Distributed Algorithms for Secure Multipath Routing in Attack-Resistant Networks

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1 1 Diriu Algorihm or Sur Muliph Rouing in Ak-Rin Nwork Prik P. C. L, Vihl Mir, n Dn Runin Ar To proivly n gin inrur rom rily joprizing ingl-ph ion, w propo iriu ur muliph oluion o rou ro mulipl ph o h inrur rquir muh mor rour o moun uul k. Our work xhii vrl imporn propri h inlu: (1) rouing iion r m lolly y nwork no wihou h nrliz inormion o h nir nwork opology, (2) rouing iion minimiz hroughpu lo unr ingl-link k wih rp o irn ion mol, n (3) rouing iion r mulipl link k vi lxiogrphi opimizion. W vi wo lgorihm rm h Boun-Conrol lgorihm n h Lx-Conrol lgorihm, oh o whih provi provly opiml oluion. Exprimn how h h Boun-Conrol lgorihm i mor iv o prvn h wor- ingl-link k whn ompr o h ingl-ph pproh, n h h Lx-Conrol lgorihm urhr nhn h Boun-Conrol lgorihm y ounring vr ingl-link k n vriou yp o muli-link k. Morovr, h Lx-Conrol lgorihm or prominn proion r only w xuion roun, implying h w n rii miniml rouing proion or igniinly improv lgorihm prormn. Finlly, w xmin h ppliiliy o our propo lgorihm in piliz niv nwork rhiur ll h k-rin nwork n nlyz how h lgorihm r riliny n uriy in irn nwork ing. Inx Trm Rilin, uriy, muliph rouing, opimizion, mximum-low prolm, prlow-puh, krin nwork. I. INTRODUCTION In onvnionl rouing proool uh OSPF [29] n RIP [26], nwork l l-o ph o rou rom our o ink. Whil h proool livr iinly, h u o ingl ph i vulnrl o gnrl ilur n uriy hr. For inn, inrur n irup h ion imply y king on o h inrmi link long h uiliz ph. Thi ingulriy nl inrur o rily vo hir rour o king h only ph. Suh nwork n pro vi ur muliph pproh in whih r ipr ro mulipl ph in or h ink. Eh ph onvy porion o rom h our, n h ink ml h rgmn riv rom h vriou ph. I om ph il o livr, hn long h l o ilur i mo, h ink n ill rovr ll uing runn rouing [27] or hrhol r hring [25]. Thror, o uully ompromi h ion, inrur mu uvr uiin numr o rouing ph n hn rquir mor rour hn ho P. L n V. Mir r wih h Dp o Compur Sin, Columi Univriy (mil: {pl,mir}@.olumi.u). D. Runin i wih h Dp o Elril Enginring, Columi Univriy (mil: nr@.olumi.u). A onrn vrion o hi ppr ppr in IEEE INFOCOM 05 [22]. n o k ingl ph. W poin ou h uing mulipl ph n ompli h pk-rorring prolm [31]. Howvr, i n rmi vi ophii oing oluion (.g., [8]) or non-rl-im rnr or nr pruring hniqu (.g., [24]) or rl-im rnr. In iion, mor rn ppliion-lyr rhiur uh ovrly nwork (.g., RON [3] n SOS [20]) provi mor promiing plorm or ploying muliph rouing ompr o onvnionl lyr-3 rhiur. Thror, i i il o op h ur muliph pproh o proivly omplih rouing rilin. On mjor hllng i o ign iriu oluion h implmn h pro o ling h lloion ro mulipl ph hrough nwork. Th iriu oluion nhn riionl nrliz oluion or ur muliph rouing uh [4], [6], [18] in irn wy. Fir, i o no rquir ny nwork no o hv ull knowlg o h nir nwork opology. I i hror qu or nrliz pr ym, uh RON [3], who no r lo in irn omin n r on minir inpnnly. In iion, i llow nwork no o lolly i uriy o, nwih onrin, n hoi o rou, n hu improv lxiiliy ompr o h nrliz pproh. To hrriz h lloion ro mulipl ph, our primry uriy ojiv i o minimiz h mximum mg inurr y ingl-link k (or ilur), i.., n inrur ompromi long ingl link in givn nwork. Thr r wo ron o juiy our prliminry nlyi on ingl-link k. Fir, hr r mny k n ilur nrio whr ingl-link ilur i likly o u h mjoriy o prolm, h nwork n on rpir, or rou r ju, o oun or h ilur or uqun oug our. Nvrhl, w ill wn o miig h mg o ingl-link ilur in i n u vr hroughpu lo in high-p nwork wihin only w on. For xmpl, 10-on oug o n OC-48 link n inur lo o 3 million 1-KB pk [23]. Son, our xprimn how h our oluion h i ign or prvning ingl-link k provi unil rilin o mulipl imulnou k wll. Thu, our nlyi n rv lin or uur work h ou on mulilink k. Unlik riionl lo-lning oluion h minimiz h mximum link uilizion (i.., h mximum rio o h link hroughpu o h link nwih), our ojiv i o gurn rilin uing ll vill nwork rour. Givn irn ion rquirmn, w k o minimiz h wor- ingl-link k whil ining h ir

2 2 hroughpu r wih h proviion nwork nwih. In hi ppr, w vi iriu ur muliph oluion h rmin h muliph rou o mximiz h uriy wih rp o ingl-link k. Our work i uil or wo ion mol, nmly: Fix-r ion: ion h wih o n rom h our o h ink pr-rmin r; n Mximl-r ion: ion h wih o n rom h our o h ink h r llow y ll vill ph in h unrlying nwork. Givn h ov ion mol, w ir propo iriu oluion ll h Boun-Conrol lgorihm, whih provly minimiz h mximum hroughpu lo whn link i k. W ormul hi oluion mximum-low prolm h n olv in iriu hion on h xnion o h Prlow-Puh lgorihm [16]. Uing h Boun-Conrol lgorihm uiling lok, w vi highr-omplxiy, u mor rilin iriu oluion ll h Lx-Conrol lgorihm. I n no only gin h wor- link k, u lo gin link k h o no u h wor mg u r ill vr (.g., h on n hir wor- link k). To hiv hi propry, h Lx-Conrol lgorihm r h o inurr y h link k vnly poil ovr ll h link in nwork, or quivlnly olv lxiogrphi-opimizion prolm [13], in iriu mnnr. By imulion, w vlu h rilin o h Boun- Conrol n Lx-Conrol lgorihm gin irn yp o k on ingl or mulipl link. In omprion o inglph lrniv, our rul ini h h Boun-Conrol lgorihm unilly r h o o h wor- ingl-link k (.g., y 78% in 200-no, 1000-link nwork). Alo, h Lx-Conrol lgorihm n urhr ru, y mor hn 50%, h numr o link h inur vr mg u o ingl-link k, n uh ruion i rliz r only hr or our irion. Whil h rilin nhnmn o h Lx-Conrol lgorihm ovr h Boun- Conrol lgorihm om h xpn o highr omplxiy, our imulion rul how h w n limi hi inr in omplxiy wihou muh lo in rilin y xuing only h ir w irion o h Lx-Conrol lgorihm. Finlly, w monr h ppliiliy o oh Boun- Conrol n Lx-Conrol lgorihm in n k-rin nwork (.g., SOS [20]), piliz nwork h pro n ho wih niv rhiur. Uing [7] our ounion, w nlyz how our propo muliph lgorihm n ploy o provi rouing rilin n in h mnim ur h nwork gin mliiou k. Th ppr pro ollow. In Sion II, w ormul h ur muliph pproh. Sion III n IV prn h Boun-Conrol n Lx-Conrol lgorihm, rpivly. In Sion V, w rpor vrl xprimn h vlu h lgorihm unr irn l o link k. Sion VI iu how o pply our lgorihm in n k-rin nwork n prn imulion rul o hir prormn. Sion VII rviw rl work. Sion VIII iu h pril iu o our work n ugg uur irion. Sion IX onlu. TABLE I MAJOR NOTATION USED IN THIS PAPER. Din in Sion II: N o no L o link G nwork (N, L) our no ink no L(u) o ougoing link l L o no u N X ion hroughpu rom our o ink x l proporion o ion rri y link l L x proporion vor (x l, l L) l uriy onn o link l L l k o l x l o link l L minimiz wor- k o p(l) piy o link l L in mximum-low prolm l low o link l L in mximum-low prolm low vor ( l, l L) in mximum-low prolm ruling mximum-low vlu B l nwih o link l L l rion oun o link l L non-inring k-o qun lxiogrphilly opimiz Din in Sion III n IV: low vlu ro y our U uiinly lrg vlu G riul nwork wih rp o Din in Sion VI: A o poin (AP) T o rg P o ph wn AP n rg G k-rin nwork (A, T, P) A(j) o AP rom whih rg j T i rhl T (i) o rg h n rhl rom AP i A C u vn h no u A T i ompromi D u vn h no u A T i unr DoS k P (E) proiliy h vn E our loking proiliy o AP i A p i II. PROBLEM FORMULATION In hi ion, w ormliz h ur muliph pproh minimx-opimizion prolm n hn i quivln mximum-low prolm. Thi ormulion will lo u lr whn w inlu link-nwih onrin n lxiogrphi opimizion. No h h ollowing ormulion i gnrlly on [1], [2], [4], [5], [13], [15], [16], [25]. To i our iuion, Tl I ummriz h mjor noion h w u in hi ppr. Our iuion rli on h onp o h mximum low n h minimum u [2]. Givn nwork wih numr o no n link, h mximum-low prolm i o rmin h mximum low h n n rom our no o ink no uj o h piy onrin (i.., h link h low oun y h link piy) n h lowonrvion onrin (i.., h n low nring ny no xp h our n h ink qul zro). Suppo h w priion h no ino wo S n S, whr S n S. A u rr o h o link ir rom S o S. A minimum u i h u h h h minimum piy (i.., h minimum um o pii o ll link in h u). Th mx-low min-u horm h h mximum-low vlu qul h piy o h minimum u.

