Balanced Sequences and Optimal Routing
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Balanced Sequences and Optimal Routing Eitan ALTMAN Bruno GAUJAL and Arie HORDIJK. N˚ 3180 Juin, 1997. THÈME 1. ISSN 0249-6399. apport de recherche.
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(248) v©gnUg¡ ¢r¼xwYhgagaY pa` ! #
(249) % r}B`mgapacBhgnY~,
(250) v©gnUgaUBYKpabtgnY` 3 y : 3 y # 3 h¦ ¡, 8 ypaY|Bxwbd Y¯bdx{gaY pmbtgnv©}Yx{j¿gnU Yµ´ m`8jHgaUBY¯x{YgmgnY pm` y # y Y}Yg8b ¢r}pm~ 8 r}Hbtxw|BUBbdCYhg ! y # #
(251) %
(252) v©gnUHpabtgnY` 3 y # 3 y # 3 h¦ Yhg$cB`]`aUBr gnUb(g 8 à v{`]btxÉbd Y ~R¦zfv{B Y 8 v{` bdxwbdBY ~àyfgnUBY$cBXCYp
(253) rd¤R´ ¶ ` v{bdvwgaY pmtbdxCrd¤xwYBdgaU M v{`c¯} r}pU} : yl¤&r}p `mr}XZYC} ¦ ir y ¤&r}p 8 ygnU YcBXCYp]rd¤ ´? y ¶ `j& M paY`a|à¦Ý´ ¶ ` VO vw³`acBUHbt³v{gnY pktbdx vw`¬Y v©gnUBYp } µr}p} } : Zv©¤ } vw`
(254) rl~B~ObdB~.} r}pc} : v{`} v{`
(255) YdY à¦ÝTVUBv{`i|Bpmr}Y `
(256) gnUBbtg 8 C v{`
(257) bdxwbdBY ~ঠir yfbd`a`mcBXZYignUBbtg y # # ÒbdpmY
(258) gaUBpaYY]~Bv\[CY paYg cBXCYp ¦q½¹Yibd`m`acBXZYigaUbtg y ¦ ½·Y
(259)
(260) v{x{xBgnpmjgnr rdB`mgapacBh3g bº3 `aY ±3cBY BYCr
(261) v{gaUgaUBY `mY¬pmY `m|CYgnv©}YipnbtgaY `Òr}µgnU Yibtxw|BUBbd3 CYhg !3 #
(262) # 3 % ¦ K c 7 C6
(263) 7 1k G R UR7
(264) -C 4r F TVUBYpaYÝYhefvw`kgn`;b¬|btvwp rd¤ r}B`mY c gnv©}YÝ´} ¶q
(265) v©gnUBrV´
(266) a¶Òv{SYg¡ ¢Y Y`av{BY y ¦;TVUBv{`XY bdB` gnUb(g gnU Y8`aY±cBY BY `]´) ¶r}p ´T } ¶8bd|B|SY:bdp¦, ¤;´)} ¶8bd|B|SY:bdpm` yCgaUBY KbZ´ ¶3 v{`]BY 3 Y `a`nbtpav{x{jH`mcBpapmr}cBB~ Y ~ jMg¡ ¢rZ´} ¶¡` ¦. Ú Ú ød£hâ& ä.
(267) [. %-G0$
(268) ~-G j" K c 7 C6
(269) 7
(270) G)
(271) ~ R UR7
(272) -C 4r F. TVUBY5 paYYhefvw`kg$bZ|bdv{p
(273) rd¤= r}B`mY c gnv©}Yº´
(274) a¶º
(275) v©gnUOBr´ ¶$vwOCYhg¡ YY ঠTVUBvw`¬`aY±cBY BY8v{`irt¤=gaUBYº¤&r}paX
(276)
(277) a¶n¦ ir y
(278) ² v{`MBrdg¯|Cr}`m`av{Bx{YKCY:bdc `aYHrd¤ºgaUBYK|BpmY `aYB YHrt¤]?´ TG}¶KbtB~
(279) a£¡pmY }cBxwbdpmv{g¡jd¦ ² 5 v{XZ|Bxwv{Y `MgaUBYKYPe v{`mgaY BY®rd¤i´ 5@|7y G 5@|7y ¶³pj }£¡pmY }c xÉbdpmv{g¡j
(280) UBv{U,vw`Mv{Br}XZ|btgnv{Bx{Y.
