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Synchronizing Periodic Clocks in Kahn Networks

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(1)Synchronizing Periodic Clocks in Kahn Networks Albert Cohen, Marc Duranton, Christine Eisenbeis, Claire Pagetti, Florence Plateau, Marc Pouzet. To cite this version: Albert Cohen, Marc Duranton, Christine Eisenbeis, Claire Pagetti, Florence Plateau, et al.. Synchronizing Periodic Clocks in Kahn Networks. [Research Report] RR-5603, INRIA. 2005, pp.38. �inria-00070404�. HAL Id: inria-00070404 https://hal.inria.fr/inria-00070404 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Synchronizing Periodic Clocks in Kahn Networks Albert Cohen, Marc Duranton1 , Christine Eisenbeis, Claire Pagetti, Florence Plateau2 and Marc Pouzet2. N° 5603 2005. N 0249-6399. ISRN INRIA/RR--5603--FR+ENG. Thème COM. apport de recherche.

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(60) ÃÑG v ŽG{_x‡ u yzp {‰c~Ue¶‡Çv_± w ³”“”vrUe rUa‡{’r ´v_~|cp2yŠ(v) x (0 + 1)nω uv_~U~Ue6pt‰Gv4x‡eÇc{4u¼rt„†oº´iyŠrtartace cy†{4qy†uSxZwcdgGe6~Up´iacvPpse›y†x4rUe~Ue6psr2±²v4~”psoZx‡u|a‡~tv4xPo ´{_p2¶‡~|psricy†pUuv’‘4e~Ue6Zo & ¡ A (Y³ ® e›cexcv_rte,Zo Q rtace pteriv±yŠxc¶‡xcyŠrte,‰+e~UyŠvqqyzuScyŠxG{~Uoº´2v4~U‡pPrta‡y†p pter”uv_y†x‡uyzqe,´iyŠrtartace,~|{’rUyŠv4x‡{„Ɛqyz{_cy†uSxZwcdˆ+e~|p 2 & ¡ A (Y³ ŒOer qe6xcvrUeºrUace „†exc¥_rta v± ³ ŒHer qe6xcvrUeºrUace xZwcdgGe6~gv_± ' pgyŠx w {x‡ |w| rtace xZwcdgGe6~,|w| v±ž_pSyŠx w ³ŒOer w[n] qex‡wvrte9rUace n|w|°0rU1a „†ertrte~ˆv± w ±²v4~ n ∈ N {xG w[1..n] rtace‰c0~Ue¶cÇ v_±„†ex‡¥rta n v± w ³º•–x rUace~te6d{y†xcyŠx‡¥´e´iy†„Š„?{_„Š´{Foqp›uv_xGpsyzqe~,y†xq¶‡x‡y™rUe‰Ge6~ty†vqqy†u‡yŠx‡{_~toN´v_~|cp ´iyŠrtaDyŠxq¶Gxcy™rUegxZwcdgGe6~ v_± '  p”v_~Xe6ÀPwcy†‘’{„†exPrt„†o_Ž+´iy™rUaN‰+e~UyŠvq (v) uv_xPr|{y†xcyŠx‡¥º{’rX„†e6{4p‚rXv_xce ' ³ `acyzp uv_~U~teIps‰+v_xGcprUv~te6dv’‘Py†xc¥grtace,y†xPrte6¥_e~xZwcdgGe6~Up±²~Uv_d Q2 {_x‡uv_xGpsyzqe~Uy†xc¥gv4xc„†o Q2 − N ³ `ace6~tegy†p›{_xNyŠxc¶‡xcyŠrtexZwcdˆ+e~Sv±?~Ue‰‡~teIpse6x4r|{’rUyŠv4x‡p±²v_~›{x yŠxc¶‡xcyŠrte9‰Ge6~ty†vqqy†uˆcy†x‡{~Uo¾´2v4~UÆ³S•–xq° cee6HŽ (0101) y†peIÀPw‡{„Prtv (01) {x‡ˆrtv 01(01) ³Oš‡v_~trtwcx‡{rte6„Šo4Ž’rtace6~teeÇqyzp‚r|pW{ x‡v_~Ud{_„Z~te6‰c~teIpse6xPrU{’rUyŠv4x ±²v4~{xZo y†xq¶‡xcyŠrtei‰+e~Uy†vZcy†ucy†x‡{_~to,´v_~| w HµyŠry†pWrtace ÂÃwcxcyzÀ4w‡eIÄptacv_~trteIp‚rW~Ue‰c~Ue6ptexPrU{rty†v_xgv±‡rUace±²v_~Ud Žc´ia‡y†u|ay†p”e6ÀPwcy†‘F{_„Še6xPrrUv9rtace,~Ue‰c~Ue6ptexPr|{’rty†v_x´iy™rUartace ptacv_~trteIp‚ri‰c~Ue¶‡Ç{x‡‰+e~UyŠvqÆ³ u(v)® e qex‡vrte,Zo v rtace,‰‡~te¶cÇ ~Ue„z{’rty†v_xO³`aZw‡p„†er w {xG w0 +eXr‚´vcy†x‡{~Uoº´v_~|cpŽ w v w0 ⇒ ‡  Š y ‡ x {_~to¾´v_~|½Â²‰+v4pUpsy†c„†oRyŠxc¶‡xcyŠrteIÄŽHpswGu|aRrta‡{r w.v = w0 ³,šcv4~Xy†x‡psrU{_x‡ue4Ž 010101 y†p›{ ‰c~te¶cÇRv± ∃v ³ (01) ŒOer qexcv_rteSrta‡eS‰+v4ptyŠrty†v_xv±OrUace °Lrta y†x ³ ® e,aG{F‘_e Ž 1 = 1 [1.w]p = [w]p−1 + 1 yŠ± p > 1[w] Ž‡{_x‡p [0.w] = [w] + 1 ³Wšµy†x‡{_„Š„†o_Žcp„†er”wG1piqew¶‡x‡e›rtacDe  Ð[ |Î[1.w].  $ Í;Î ~te6„†{rty†v_x  qe¶‡xceIZo p. p. w1  w2 ⇐⇒ ∀p ≥ 1, [w1 ]p ≤ [w2 ]p .. OQP ^O6R.

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(79) Ð Z  [Ð  p = |u0| − |u|  /Í n = ŠÎ (|v|, |v0 |) ) Ï Í |u| ≤ |u0 | +)²F   Í     Í w = u.v[1] · · · v[p]. (v[p + 1] · · · v[p + |v|])n/|v| / w0 = u0 .(v 0n/|v | )   010(001100) / # |Î|Ñ.    Í 10001(10) M : Z zË +Sс/ Í T01000(110000) Ì $Í #ђÐUÎ T) Ï  )  [Ð 10001(101010)  Q ¼Ë  /Í ˼1Ò 0 Q Ë›Ï  U! T)²FÏ 9Ë ! ÍGÒ # Ð9Ñ #  Ë  w =    Í  a(b) Í w0 = a0 (b0 ) Z K)²Ï |a|1 = |a0 |1 = max(|u|1 , |u0 |1 ) / |b|1 = |b0 |1 = ŠÎ (|v|1 , |v 0 |1 ) . / $     Í $   C+Ë¼Ò Ñ’EË |u|1 ≤ |u0|1 Í |v|1 ≤ |v0 |1 p = |u0|1 −|u|1 r = [v]p Í n = †Î (|v|1 , |v0 |1) 0. ^3^`_!ab

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