3 3 W r inr in onn, ir, n yli nwork h i viw grph G = (N, L), whr N i h o no n L i h o ir link. Our nlyi i on ingl ion wih our no n ink no. W mphiz h our nlyi n gnrliz o homognou l o mulipl ion y mpping our n ink o h ingr n gr poin o h nwork, rpivly. Suppo h our n o ink wih ion hroughpu givn y X (y, in M/). W l x l, 0 x l 1, h proporion o h nir ion rri y link l L (i.., x l qul h hroughpu o link l ivi y X) n l x = (x l, l L) h orrponing proporion vor. Our nlyi minly ou on ingl-link k, u w lo r ingl-no k in Sion VI. In hi ppr, w ou on hr mol in whih h mg u o h k on link l L no only i proporionl o h hroughpu n ovr link l, u lo pn on ohr or uh h liklihoo h n k n uully ring own link l. W hrriz uh mg n k o l = l x l, whr l, whih w rm h uriy onn o link l, pii h vulnriliy o link l. Inuiivly, h k o i u o mur h l o hroughpu lo u o ingl-link k. No h l n hv vrl phyil inrprion, uh h proiliy h link l i uully k givn h h inrur mp o k link l [5], h ilur proiliy o link l [4], or h proporion o lo o rvring link l whn i i k. To nur h vry link l h onin inrprion o l, vry no h o lir l wih rp o n gr-upon iniion o n k. Alo, o nl u o inrpr l proiliy or proporion, w rquir h 0 l 1 or vry link l L. Wih n gr-upon k mol, vry no u n hn rmin in vn l or h o i own ougoing link l L(u), whr L(u) i h o ll ougoing link o no u, uing vulnriliy moling [10], iil murmn o rliiliy inx [15], or uriy monioring ym [25]. W poin ou h i n ur im o l i no vill, w n l = 1, mning h link l h ll i lo whn i i unr k, n our nlyi ill ppli o hi wor- nrio. A. Minimx Opimizion To miig h wor mg u o ingl-link k, our ojiv i o i il proporion vor x h minimiz h mximum k o ovr ll link in h nwork. Thi n viw h ollowing minimx opimizion prolm 1 : = min mx l = min mx x l L x lx l l L uj o 0 x l 1, l L. (1) Prolm 1 n olv in polynomil im vi linr progrmming, u hi i nrliz oluion n rquir h inormion o h nir nwork opology. To implmn 1 All prolm prn in hi ppr r unr low-onrvion onrin, lhough h onvnion i omi or rviy. iriu oluion, w n ir rnorm h prolm ino mximum-low prolm y ing h piy o vry link l, no y p(l), h riprol o l [1], n hn olv or h mximum low uing h iriu Prlow-Puh lgorihm [16], whih i ummriz ollow. Sour ir inii h lgorihm y puhing h mximum poil low o i nighor no. All no xp our n ink hn mp o puh h low owr ink long h im hor ph unil h ruling mximum low rh ink. Any x low i puh k o our. In [16], i xplin how o implmn h Prlow-Puh lgorihm in iriu n ynhronou hion. W rr rr hr or il iuion. For ompln, w inlu h puo-o o h Prlow-Puh lgorihm in Appnix I. L = ( l, l L) h low vor whr l no h low o link l, n h n low nring ink. Prolm 1 n hu mpp o h ollowing mximum-low prolm: = mx uj o 0 l 1/ l, l L, (2) whr h oluion o Prolm 1 n 2 r rl y = 1/ n x l = l /, l L. To illur oh prolm, Figur 1() pi nwork whr l =1 or ll link l. From h Prlow-Puh lgorihm, w know h mximum low i = 2 n hu h wor k o i minimiz =0.5. Alo, h lgorihm rurn h orrponing vor n x. B. Minimx Opimizion wih Bnwih Conrin On limiion o Prolm 1 i h vry link i um o hv inini nwih o h i n ommo h nir ion. To inorpor h link-nwih onrin, w um h h no u pii priori nwih B l (y, in M/) or i ougoing link l L(u). W l l = min(b l /X, 1), whr 0 l 1, no h rion oun o link l h oun rom ov h proporion o h n n hrough link l or givn ion hroughpu X. W hn inorpor h rion oun ino Prolm 1 : = min mx l = min mx x l L x lx l l L uj o 0 x l l, l L. (3) Th orrponing mximum-low prolm om: = mx uj o 0 l min(1/ l, l ), l L. (4) For lriy, h rm nwih (i.., B l ) rprn h mximum moun o h n n ro link, n h rm piy (i.., p(l) = min(1/ l, l )) no h uppr oun o h link low in h rnorm mximumlow prolm. Whil h nwih B l i ix, h piy p(l) vri pning on h low vlu h rh ink. Figur 1() pi h whr w ign h rion oun l = 0.4 o h link rom no o ink n = 1 o h r. Similr o Prolm 1, w n olv l

4 4 (0.5, 1, -) (0.5, 1, -) (0.3, 1.5, 1) (0.3, 1.5, 1) (0, 0, -) (0, 0, -) (0.5, 1, -) (0.3, 1.5, 1) (0.3, 1.5, 1) (0.6, 3, 1) (0, 0, -) (0, 0, -) (0.5, 1, -) (0.4, 0.67, 0.4) (0.2, 1, 1) (0.2, 1, 1) (0.4, 2, 0.4) (0.5, 1, -) (0.5, 1, -) (0.4, 0.67, 1) (0.4, 0.67, 1) (0.2, 1, 1) (0.2, 1, 1) () = 0.5 () = 0.6 () = 0.6 Fig. 1. Opiml oluion o h hr opimizion prolm: () minimx opimizion, () minimx opimizion wih h nwih onrin, n () lxiogrphi opimizion. Evry link l h l =1 n i oi wih ripl (x l, l, l ), whr x l n l r h oluion r h opimizion prolm r olv, n l (in or () n () only) no h iniil rion oun ign o link l. No h l i irn rom i iniil vlu r h lxiogrphi-opimizion prolm i olv ( Sion IV n Figur 3 or il). Prolm 3 in nrliz mnnr vi linr progrmming. To implmn iriu pproh, in Sion III, w vlop h Boun-Conrol lgorihm, whih i uil upon h Prlow- Puh lgorihm o olv Prolm 4 n hn Prolm 3. C. Lxiogrphi Opimizion A limiion o h prviou prolm i h hy r onrn only wih how o minimiz h wor- k o, u o no mp o ru h o o vr link k. For xmpl, in Figur 1() n 1(), h k o r unvnly iriu. Spiilly, in Figur 1(), hr r ix link who k o r l 0.4 h. By vnly iriuing h o hown in Figur 1(), only wo uh link xi. Thu, w ru h numr o link whr h ingl-link k n l o vr mg. To ormliz h onp o h vn iriuion o k o, w l = l1 x l1, l2 x l1,, l L x l L, whr l 1, l 2,, l L L, non-inring k-o qun. Th iriuion o h k o i i o h mo vn i h oi k-o qun i lxiogrphilly minimiz, i.., or ny ohr non-inring k-o qun = l1 x l 1, l2 x l 2,, ll x l L, hr xi i, whr 1 i < L, uh h lj x lj = lj x l j or j < i n li x li < li x l i. L lxmin (.) h union h rurn h lxiogrphilly minimum qun. W