(281) v©gnU ´
(282)
(283) a¶CY:bdcB`mYrd¤
(284) a£¡pmY }c xÉbdpmv{g¡jd¦]TVUBYpaYh¤&r}paY² u f¦ irdgaYgnUBbtgigaUBv{`ºbdx{`arMv{XZ|Bxwv{Y `ignUBYYhefvw`kgnYB Y rd¤0´ '}¶ºbdB~Ord¤ ´
(285) )
(286) }¶¦ K c 7 C6
(287) 7 1
(288) TG
(289) lR7
(290) -/C 4r F gnU Y ¢r}pas~ r X8c `mgi r}gbtvw.b´ n¶¦ÝTVUBvw`V´ ¶v{`
(291) BY Y `m`nbdpmvwx©j.`mcBpapmr}cBB~ Y ~Kjg¡ ¢rZ´ }¶¡`¬`av{B Y$´ '}¶ Yhefv{`mga` j }£¡pmY }c xÉbdpmv{g¡jd¦¾TVUBv{` }pmr}cB|¯v{` Y Y`a`abdpav{x{j¯`acBpmpar}c B~BY ~jZg¡ ¢r´
(292) a¶ `¢`mvwBY¬´
(293) '
(294) a¶iYhefv{`mgn`y BY Y `m`nbdpmvwx©jO`acBpmpar}c B~BY ~³jMg¡ rµ´ ¶ ` ¦¾½·Y}YgVgaUBY`aY ±cBYB Yº´
(295) TG
(296) }¶¦ & r u
(297) GTG
(298) Gk F irx{YgmgnY pbdpmr}cBB~HgaUBvw``mY ±cBY Y¯ bd®CYZb´ n¶µSY :btcB`aYb`aY±lc Y BY
(299) v©gnU®¤&r}cBpºx{YgagaY pm`btB~®Br ´ ¶ Yhefv{`mgn`;´
(300) )
(301) a¶¦ ¬r}BY¬:bdCYib]´
(302) a¶
(303) `mvw Yib`mY ±cBY Y]
(304) v©gnUgnUBpmY Y¬xwYhgagnYpa` bdB~Brº´
(305) a¶ybtXYx{j ´T}¶iYhefv{`mgn`¦ÒTVUBY pmY¤&r}pmYdyfgaUBYjUb:}YignrSYig¡ ¢r´ ¶ ` ¦¾TVU Y BrdgaYigaUbtg gaUBYi¤&rdxwx{r
(306) vwBg¡ rµx{YgmgnY pm` :btBBrdg¢CYib]´ n¶ M&CY:bdcB`mY]rd¤CgnUBY¬Yhefv{`mgaY BY$rd¤C´
(307) '
(308) }_¶ OPyfBrdp bd´ }¶ M SY bdcB`aYird¤RgaUBY¬Yhefvw`kgnYB Y rd¤V´
(309) GTn_¶ O8`mr¿v©gZv{`b³´
(310) a¶yqgnU Y ¤&r}xwx{r Y~? Y Y`a`abdpav{x{j j bd¹´ }¶¦ H gµgaUBv{`|Sr}v{g:yÒ YUb:}YgnU Y `aY±cBY BY´ e
(311) )
(312) TG
(313) )
(314) ¶¦ Ç rdgnU ObdpaYºBY Y`a`nbtpav{x{jM´ ¶ ` ¦ T=rHYB~¼gnUBY¯|Bparlrd¤ïy¾BrtgnYMgnUBbtgµbO´ n¶¯ Y Y`a`abdpav{x{j«Ubd`gnU YM`aY±lc Y BY´
(315) '
(316) }¶r}¹v{ga`µpav{}Ug bdB~KgaUBY`abdXZYµ`aY ±cBYB Yr}Kv©gn`ºx{Y¤ g ¦ I;v{bdx{x{j.BrdgnYgnUbtgignUBYµ`mY ±cBYB Y8´
(317) G)
