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"O ; Table of contents  $Circulant graphs$ $ Permutation routing$ $ Routing algorithms (1)$ $GRouting algorithms (2)$ $Static algorithms$ $ESimulation program$ $Simulation results$ $LTwo-terminal routing$ $TStatic two-terminal routing$ $XDynamic two-terminal routing$ $ZDynamic two-terminal routing$ $\Dynamic two-terminal routing$ $5 Conclusions   $ $/'(,-.3 4 6FHPWY[]P   0` 3ff` ` ___>?" dF@0?n2d@uK FA@ " d`  n?" dd@   @@``PR   @ ` `PBp>> L0  " bZ( = j2  B?   Nx.gֳgֳ ?P . T Click to edit Master title style! !.  HX.gֳgֳ ? . RClick to edit Master text styles Second Level Third Level Fourth Level Fifth Level!     S  T.gֳgֳ ?`p . ^*  T.gֳgֳ ?`  . `*  T.gֳgֳ ?`@ . `*N  6޽h? ? 3f3ff $Generic (Standard)  0 L0  "    ( |  vB  N1?44j2  B?   N/gֳgֳ ?  / T Click to edit Master title style! !  H  H BJ f >  H < J fP > A H <$J ``0,$ 0  Circulant graphs&`4 $   r H S %JPHAp  J U H N9Jgֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$  H T4 n ............ the number of nodes h1, h2 .... jump offsetst?#?$  d BJ E6 h a part of a graph G(16; 1, 2, 4)!!!$F  d C A k30`  d 0,J4 na graph G(8, 2, 3)* $F d C A k6~T D~| d BĒJ   4 a 2-jump circulant graph is denoted by G(n, h1, h2)55$ r d S  JP Pp  J  d BXJ P+  4 nodes are labeled with integers 0, 1, ..., n-14%5$  d BJ B   the node x is connected with x + h1 (mod n) x  h1 (mod n) x + h2 (mod n) x  h2 (mod n)l     $ U d NJgֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H d 0޽h ? 3f!fy___PPT10Y+D='  = @B +7   0 L0    h_ (  h h s *J@ F$ h c $xJ f  P the routing is finished when all packets have reached their destinations 029Q$B h 0J`f h at each node of the network we have one message (packet) with exactly defined destination node,H;.  9i$ h 0Jki ?w the destination of a packet is defined by a permutation; a packet from the node x in destinated to the node p(x),|:>  (r$ $  h c $D @p,  0 at each routing step a packet can move from the current node to one of the neighbouring nodes, BUT only one packet can traverse each link ^9hM $r  h S  JYYj     h T  ?"f  3The aim is to minimize the number of routing steps.44!3$ U  h Nx gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H h 0޽h ? 3f!f`  0 L0 l@(  l  l s *l0 @ F$  l Z1 gֳgֳ ?2ee   _ l T3  ?"P 7Routing algorithm = preprocessing phase + routing phaseX8!! ! !!!7$   l T<  ?". P {)calculating the global routing parameters**)$  l TA  ?". 4P u#several routing steps in each node$$#$ ~ l H ?"PG .~ l H ?"PG .B l N D?">Qa.G l TF  ?"@v  3At each routing step the routing algorithm has to: D4 3$ P l TN  ?"~ l  H determine the packets to proceed towards their final destinations,8+xx !?H$  l TV  ?"0 *f   reserve the appropriate links,*!! $  l T[  ?" f6  send the selected packets and* !$ @ l T`  ?"0Qf & prevent and/or solve the collisions. J'!!&$ U l Np gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H l 0޽h ?/@lll lll 3f!f   0 L0     (    s *lv @ F$  Zw gֳgֳ ?2ee   ~  H ?"PG .  T$z  ?"~  oSTATIC ROUTING ALGORITHMS!$   Tt  ?"  pDINAMIC ROUTING ALGORITHMS!$   T̃  ?"F k| ~(K - THE QUALITY OF THE ROUTING ALGORITHM))!($ <  T  ?" tAll the parameters needed to solve the given permutation routing problem are calculated in the preprocessing phase.t$ S  T  ?"  The routing parameters are calculated during particular routing step; the calculations are made based on the current packet configuration.$   T,  ?"U :v.s.$ 5  T  ?"PuF 7K = AVG ( ).8 ,,,7$ .  T  ?"d  4 number of routing steps . maximum packet distance*54$ U  N gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H  0޽h ?  3f!fy___PPT10Y+D='  = @B +F  0 L0 UM-3|(  | | s *l @ F$: | 3 g : | 3  ? : | 3  : | 3 R: | 3 )4B  | # jJ 44B !| # jJh4B "| # jJ  4B #| # jJ! I : $| 3 !I ~~ %| 6/ I c RouteOffDDE    @ &| C jJI ~~: '| 3 ! x!  (| 6/"   c RouteOffDDD    @ )| C jJ x! : *| 3 !x  +| 68   c RouteOffDDC    @ ,| C jJx : -| 3 !4xh .