hn xpr h lxiogrphi-opimizion prolm : uj o = lxmin x = lxmin l1 x l1,, ll x ll x x = rg min mx x lx l, l L 0 x l l, l L. (5) Hn, h orrponing mximum-low prolm i: uj o = lxmin = lxmin, l 1 l1,, l L ll = rg mx 0 l min(1/ l, l ), l L. (6) In Sion IV, w propo h Lx-Conrol lgorihm o r hi prolm. By xning h Boun-Conrol lgorihm n ing h rion oun o h link pproprily, h Lx-Conrol lgorihm n rmin h lxiogrphilly opiml oluion or Prolm 5 n 6 in iriu hion. Thi yp o lxiogrphi-opimizion prolm w ir nlyz in [13], rom whih our Lx-Conrol lgorihm h wo min iinion. Fir, whil h nlyi in [13] um no link-nwih onrin, w xpliily inorpor hi onrin ino our lgorihm. Furhrmor, our lgorihm llow iriu implmnion, whil h oluion in [13] i nrliz n rquir h knowlg o h whol nwork. III. BOUND-CONTROL ALGORITHM Thi ion prn h Boun-Conrol lgorihm, whih olv Prolm 3 n 4, in whih rion oun l i impo on vry link l L. W ri how i opr n how i uppor oh ix-r n mximl-r ion mol ri in Sion I. W rr rr o Appnix II- A or h proo o i orrn. Hr, w l h low vlu h our ro o h nwork in h Boun-Conrol lgorihm. W lo l U uiinly lrg vlu h pproxim ininiy. For inn, U n h lrg vlu h n pro y h implmnion. A. Dripion o h Boun-Conrol Algorihm Th i o h Boun-Conrol lgorihm i o rply olv mximum-low prolm vi h Prlow-Puh lgorihm n ju h link pii unil h mximum-low rul onvrg o h opiml oluion. Th Boun-Conrol lgorihm i prn in Algorihm 1. In Algorihm 1, our ir ro uiinly lrg vlu = U o inii h Boun-Conrol lgorihm (lin 1). Nx, ll nwork no xu h Prlow-Puh lgorihm uj o h link-piy onrin p(l) = min(1/ l, l ) = 1/ l or vry link l L (lin 2-5). By hking h moun o low h h n n ou, our n rmin h mximum-low rul. Sour hn ro h ompu mximum-low rul rprn y o h nwork (lin 7-8) o h vry nwork no n ju h pii o i ougoing link (lin 9-11). Arwr, ll no xu gin h Prlow-Puh lgorihm unr h nw link pii (lin 12). Th lgorihm ir in h rp-unil loop (lin 7-12), n rmin i h

5 5 (0.5, 1, 1) (0.5, 1, 1) (0.56, 1, 1) (0.56, 1, 1) (0.5, 1, 1) (0.56, 1, 1) (0.5, 1, 0.4) (0.44, 0.8, 0.4) (0.5, 1, 1) (0.5, 1, 1) (0.44, 0.8, 1) (0.44, 0.8, 1) () 1 Prlow-Puh: mximum low = 2 () 2n Prlow-Puh: mximum low = 1.8 (0.58, 1, 1) (0.58, 1, 1) (0.58, 1, 1) (0.42, 0.72, 0.4) (0.4, 0.67, 0.4) (0.42, 0.72, 1) (0.42, 0.72, 1) (0.4, 0.67, 1) (0.4, 0.67, 1) () 3r Prlow-Puh: mximum low = 1.72 () Opiml oluion: =1.67, =1/ =0.6 Fig. 2. Exmpl o h Boun-Conrol lgorihm in Algorihm 1 or h nwork hown in Figur 1. Evry link l h l =1 n i oi wih ripl (x l, l, l ). Th igur illur: ()-() h low vlu r h ir hr xuion o h Prlow-Puh lgorihm (lin 5 n 11) n () h opiml oluion rurn rom h Boun-Conrol lgorihm. Algorihm 1 Boun-Conrol 1: our ro = U o ll no u N 2: or ll u N o 3: or ll l L(u) o 4: no u p(l) = min(1/ l, l ) 5: ll no run Prlow-Puh 6: rp 7: our o h mximum-low rul 8: our ro o ll no u N 9: or ll u N o 10: or ll l L(u) o 11: no u p(l) = min(1/ l, l ) 12: ll no run Prlow-Puh 13: unil our in h qul h mximum-low rul mximum low oin rom h Prlow-Puh lgorihm qul h low vlu h h ju n ro (lin 13). Th opiml vlu i givn y. Figur 2 illur how h Boun-Conrol lgorihm work. B. Diuion o h Boun-Conrol Algorihm In ul implmnion, w n uppor oh ix-r n mximl-r ion mol ( Sion I) y rmining h il ion hroughpu X n hn h rion oun l in iriu hion. Sour ir inii h Prlow- Puh lgorihm o i h il ion hroughpu X uj o h nwih onrin B l or ll l L, n hn ro X o ll h no in h nwork o h hy n piy h rion oun l or hir oi link l. Th ix-r ion mol i hu provi y ning h ix r X. I X i h mximum low rurn rom h Prlow-Puh lgorihm, hn w n hiv h mximum uriy unr h mximum ion hroughpu uing h Boun-Conrol lgorihm. Thu, h mximl-r ion mol i uppor. W n urhr nhn h iiny o h implmnion o h Boun-Conrol lgorihm vi iion rh o lo h opiml vlu in h Boun-Conrol lgorihm ollow. Suppo h low n high no h lowr n uppr oun, rpivly. Sour ir iniiliz low o zro n high o wi h mximum-low rul rmin y h ir xuion o h Prlow-Puh lgorihm (i.., lin 5 o Algorihm 1). I hn ro = ( low + high )/2 o h nwork. I h nx xuion o h Prlow-Puh lgorihm rurn h mximum low l hn, hn our ign h mximum-low rul o high. Ohrwi, h rul i ign o low in. Sour rply rh or, n h lgorihm rmin i h mo rnly ro vlu n h l mximum-low rul r qul (or irn y om olrn vlu pning on h implmnion). Wih iion rh, h omplxiy o h Boun-Conrol lgorihm i O(pT ), whr p i h numr o priion igi riing ll poil low vlu n T i h omplxiy o xuing h Prlow-Puh lgorihm. For inn, i h Boun-Conrol lgorihm implmn h iriu n ynhronou vrion o h Prlow-Puh lgorihm [16], i inrou O(p N 2 L ) mg n k O(p N 2 ) im o onvrg. IV. LEX-CONTROL ALGORITHM In hi ion, w prn h Lx-Conrol lgorihm, whih olv h lxiogrphi opimizion pii in Prolm 5 n 6. W xplin how h Lx-Conrol lgorihm i xn