(318) }¶µv{`$BY Y `m`nbdpmvwx©j `ac papardcBB~BY~.jMg¡ ¢rZ´ ¶ ` ¦qTVUBYpaYh¤&r}paYºgnU Y ¢r}pms~ r vw`
(319) |SY pav{rl~Bvwrd¤gnU Y¤&rdpab X
(320) TG
(321) k ¦ . . . . SUT . 6Z . s#b f0Z. . . . TVUBYZ:bt`aY u v{`º}Ypmj®`av{XZvwxwbdpºgnrgaUBYµ:bd`mY u ¤&r}pº~ vw`kgnv{Bgpnb(gnY `¦52ir ¢Y}Yp
(322) UBY¿g¡ ¢r pnb(gnY `¬bdpmYºY ±cbdx ygnUBv{`
(323) :bd`mYv{`
(324) XZr}paYºr}XZ|Bxwv{:b(gnY ~R¦ ?A@ ÁÅ Á 1 - -J C 1;L 3 y # 3 # 3 # 3 to ~?$&0-C L -J C Ll3
(325) C $&7 } 3 y # 3 # 3 # 3 du 6'
(326) # '
(327) # '
(328) # '
(329) F G-/$6$KHF
(330) ½·Yµ`acB|B|Sr}`mYgaUbtg 3 y 3 3 3 bdB~H ¢Y`mUBr gaUbtg$gaUBY pmYv{`ºr}Bx©jKrdBYµbdxwbdBY ~ ¢r}pa~Z
(331) v{gaU¤&pmY ±cBY v{Y ` 3 y 3 3 3 btB~gaUBr}`mY
(332) ¤&pmY ±cBYB v{Y `Vbtpa}Y 6'
(333) # '
(334) # '
(335) # '
(336) h¦ TVUBY|Bparlrd¤vwdr}x{dY `$~ v [CY pmY g¬`mgnY|B` ¦ K qr $&7 el 7 C6
(337) . TG t$&- .) l$&- 1k.) C$&- 1T.TG RF TVUBY pmYµYhefv{`mgn`ºg¡ ¢r r} `aY c gnv©}Y8´ }¶¡`]
(338) v©gnUH r´
(339) a¶v{HCYhg¡ YY ®CY:bdc `aY y ¦ºTVUBYpaYh¤&r}paYty Y v©gnU Y p¯´ '}¶.r}p´ T}¶Hrdp´ T.)}¶Hrdp?´ T.)¶.r}p´ .T}¶.Yhefvw`kg:¦ , ¤º´ '}¶KYhef3 vw`kgn`y]3 gnUBYày$b·´ n¶Kv{` `ac papardcBB~BY~.jMg¡ ¢rZ´ }¶¡` ¦ K qr $&7 el 7 C6
(340)
(341) )
(342) I;v{pa`mg yf ¢Yº`aUBr ,gnUBbtg v©¤=b8`aY±lc Y BY$´
(343) 5
(344) a¶]Yhefv{`mgn`y gnU Y ² u f¦ ,¡ ~BY Y~ày gnU Y]¤ bdg¢gnUb(g¢´ T.)}¶ r}p ´ T}¶ºrdp ´ T.'}¶$r}p ?´ .TG5&|7¶$y Yhefv{5&`m|7giy Xbt»dY `V´
(345)
(346) a¶btB~´
(347)
(348) a¶vwXZ|Sr}`a`mvw xwYt¦ $OgaUBYrdgnUBYp
(349) UbdB~àyv©¤ ² yCgaUBYYhefv{`mgaY BY5 rt¤;?´ ¶8v{`$BY Y`a`nbtpmjHj £ paY}cBxwbdpav©g¡jHbd ~Kvw`ºv{Br}XZ|btgnv{Bx{Y8
(350) v©gnU gnU YYPefvw`kgnY Y8rt¤ ´
(351)
(352) a¶SY bdcB`aYrd¤
(353) a£¡paY}5 cBxwbdpav©g¡j}¦ ir yv{¤CBr`aY±lc Y BYird¤SgnU Y
