregulr network is generted rndomly, while we present the verge results of 10 irregulr networks. We consider three STs for ll simultion results relted to multiple spnning trees nd the uniform trffic pttern for ll simultion results. We do not consider routing tle storge nd look-up for previous methods ( nd ), nd just present the performnce comprison with them. Fig. 10 presents the performnce evlution of irregulr networks with 32 switches/nodes nd 64 links. The up*/down* routing scheme reches the sturtion point very quickly when the pplied lod reches 0.1. The extended Duto s

7 protocol reches the sturtion point when the pplied lod hs reched The dptive routing lgorithm A with the up*/down* routing scheme s the seline routing works etter thn nd, whose sturtion point is out 0.26, while the ccepted trffic is lso pprently etter thn the previous two methods. The most recent work (), sed on multiple spnning tree, works etter thn A, lthough it is not fully dptive. The sturtion point of is little lter, which is t The dptive routing lgorithm A is sed on the flow control scheme, which uses the multiple spnning tree sed routing s the seline routing scheme. The A routing lgorithm pprently works etter thn,, A, nd on ccepted trffic nd ltency to deliver pcket in ll cses. Figs. 11 nd 12, present the performnce evlution on irregulr networks with 64 switches nd 128 links, 128 switches nd 256 links. Fig. 11 presents the performnce comprison of the proposed lgorithms with previous methods. The lgorithm does not work etter thn nd A on ltency to deliver pcket when the injection rte is very low, just like tht in the networks with 32 switches. As shown in Fig. 11, the fully dptive routing lgorithm A still works etter thn when the injection rte reches The reson should e tht it is esy for nd A to reserve dptive chnnels when the injection rte is not high enough. The order for the lgorithms to rech the sturtion points is still,, A, nd A. Fig. 12 provides performnce comprison of the new lgorithms with previous ones in networks with 128 switches nd 256 links. The situtions re still the sme. The lgorithm works worse thn A when the injection rte is less thn 0.05, which consistently works etter thn,, nd A fter tht. The A lgorithms work even etter in ll cses. The proposed method voids routing tle look-up y implementing the routing tle look-up through some very simple extr logic. The detils for the router rchitecture re not presented in this pper. Therefore, the sclility is not prolem for the proposed method. In ll simultion results for nd, the extr ltency to look up the routing tles is not included. VI. CONCLUSIONS A new flow control scheme ws proposed for irregulr networks, which provides series of fully dptive routing lgorithms for VCT-switched (or pcket-switched) irregulr networks. Our methods provide dedlock-free dptive routing scheme using unfixed multiple spnning trees, tht is, pcket cn chnge its spnning tree t the intermedite nodes in order to reduce ltency to deliver it. The fully dptive routing frmework is pplied to up*/down* routing, multiple spnning tree, nd the unfixed multiple spnning tree routing schemes. Simultion results show tht the proposed flow control scheme cn provide very effective fully dptive routing lgorithms without ny virtul chnnels, nd ring 10% improvement when compred to the originl routing lgorithms. REFERENCES [1] N. J. Boden, D. Coen, R. E. Feldermn, A. E. Kulwik, C. L. Seitz, J. Seizovic nd W. 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