6 6 (0.4, 0.67, 1) (0.4, 0.67, 1) (0.6, 1, 0.6) (0.4, 0.67, 0.4) (0.3, 1, 0.3) (0.3, 1, 0.3) (0.1, 0.33, 1) (0.3, 1, 1) (0.3, 1, 0.3) (0.3, 1, 1) (0.3, 1, 0.3) (0.1, 0.33, 1) (0.6, 2, 0.6) (0.4, 1.33, 0.4) (0.3, 1.5, 0.3) (0.3, 1.5, 0.3) (0.2, 1, 0.2) (0.2, 1, 0.2) (0.3, 1.5, 0.3) (0.3, 1.5, 0.3) (0.2, 1, 0.2) (0.2, 1, 0.2) (0.6, 3, 0.6) (0.4, 2, 0.4) () 1 Boun-Conrol: mximum low=1.67 () 2n Boun-Conrol: mximum low=3.33 () 3r Boun-Conrol: mximum low=5 Fig. 3. Exmpl o h Lx-Conrol lgorihm in Algorihm 2 or h nwork hown in Figur 1. Evry link l h l = 1 n i oi wih ripl (x l, l, l ). Ar vry xuion o h Boun-Conrol lgorihm (lin 1 n 11), h no iniy h riil link (in h rrow) n ju h rion oun l oringly (lin 6-10). rom h Boun-Conrol lgorihm. I proo o orrn n oun in Appnix II-B. A. Dripion o h Lx-Conrol Algorihm To unrn h Lx-Conrol lgorihm, uppo h or priulr mximum-low prolm, w hv oun h mximum low n h minimiz wor- k o = 1/. Th nwork will hn oniu o riil link, in h link l L who k o nno urhr r wihou inring. Th i o h Lx-Conrol lgorihm i o irivly olv mximumlow prolm uing h Boun-Conrol lgorihm n iniy iionl riil link unil h lxiogrphilly opiml oluion i oin. Bor riing h lgorihm, w in h riul nwork G =(N, L ) wih rp o h mximum low ollow [2]. Suppo h h mximum low i olv n h link l L rri low l. To onru L, or h link l L ir rom no u o no v, whr u, v N, i p(l) l > 0, w inlu orwr link rom u o v ino L, n i l > 0, w inlu kwr link rom v o u ino L. Algorihm 2 Lx-Conrol 1: ll no run Boun-Conrol 2: our o h ompu mximum low 3: whil < U o 4: our ro o ll no u N 5: or ll u N o 6: no u run onniviy-hking lgorihm on G 7: or ll l L(u) o 8: i l i riil link hn 9: no u l = 1/U 10: no u l = l / 11: ll no run Boun-Conrol 12: our o h ompu mximum low Algorihm 2 ummriz h Lx-Conrol lgorihm. All no ir run h Boun-Conrol lgorihm o minimiz h wor- k o uj o h piy onrin p(l) = min(1/ l, l ) or ll l L in h rnorm mximum-low prolm (lin 1). Sour hn ro h ompu mximum low (lin 4). Eh no run onniviy-hking lgorihm (.g., h rh-ir rh) on G (lin 6-8). I i nighor in G r no rhl in G, hn h orrponing link wn il n i nighor in G r lying on minimum u n hn r riil ( Appnix II-B). I moii l n l or h po riil link l (lin 9-10) o h p(l) i ju o oun only h proporion o low urrnly rri. Hr, w 1/ l o uiinly lrg vlu U (in in Sion III) o h i o no p(l). Th lgorihm irivly inii h riil link (lin 3-12, ollivly in lxiogrphi irion), n rmin whn h mximum low ompu rom h Boun-Conrol lgorihm qul U. Figur 3 pi how h Lx-Conrol lgorihm ompu h lxiogrphilly opiml oluion. B. Diuion o h Lx-Conrol Algorihm Th omplxiy o h Lx-Conrol lgorihm i omin y h xuion o h Boun-Conrol lgorihm. Sin h lxiogrphi irion iovr l on riil link, h Lx-Conrol lgorihm h omplxiy h i O( L T ), whr T i omplxiy o h Boun-Conrol lgorihm. In o loing ll riil link, w n imply prorm pr-rmin numr, y k, o lxiogrphi irion o iniy u o riil link in orr o gin prormn ni in ul implmnion. Sin h lr lxiogrphi irion mp o iniy h riil link wih mo k o, h mo unil uriy improvmn our uring rlir lxiogrphi irion. Wih hi moiiion, h omplxiy o h Lx-Conrol lgorihm i ru o O(kT ). V. EXPERIMENTS In hi ion, w prorm n xniv xprimnl uy on h propo lgorihm vi imulion. W onir hr nwork ing, h o whih onin 200 no, onn y 600, 800, n 1000 link, rpivly. W u BRITE [28], nwork opology gnror, o onru 50 xprimnl opologi or h nwork ing. All no wihin opology r rnomly onn n pl in rngulr wo-imnionl pln. W i h no lo o n rh rom h origin (i.., h oom l-hn ornr o h pln) o our n ink, rpivly. To onru ir yli opology, or h link wn ny wo

7 7 Minimiz wor- k o no, 600 link 200 no, 800 link 200 no, 1000 link Numr o xuion o h Prlow-Puh lgorihm no, 600 link 200 no, 800 link 200 no, 1000 link Rouing ovrh no, 600 link 200 no, 800 link 200 no, 1000 link Fig Proporion o h mximum poil ion hroughpu Proporion o h mximum poil ion hroughpu () Minimiz wor- k o () Numr o xuion o h Prlow-Puh lgorihm Exprimn 1: Anlyi o h Boun-Conrol lgorihm irn ion hroughpu Proporion o h mximum poil ion hroughpu () Rouing ovrh no u n v, w ir i rom no u o no v i no u Eulin in o h origin i l hn h o no v. Morovr, h link l i uniormly ign uriy onn l wn 0 n 1 n nwih B l wn 1 n 5. W hn nlyz h vrg prormn o h lgorihm ovr h 50 opologi. Our xprimn ou on hr mri, nmly: (1) k o (in in Sion II), whih mur h rilin o h propo lgorihm owr vriou yp o link k, (2) numr o xuion o h Prlow-Puh lgorihm, whih mur h mg omplxiy n h onvrgn im o h propo lgorihm. (3) rouing ovrh, whih i in h rio o h vrg hop oun rom our o ink in our muliph pproh o h minimum hop oun in ingl-ph rouing. W n ompu h rouing ovrh ollow. L r(u) h hop oun rom no u o ink n l uv L h link ir rom no u o no v. Rll h x l no h proporion o h ion rri y link l. Th vrg hop oun o h muliph rouing i hu givn y h ruriv quion r() = x lu u:l u L [1 + u:lu L x lu r(u)], whr r() i iniiliz o zro. W hn ivi r() y h minimum hop-oun in ingl-ph rouing o oin h rouing ovrh. Exprimn 1 (Anlyi o h Boun-Conrol lgorihm irn ion hroughpu): Thi xprimn ui how h Boun-Conrol lgorihm pro gin h wor ingl-link k vriou ion hroughpu. For h opology, w u h Prlow-Puh lgorihm o rmin h mximum poil ion hroughpu uj o h linknwih onrin. W hn lul h hroughpu r h r givn y irn proporion o h rmin mximum ion hroughpu o r oh ix-r n mximl-r ion mol ( Sion I). Finlly, w ign h ppropri rion oun o ll link ( Sion III- B). Hr, w vlu h gr o rilin on h minimiz wor- k o. Figur 4 pi h prormn mri irn ion hroughpu, n Tl II how h wor- k o whn ingl ph wih h minimum hop-oun i u. Figur 4() how h h Boun-Conrol lgorihm unilly ru h wor- k o whn ompr o h ingl-ph pproh (.g., rom 0.78 o 0.17, or y 78%, or h 1000-link nwork h u h mximl-r TABLE II EXPERIMENT 1: WORST-CASE ATTACK COST WHEN A SINGLE PATH WITH THE MINIMUM HOP-COUNT IS USED. Nwork ing Ak o 200 no, 600 link no, 800 link no, 1000 link 0.78 mol or h mximum ion hroughpu). Spiilly, w orv wo kin o r-o. Fir, h ion hroughpu inr, link xprin ighr rion oun in gnrl. Thi l o mor Prlow-Puh xuion n highr wor k o. Son, whil nwork wih mor link in mllr wor- k o, i lo inur mor mg in running h Boun-Conrol lgorihm wll highr rouing ovrh. Exprimn 2 (Anlyi o h Lx-Conrol lgorihm irn numr o lxiogrphi irion): Thi xprimn onir how h Lx-Conrol lgorihm prvn h vr ingl-link k whn i xu irn numr o