| 6 j4 c RouteOffDDB    @ /| C jJ4xh: 0| 3 !x  1| 6X   c RouteOffDDA    @ 2| C jJx ^B | 6DjJ7^B | 6DjJy^B | 6DjJ  ^B  | 6DjJ0 W   | c $ H XK=1.98 $  | c $ aHH XK=1.37 $  | c $d H  XK=1.35 $  | c $  H  XK=1.20 $ | c $\ x H_ XK=1.14 $XB |@ 0DXB |@ 0DXB |@ 0DP P XB |@ 0D   | 0 c ;in certain circumstances the odd/even criteria is disobeyed<;<$ | 0 Q 8 o)the long hops are not privileged any more***$ | 0  @Ht  ;the odd/even criteria is replaced by the distance criteria <;<$ | c $    .the strategy for choosing packages is improved /./$r | S h 2ee   U 3| N gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H | 0޽h ? 3f!fy___PPT10Y+D='  = @B +g   0 L0 v n  (    s * @ F$  ZL gֳgֳ ?2ee   ~  Z ?"0 L  ROUTE = program used to simulate a permutation routing in a given network with a given algorithm$ ^  c 6AjavalogoET   Z ?"W  u# network, permutation, algorithm.$$#$   Z ?"v XINPUT:$   Z`" ?"c [OUTPUT:$   Z& ?" p routing plan, routing costs.$   NX%  ?"_  WROUTE$   N !!?" @  N !!?" @  N41  ?"` p| xBWith the ROUTE program we have made about 3.000.000 simulations. B$ U  NL gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H  0޽h ? 3f!fy___PPT10Y+D='  = @B +  0 L0   LF( Ly0t0 L  L N 3f?#" ` m 3 IM L 0 P8 K = number of routing steps/longest distance o = number of routing steps - longest distance T = maximum waiting time v = waiting queue length/number of vertices$l L 0A ??    L s *̬ @ F$  L Z( gֳgֳ ?2ee   U  L N gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ H L 0޽h ? 3f!fy___PPT10Y+D='  = @B +   0 L0  (    s * @ F$  Z gֳgֳ ?2ee   U  Nx gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$    NB  ?"t + There is only one package in the network,,,$    N  ?"Q9   The aim is to route the package along the shortest path connecting source and destination node and to minimize the number of routing steps"!$    N  ?"7  m  = $ ?   N  ?"F c2b  Q If there are more then one shortest paths the algorithm can choose among them+'RR$ H  0޽h ? 3f!fy___PPT10Y+D='  = @B +3   0 L0 B: 8(  8 8 s *~ @ F$ 8 Z, gֳgֳ ?2e   U 8 NH gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ K 8 N,  ?"M4i ] A static algorithm calculates a routing path before the package leaves the source node5)^^$ , 8 NL  ?"`l|  > During the routing, pre-planed routing strategie is used)??$ " 8 N  ?"c j  Zadvantage: less complicated routing algorithms at each node[$ .  8 Nİ  ?"9 j; fdisadvantage: deficiency of flexibility (problems in heavy-weighted networks due to collisions)g$ H 8 0޽h ? 3f!fy___PPT10Y+D='  = @B +  0 L0 @E(  @ @ s * W@ F$ @ Z gֳgֳ ?2e   U @ N,˴ gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$  @ N8ϴ  ?"M4 ; A dynamic algorithm calculates the routing path on-the-fly<<<$ : @ NTٴ  ?"J  radvantages: - algorithm can solve or even avoid potential routing problems - identical nodess$  @ N  ?" w  ]!disadvantages: - slow decisions"$ H @ 0޽h ? 3f!fy___PPT10Y+D='  = @B +s  0 L0 z H(  H H s *$:/W@ F$ H Zl gֳgֳ ?2e   U H N gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$  H N  ?"M*  JSolving the  slow decisions problem:&$   H N4  ?"H@8___PPT9 eAlgorithm for calculating the shortest paths in circulant graphs using classical strategy (CS)Gff$ @`P  H NH  ?"   ~2) Algorithm for calculating the shortest paths in circulant graphs using the restricted shortest path algorithm (RSP)$ @`H H 0޽h ? 3f!fy___PPT10Y+D='  = @B +s  0 L0 zP(  P P s * W@ F$ P Zx" gֳgֳ ?2e   U P N gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$  P N/  ?"M* `$Comparation - CS v.s. RSP approach:%$ + P N;  ?"-  YOur RSP approach has in more than 11% off all cases larger set of possible routing paths.Z$ @`H P 0޽h ? 3f!fy___PPT10Y+D='  = @B +C   0 L0 R J ` ( 8}@   s *M/@ F$  Z|O/gֳgֳ ?2ee  /   NR/  ?" &  " U  Nh^/gֳgֳ ?`W G,$  0 ROUTING IN CIRCULANT GRAPHS.F!$ R  Nb/  ?"M$5i d We have designed and evaluated several permutation routing algorithms for circulant graphs6/ee$ ?  Nn/  ?" 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