(354) ¤&rdpaX ´
(355)
(356) a¶
(357) YPefvw`kgn` ygnU Y ZgnUBYpaY
(358) Yhefv{`mgÒg¡ r rdB`aY c gav{dY
(359) ´
(360) a¶¡`¾
(361) v©gnU r}BY
(362) ´ .}¶]bdB~ r´ n¶ivwSYg¡ ¢Y Yà¦ÝTVUBvw`V`mY ±cBYB PY @y1 ¢v{` rd¤ gaUBY]¤&r}pmXO ?´ (
(363) Gj.' {
(364) ̶¦ irdgaY]gnUb(g gnU YcBXCYpa` + # E# } # ;Xb:jSYY±cbdx0gnr YparBc g + : :p} :< j
(365) a£¡pmY }c xÉbdpmv{g¡jd¦ . . /. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ø&ÙÚ0Ø&Û.
(366) [}[.
(367)
(368) . !#"%$&' )(. VT UBY pmYbdx{`arYhefv{5`mg¬g¡ r r} `aY c gnv©}Y´ n¶¡`V
(369) v©gnUOBr´ .}¶$vw³CYhg¡ YY ঠTVU vw`¬`aY±lc Y BYz PySv{`
(370) rd¤ gnU Y¤&r}paX¬´ G
(371) n¶¦ irdgaYgnUb(g+ # E# } # # M # ² bdpmYµvwgaY }Ypa`ºgnUb(gº:bd®~Bv\[CY pjHbtgXZr}`kgr}BYdy gnU Y]x{Y BdgaUrd¤ @yPy$,O:²Z @y a # $ ² : M : 2²-: M : u RO:² P¦qTVUBvw` vw`¢vwXZ|Sr}`a`mvwBx{Y zj n£¡paY}cBxwbdpav©g¡j}¦ K 7 6
(372) 1
(373) TG
(374) )
(375) kG
(376) 5 10 L j sr FTVU Y paYµYPefvw`kgn`$g¡ ¢rOr}B`mY cfgnv©}Y8´ ¶ `
(377) v©gnUKBrµ´ .¶ºvw³CYhg¡ YY ব¤&par}X gaUBY| parlrd¤qrd¤ YXZXbMf¦w[: y Y»l r gaUbtgibµ´ n¶vw`¬BY Y `m`nbdpmvwx©j `ac papardcBB~BY~·jKgaUBYµ`aY±lc Y BYZ´
(378) G}¶n¦8TVUBYpaY¤&rdpaYµ YZUb:dYdi´
(379) GTG
(380) G)
(381) G
(382) ¶ny=
(383) UBYpaY v{`b `aY±cBY BYgaUbtgVr}gbdv{B`
(384) Br´ .}¶]bdB~Br´ ¶n¦ § bdBBrdg `mgbtpmgV
(385) v©gnUObd´ }5¶ M&CY:bdcB`mYrd¤à´
(386) TG
(387) aK¶ O :btBBrdg
(388) `kgbdpkgV
(389) v{gaU³b´
(390) a!¶ M&CY:bdcB`mYrd¤à?´ TG}_¶ O :btBBrdg
(391) `kgbdpkgV
(392) v{gaU³b´ n¶ºbd ~O bdBBrtg
(393) `mgbtpmgV
(394) v©gnUOb ´.}¶$j rdB`mgapacBhgnv{r}à¦ÒTVUBY pmY¤&r}pmY §Ubd`VgnrZSYYXZ| g¡j}¦ K .r L407 17 '
(395) P Lj7 -G $ s$K 7 -G $ 1
(396) kG
(397) T.)
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