lxiogrphi irion. W rgr ingl-link k vr i i ruling k o i l 25% o h wor- on. Hr, or h opology, w vlu h lgorihm uing h mximl-r ion mol ( Sion I) in whih h mximum ion hroughpu i rmin in Exprimn 1. Alo, w u h numr o link h inur vr k o h rilin mur. Figur 5 plo h ruling mri. I how h h Lx- Conrol lgorihm n ru h numr o link whr h ingl-link k r vr. Th ruion i mor lin in h 1000-link nwork (.g., y mor hn 50% in hr or mor lxiogrphi irion). Th r-o i h h rquir numr o xuion o h Prlow-Puh lgorihm inr linrly wih h numr o lxiogrphi irion. On inring i ni o h Lx-Conrol lgorihm i h i llvi h rouing ovrh wll. A poil xplnion i h horr ph inur mllr k o in gnrl, o h Lx-Conrol lgorihm pro, i inii h mor ur horr ph n hn ru h rouing ovrh. From Figur 5, h ni o h Lx-Conrol lgorihm r mor prominn in h ir hr lxiogrphi irion. Thu, in pri, i i ronl o run mll numr o lxiogrphi irion. Thi llow ym ignr o l h r-o o iminihing rurn.

8 8 Fig. 5. Numr o link inurring l 25% o h wor- k o no, 600 link 200 no, 800 link 200 no, 1000 link 10 Numr o lxiogrphi irion () Numr o link inurring vr k o Numr o xuion o h Prlow-Puh lgorihm no, 600 link 200 no, 800 link 200 no, 1000 link 0 Numr o lxiogrphi irion () Numr o xuion o h Prlow-Puh lgorihm Exprimn 2: Anlyi o h Lx-Conrol lgorihm irn numr o lxiogrphi irion. Rouing ovrh no, 600 link 200 no, 800 link 200 no, 1000 link 1.6 Numr o lxiogrphi irion () Rouing ovrh Avrg k o Fig no, 600 link 200 no, 800 link 200 no, 1000 link 0 Numr o lxiogrphi irion () Avrg k o unr h uniorm ingl-link k Avrg ggrg k o no, 600 link 200 no, 800 link 200 no, 1000 link 0.05 Numr o lxiogrphi irion () Avrg ggrg k o unr h uniorm 10-link k Exprimn 3: Anlyi o h Lx-Conrol lgorihm uj o irn l o uniorm link k. Avrg ggrg k o no, 600 link 200 no, 800 link 200 no, 1000 link 0.3 Numr o lxiogrphi irion () Avrg ggrg k o unr h uniorm 50-link k Fig. 7. Avrg ggrg k o no, 600 link 200 no, 800 link 200 no, 1000 link 0.2 Numr o lxiogrphi irion () Avrg ggrg k o unr h proporionl 5-link k Avrg ggrg k o no, 600 link 200 no, 800 link 200 no, 1000 link 0.5 Numr o lxiogrphi irion () Avrg ggrg k o unr h wor- 5-link k Exprimn 4: Anlyi o h Lx-Conrol lgorihm uj o h proporionl n wor- muli-link k Exprimn 3 (Anlyi o h Lx-Conrol lgorihm uj o irn l o uniorm link k): Alhough our nlyi onnr on h wor- ingl-link k, in h Lx-Conrol lgorihm k h mo ln iriuion o k o o ll link, w nviion h i lo minimiz h vrg k o unr uniorm link k, i.., n inrur uniormly k ingl or mulipl link h rry ion. In hi xprimn, w invig hi ponil ni y oniring irn l o uniorm link k. In h xprimn up, w l h uriy onn l h proporion o lo o rvring link l h i ing k ( Sion II), o h k o o link l (i.., l = l x l ) rprn h ul proporion o lo or h ion. For h ingl-link k, w ompu h vrg k o y iviing h ol k o o ll link y h numr o link h rry. For muli-link k, w look h moun o rmining h ully rh h ink in orr o ompu h ggrg k o. Thn w imul 50 muli-link k or h opology o oin h vrg ggrg k o. Hr, w ou on h mximl-r ion mol in Exprimn 2. Figur 6 illur h k o inurr y h uniorm k on on, 10, n 50 link. I how h h Lx-Conrol lgorihm n miig h hr o uniorm link k. For inn, givn h 50 ou o 1000 link r k, h vrg ggrg k o i ru y 40% (or rom 0.75 o 0.45) wih our or mor lxiogrphi irion. Thror, pr rom h wor- ingl-link k, h Lx-Conrol lgorihm lo nhn h roun o h nwork uj o vriou l o uniorm link k. Exprimn 4 (Anlyi o h Lx-Conrol lgorihm uj o h proporionl n wor- muli-link k): Th inl xprimn h Lx-Conrol lgorihm unr

9 9 h proporionl n wor- muli-link k. In h proporionl muli-link k, n inrur k numr o link uh h h proiliy h h link i k i irly proporionl o i k o. In h wor- mulilink k, howvr, h inrur rminiilly k h link wih h high k o. W u h m ing in Exprimn 3 o vlu h Lx-Conrol lgorihm on h mximl-r ion mol. Figur 7 illur h vrg ggrg k o whn iv link r k. I how h in gnrl, h Lx- Conrol lgorihm n ru h vrg ggrg k o in oh proporionl n wor- k. For inn, in 1000-link nwork, h k o i r rom 0.3 o 0.23, or y 23%, in h proporionl 5-link k, n rom 0.59 o 0.52, or y 12%, in h wor- 5-link k. Alo, roun our lxiogrphi irion r uiin o hiv uh ruion. Summry: Th xprimn how h h Boun-Conrol lgorihm igniinly pro gin h wor- ingllink k, n h h Lx-Conrol lgorihm provi iionl proion y ruing h numr o link wih vr k o. Morovr, h Lx-Conrol lgorihm ivly n gin h uniorm, proporionl, n wor- muli-link k, wih h mjoriy o ni ourring wihin h ir w lxiogrphi irion. VI. APPLICATION IN ATTACK-RESISTANT NETWORKS To urhr xmin h ppliiliy o oh Boun-Conrol n Lx-Conrol lgorihm, w onir hir u in n krin nwork [7], piliz nwork h pro n ho y urrouning hm wih niv rhiur. On xmpl i Sur Ovrly Srvi (SOS) [20], whih onru h niv rhiur wih o ppliionlvl ovrly no lyr op h unrlying nwork rhiur. Aoring o [7], n k-rin nwork houl iy wo ruil u onriing riri rm (1) riliny: h nwork houl or lrn ph in h o no ilur, n (2) uriy: h nwork houl onin h mg u y ompromi no. In hi ion, w propo how o ln h wo riri uing our muliph oluion n vlu hir r-o vi imulion. A. Ovrviw Following [7], [20], w ir ovrviw h rhiur o n k-rin nwork illur in Figur 8. To ommuni wih pro ink, our ir onn o n poin (AP), whih uhni inoming pk. Auhni pk r hn rou rom h AP hrough n inronnion nwork o rg (.k.. r rvl [20]), whih rly pk o ink. Inuiivly, h AP n h rg n viw n nry poin o h k-rin nwork n o ink, rpivly, uh h no pk n rh ink wihou proprly going hrough n AP ollow y rg. In gnrl k-rin nwork, our n rh ink uing mulipl ph hrough irn AP n rg. Alo, h AP union inpnnly rom on nohr [7]. Thu, w n ploy our iriu muliph oluion in hi yp Fig A A... A poin inronnion nwork Exmpl o n k-rin nwork. T T... T rg... o k-rin nwork o urhr pro h unrlying ur ommuniion. In orr o ur h rg n hn h n ho, h inii o h rg r hin rom h puli n known only o mll o no (.k.. on [20]) whih r in urn known only o h oi AP. Howvr, i n AP i ompromi, hn n inrur n iniy n hn k h oi rg hrough h ompromi AP. Thi po r-o iu wn riliny n uriy: or riliny, h AP houl oi wih uiin rg o o l n lrn rg i on rg om unvill, whil or uriy, h AP houl no ign oo mny rg o o uppr h numr o rg ing k whn n AP i ompromi. To r h r-o wn riliny n uriy, [7] ormul n ignmn prolm, nmly, givn ix lloion on h AP, h ojiv i o in n opiml ph ignmn wn h AP n rg h minimiz h loking proiliy, in h proiliy h rqu nno rh pro n ho u o h k on h AP n rg. Howvr, h xin o polynomil-im opiml lgorihm o hi prolm rmin n opn iu. In iion, h pproximion lgorihm in [7] o no k ino oun lo lning on h AP n rg. Thror, w r h prolm rom nohr prpiv in whih our gol i o ign rnmiion r h AP in o ining n opiml ph ignmn. W ormliz h prolm in h ollowing iuion. B. Mol W ir ormul h prolm on [7]. W onir nwork grph G = (A,T, P), whr A i h o AP, T i h o rg, n P i h o ir ph rom h AP o h rg. W l A(j) n T (i) h o AP n rg, rpivly, uh h i hr xi ph in P rom poin i A o rg j T, hn i A(j) n j T (i). W ou on wo yp o k: (1) ompromi k, in whih n inrur oin unuhoriz o no, n (2) nil-o-rvi (DoS) k, in whih n inrur prvn no rom proviing lgiim rvi, or xmpl, y looing h viim no wih hug ri. L C u n D u h rpiv vn h no u A T i ompromi n i vulnrl o DoS k, n C u n D u

10 10 no hir omplmn. L lo P (E) h proiliy h vn E our. In pri, h k proilii P (C u ) n P (D u ) n im vi h pproh ri in Sion II. Sin h inii o h rg r hin rom h puli, w um h n inrur n only moun h k hrough h AP. I ollow h P (C j ) = 0 n P (D j C i1,, C ik ) = 0 or ll j T n i 1,, i k A(j). W um h i AP i i ompromi, hn n inrur will iniy ll rg j T (i) n lunh DoS k on hm rom h ompromi AP i. Thi impli P (D j C i ) = 1 or ll j T (i). Finlly, w um h C i n D i r muully inpnn or ll i A. An AP i A i i o lok i i i ompromi, i i h viim o DoS k, or i oi rg r ll viim o DoS k. Thu, h loking proiliy p i AP i i givn y p i = 1 P (D i )P ( j T (i) D j). (7) No h h vn j T (i) D j impli h AP i i no ompromi. To ompu P ( j T (i) D j), w no h rul o h inpnn umpion, h proiliy h ll k rg j 1,, j k T r no vulnrl o DoS k i givn y P (D j1 D jk ) = i T (j 1) T (j k ) P (C i). (8) W hn vlu P ( j T (i) D j) vi h inluion-xluion prinipl [9]. To pply our lgorihm, w xn h ingl-link k mol in Sion II o ingl-no k mol (.g., vi no pliing [2]). W l x u, whr 0 x u 1, h proporion o rri y no u, whr u A T n n AP or rg, n x = (x u, u A T ) h proporion vor. Th k o o no u i hu in u = u x u, whr u i h uriy onn o no u. Uing h loking proiliy our mur, or vry AP i A, w i = p i, iniing h h k o o AP i i qunii h xp proporion o lo whn AP i i lok. In onr, in h loking proiliy i lir only h lyr o AP, or vry rg j T, w j = 0. To rmin h uriy onn, h AP n inpnnly onul i h rg or h ompromi proilii o hir oi AP. I hn ompu h loking proiliy on Equion 7 n 8. Furhrmor, or lo lning, h o h AP n rg n ign il nwih onrin n rmin i rion oun u, whr 0 u 1 or u A T, uing ihr h ix-r mol or h mximl-r mol. Givn h ov ormulion, our primry ojiv i o i h lloion x = (x u, u A T ) h minimiz h wor- k o o n AP uj o nwih onrin. Thu, w hv h ollowing opimizion prolm: = min mx i = min mx x i A x ix i i A uj o 0 x u u, u A T. (9) W n urhr xn Prolm 9 o lxiogrphi opimizion prolm. W n hn rily oin h opiml lloion uing h Boun-Conrol or Lx-Conrol lgorihm. * AP, 20 rg 50 AP, 20 rg 100 AP, 20 rg Conniviy A* AP, 20 rg 50 AP, 20 rg 100 AP, 20 rg Conniviy () Minimiz wor- k o () Aggrg k o Fig. 9. Evluion or k-rin nwork: k o v. onniviy. Normliz * AP, 20 rg 50 AP, 20 rg 100 AP, 20 rg Conniviy Normliz A* AP, 20 rg 50 AP, 20 rg 100 AP, 20 rg Conniviy () Normliz minimiz () Normliz ggrg k o wor- k o Fig. 10. Evluion or k-rin nwork: normliz k o v. onniviy. C. Evluion W onu imulion uy on hr k-rin nwork ing, h o whih h 30, 50, n 100 AP, rpivly, oghr wih 20 rg. W in onniviy h numr o rg o whih h AP i onn. For h nwork ing n onniviy, w onir h vrg rul ovr 50 opologi. Wihin opology, h AP i onn rnomly o irn rg oring o h onniviy n i ign h ompromi n DoS proilii uniormly rnom wn 0 n Morovr, w um h h AP n i ph o h rg hv inini nwih n h h rg h h m nwih. W hn rmin h rion oun o h AP n rg uing h mximl-r mol. W ir nlyz h minimiz wor- k o wll h ggrg k o A o h iv AP h hv h high k o. W ompu n A rpivly vi h Boun-Conrol lgorihm n h Lx-Conrol lgorihm wih hr lxiogrphi irion. Figur 9() n 9() plo n A vru h onniviy, rpivly. Iniilly, h rilin o n k-rin nwork inr wih h onniviy n oh k o r. A h onniviy urhr inr, h rg r h o mor AP h n ompromi. Thu, hy om mor vulnrl o DoS k, n h k o inr. Suh inr i mor vr whn nwork h mor AP (.g., 100 AP). Thi how h nwork wih mor AP n highr onniviy o no nrily or r proion. To vlu h ivn o h muliph lgorihm, w normliz o h rpiv o whn no Boun- Conrol lgorihm i u (i.., only h mximl-r mol i ii), n w lo normliz A o h rpiv o whn only h Boun-Conrol lgorihm i u. Figur 10() plo h normliz n how h h Boun-Conrol

11 11 lgorihm n ivly ru h wor- k o o n AP (.g., y l 80% or 100 AP n 20 rg). In Figur 10(), w plo h normliz A n orv h h Lx-Conrol lgorihm ru h ggrg k o low onnivii. A h onniviy inr, h AP i ign mor rg n i l onrin y h nwih rquirmn. Thu, h Boun-Conrol lgorihm immily minimiz h wor- k o ll AP (i.., whr h minimum u li) n h h m rul o h Lx-Conrol lgorihm. VII. RELATED WORK Muliph rouing w ir ui in [27], in whih i rnmi ovr mulipl ijoin ph mn o provi lo lning n rouing rilin. Runny n o h rnmi o h h rivr i n ully ronru h in h prn o mor lo. B on hi inuiion, w pply muliph rouing o r h prn o link k uing opimizion mol. On poil opimizion mol or muliph rouing i on minimx opimizion. Prviou ui onir h lo-lning prolm (.g., in [1], [17]), muliph oluion o om link k (.g., in [5], [6], [18]), n h nworkinruion prolm (.g., in [21]). No h h ov ui ou on nrliz lgorihm h um h knowlg o h nwork opology. W xn yon h prviou ui y viing iriu oluion h n hnl linknwih onrin. Anohr poil opimizion mol i on lxiogrphi opimizion, whih h n ui in [11], [13] or nwork ing. Whil [11] onir only h lxiogrphi opimizion o h low o h link h o h our no, [13] xn [1] o lxiogrphilly opimiz h low o ll h nwork link in nrliz mnnr. Spiilly, h i o [13] i o olv h minimx prolm vi h mximum-low prolm or givn nwork, iniy h minimum-u link, n rurivly olv h minimx prolm or h unwork pr y ho link. Our Lx-Conrol lgorihm uppor h iriu implmnion in h prn o link-nwih onrin. Anlyil ui rgring ur muliph rouing n oun in [4], [5], [6], [18], [25], [32], in whih h vulnriliy o h no i hrriz y n k (or ilur) proiliy. In priulr, [25], [32] onir ijoin ph mong nwork no, whil [4] rlx hi ijoinn rquirmn n propo rilin rouing hm uing wo non-ijoin ph. Our lgorihm, in [5], [6], [18], xplor highr gr o nwork ivriy y uing ll h ph (ihr ijoin or no) h r vill. In rm o h ppliiliy o ur muliph rouing, [30] ui h implmnion iu o proing n krin nwork (.g., SOS [20]) gin h inrur h k o ompromi mll porion o ovrly no rnom. Our work, on h ohr hn, provi n nlyil uy or proing n k-rin nwork uing wor k mol. Bi k-rin nwork, muliph rouing h lo n ppli o improv h rilin o ohr rhiur, uh nor nwork [12]. VIII. PRACTICAL ISSUES In hi ion, w r vrl pril iu o our urrn work n ugg irion or uur rrh. Runn rouing: A mnion in Sion I, w um h om rror orrion mhnim i u o rlily livr. Inuiivly, runn mg mu o rnmi o h rivr n rovr ll long lo u o il ph i mo [25], [27]. Alhough runn rouing provi rliiliy, pr o h rw nwork nwih i u o rnmi runn, n hi r h iv nwork nwih or rrying ul. Drmining h uil lvl o runny h ln h r-o wn rliiliy n iv nwork nwih i hllnging n hn rquir urhr invigion. Ful-olrn: W urrnly um h h no rmin l hroughou h xuion o h lgorihm, y in pri, no n xprin k or rnin ilur. To or ul-olrn, w n ihr rr h lgorihm, or op h l-ilizing oluion in [14], [19]. In priulr, [19] nhn h originl Prlow-Puh lgorihm o ju o h hng o link. Howvr, h wor- omplxiy o hi oluion i proporionl o h numr o jumn mulipli y h omplxiy o h originl Prlow-Puh lgorihm, ling o vr prormn grion i h jumn our rqunly. Hn, w n o onir h r-o wn rring h lgorihm n invoking h l-ilizing prour. Mulipl ion: Our lgorihm r on pr-ion i. To uppor mulipl ion, on impl pproh i o rquir h link o llo irn nwih (or rion oun) or h mulipl ion on h ppliion rquirmn. Howvr, uh n pproh my no ully uiliz h link nwih. For xmpl, w my llo h unu link nwih o on ion o ohr ion. I ion i givn mor nwih, i i uj o wkr rion oun. In Sion V, Exprimn 1 how h wkr rion oun n hiv mllr wor- k o. Thu, w hv o xmin how o llo rion oun or mulipl ion ivly o hiv h opiml oluion. Quniying vulnriliy: W urrnly um h w n hrriz h mg o n k vi uriy onn ( Sion II) or k proilii ( Sion VI). W mphiz h our lgorihm n ill ppli rgrl o h ul vlu o h prmr, lhough oun k-moling mhnim n provi r lloion ovr mulipl ph. W po hi prolm uur work. IX. CONCLUSIONS W prn iriu ur muliph oluion h ompri h Boun-Conrol n Lx-Conrol lgorihm, oh o whih proivly om link k in iriu hion n provly onvrg o h ir opiml oluion. W u imulion o monr h rilin o oh lgorihm owr irn yp o ingl-link n muli-link k. Spiilly, imulion rul monr h h Lx-Conrol lgorihm ounr vr link k iinly

12 12 wihin h ir w lxiogrphi irion, n hn oh rouing uriy n lgorihm prormn n ivly hiv uring ul implmnion. Finlly, w ui h ppliiliy o oh lgorihm uing n k-rin nwork n xmpl. By imulion, w vlu hir prormn n nlyz how hy r o riliny n uriy unr irn k-rin nwork ing. REFERENCES [1] R. K. Ahuj. Algorihm or h Minimx Trnporion Prolm. Nvl Rrh Logii Qurrly, 33: , [2] R. K. Ahuj, T. L. Mgnni, n J. B. Orlin. Nwork Flow: Thory, Algorihm, n Appliion. Prni Hll, [3] D. Anrn, H. Blkrihnn, F. Khok, n R. Morri. Rilin ovrly nwork. In Pro. o h 18h ACM Sympoium on Opring Sym Prinipl (SOSP), Oor [4] R. Bnnr n A. Or. Th Powr o Tuning: Novl Approh or h Eiin Dign o Survivl Nwork. In Pro. o IEEE Inrnionl Conrn on Nwork Proool (ICNP), Oor [5] S. Bohk, J. Hpnh, J. L, C. Lim, n K. Orzk. Enhning Suriy vi Sohi Rouing. In Pro. o ICCCN, My [6] J. P. Brumugh-Smih n D. R. Shir. Minimx Mol or Divr Rouing. INFORMS Journl on Compuing, 14(1):81 95, Winr [7] T. Bu, S. Norn, n T. Woo. Tring Riliny or Suriy: Mol n Algorihm. In IEEE Inrnionl Conrn on Nwork Proool (ICNP), Oor [8] J. Byr, M. Luy, n M. Miznmhr. Aing Mulipl Mirror Si in Prlll: Uing Torno Co o Sp Up Downlo. In Pro. o IEEE INFOCOM, Mrh [9] T. H. Cormn, C. E. Liron, R. L. Riv, n C. Sin. Inrouion o Algorihm. MIT Pr, 2n iion, [10] W. Du n A. Mhur. Ting or Sowr Vulnriliy Uing Environmn Prurion. In Pro. o h Inrnionl Conrn on Dpnl Sym n Nwork, [11] G. Gllo, M. D. Grigorii, n R. E. Trjn. A F Prmri Mximum Flow Algorihm n Appliion. SIAM Journl on Compuing, 18(1):30 55, Frury [12] D. Gnn, R. Govinn, S. Shnkr, n D. Erin. Highly-rilin, Enrgy-iin Muliph Rouing in Wirl Snor Nwork. ACM SIGMOBILE Moil Compuing n Communiion Rviw, 5(4):11 25, Oor [13] L. Gorgii, P. Gorgo, K. Floro, n S. Srzki. Lxiogrphilly Opiml Bln Nwork. IEEE/ACM Trnion on Nworking, 10(6): , Dmr [14] S. Ghoh, A. Gup, n S. V. Pmmrju. A Sl-ilizing Algorihm or h Mximum Flow Prolm. Diriu Compuing, 10(3): , [15] B. Gnnko n I. A. Uhkov. Proilii Rliiliy Enginring. John Wily & Son, In., [16] A. V. Golrg n R. E. Trjn. A Nw Approh o h Mximum- Flow Prolm. Journl o h Aoiion or Compuing Mhinry, 35(4): , Oor [17] C.-C. Hn, K. G. Shin, n S. K. Yun. On Lo Blning in Muliompur/Diriu Sym Equipp wih Cirui or Cu-Through Swihing Cpiliy. IEEE Trnion on Compur, 49(9): , Spmr [18] J. Hpnh n S. Bohk. Prliminry Rul in Rouing Gm. In Pro. o h 2001 Amrin Conrol Conrn, volum 3, pg , Jun [19] B. Hong n V. K. Prnn. Diriu Apiv Tk Alloion in Hrognou Compuing Environmn o Mximiz Throughpu. In Pro. o IPDPS, April [20] A. Kromyi, V. Mir, n D. Runin. SOS: An Arhiur or Miiging DDoS Ak. IEEE JSAC, Spil Iu on Srvi Ovrly Nwork, 22(1), Jnury [21] M. Koilm n T. V. Lkhmn. Ding Nwork Inruion vi Smpling: A Gm Thori Approh. In Pro. o IEEE INFOCOM, April [22] P. L, V. Mir, n D. Runin. Diriu Algorihm or Sur Muliph Rouing. In Pro. o IEEE INFOCOM, Mrh [23] S. L, Y. Yu, S. Nlkuii, Z.-L. Zhng, n C.-N. Chuh. Proiv v. Riv Approh o Filur Rilin Rouing. In Pro. o IEEE INFOCOM, Mrh [24] D. Loguinov n H. Rh. En-o-En Inrn Vio Tri Dynmi: Siil Suy n Anlyi. In Pro. o IEEE INFOCOM, Jun [25] W. Lou, W. Liu, n Y. Fng. SPREAD: Enhning D Coniniliy in Moil A Ho Nwork. In Pro. o IEEE INFOCOM, Mrh [26] G. Mlkin. RIP Vrion 2, Novmr RFC [27] N. Mxmhuk. Dipriy Rouing. Pro. o ICC, [28] A. Min, A. Lkhin, I. M, n J. Byr. BRITE: An Approh o Univrl Topology Gnrion. In Pro. o MASCOTS, Augu [29] J. Moy. OSPF Vrion 2, April RFC [30] A. Svrou n A. D. Kromyi. Counring DoS Ak wih Sl Muliph Ovrly. In Pro. o ACM Conrn on Compur n Communiion Suriy (CCS), Novmr [31] D. Thlr n C. Hopp. Muliph Iu in Uni n Muli Nx-Hop Slion, Novmr RFC [32] J. Yng n S. Ppviliou. Improving nwork uriy y muliph ri iprion. In IEEE Miliry Communiion Conrn (MILCOM), Oor APPENDIX I PREFLOW-PUSH ALGORITHM W oulin h Prlow-Puh lgorihm in Algorihm 3, whil h il iuion o h lgorihm n oun in [2], [16]. Algorihm 3 Prlow-Puh 1: our puh muh low poil o i nighor 2: whil no u N {, } hving low x o 3: i no u h n miil nighor no v hn 4: no u puh x low o no v 5: l 6: no u rll i im hor in o ink 7: our rmin h mximum low Conir nwork G = (N, L) wih our N n ink N. Iniilly, our ir puh h mximum poil low owr i nighor no in G. For vry no u N {, } h h low x, i k o puh i low x o n miil nighor no v (.k.. puh oprion). By miil, w mn no v i nighor no o u in h riul nwork (in in Sion IV) wih rp o h urrn low n no v li on h im hor ph rom no u o ink. I no miil nighor i oun, hn no u up i im hor in o ink (.k.. rll oprion). Th lgorihm rmin whn ll no i our n ink hv no x low. Thn our n rmin h mximum low y hking how muh low h n ully n o ink. APPENDIX II PROOFS A. Corrn o h Boun-Conrol Algorihm To prov h orrn o h Boun-Conrol lgorihm, w ir how h xin o n opiml mximum low or Prolm 4 (in in Sion II) unr nry n uiin oniion or h vlu o l. Thn w prov h h low vlu ro y our i rily ring n oun rom low y. Thi impli h Boun-Conrol lgorihm onvrg o h opiml vlu. Lmm 1: (Exin) Thr lwy xi mximum low > 0 or Prolm 4 i n only i l C l 1 or ny u C in h nwork G.

13 13 Proo: Niy ( ): Givn > 0, uppo h hr xi u C uh h l C l < 1. Hn, h piy o h u C i givn y l C min(1/ l, l ) l C l = l C l <. Thi onri h mx-low min-u horm, whih ugg h h piy o ny u i l h vlu o h mximum low. Suiiny ( ): W wn o how h = 1 i il low or Prolm 4. From Prolm 4, i = 1, h piy o ny u C i givn y l C min(1/ l, l ) l C l 1 (rll h l i normliz n o 1/ l 1). Hn, h low = 1 i oun rom ov y h piy o ny u n i rgr il. Thi impli h opiml mximum low > 0 xi. Bor proing o h nx proo, w in iionl noion. B on Algorihm 1, w ir l (0) h low vlu iniilly ro (lin 1). For n 1, w l (n) h low vlu ro in h nh irion o h rpunil loop (lin 8). No h (n) rprn h mximum low ompu rom h prviou xuion o h Prlow- Puh lgorihm. Morovr, w l C (n) n C on o h minimum u oi wih h mximum low (n) n, rpivly. Lmm 2: (Monooniiy n ounn) For ny poiiv ingr n, w hv (n 1) (n). In priulr, i (n 1) = (n) or om n, w hv (n 1) = (n) =. Thi lmm impli h prior o h rminion o h Boun-Conrol lgorihm, h low vlu i rily ring. Furhrmor, i h vlu h h ju n ro qul h ompu mximum-low rul, h lgorihm rmin wih h opiml vlu =. Proo: W ir prov y inuion on n h (n 1) (n) or ny poiiv ingr n. B : For n = 1, (0) qul h uiinly lrg vlu U, whil (1) i h mximum low givn y h ir run o h Prlow-Puh lgorihm. Thi impli h (0) (1). Alo, (1) n r h mximum low uj o h piy onrin p(l) = 1/ l n p(l)=min(1/ l, l ) or vry link l L, rpivly. Sin h lr onrin i ighr, i no grr hn (1). Inuion hypohi: L (k 1) poiiv ingr k. (k) or om, (k+1), n r h Inuion p: W no h (k) mximum-low rul uj o h piy onrin p(l)=min(1/ l, l (k 1) ), p(l)=min(1/ l, l (k) ), n p(l) = min(1/ l, l ) or vry link l L, rpivly. By hypohi, i uj o h igh piy onrin, ollow y (k+1), n inlly (k). Thi impli h (k) (k+1). By inuion, w hv (n 1) (n) or ny poiiv ingr n. Furhrmor, i (n 1) = (n), (n) i h mximum low iying h piy onrin p(l) = min(1/ l, l (n 1) ) = min(1/ l, l (n) ) or vry link l L. Thu, (n) i il low or Prolm 4. Thi impli (n). Howvr, w hv prov h in vry irion, w hv (n). I ollow h (n) =. Thorm 1: (Convrgn) Th Boun-Conrol lgorihm onvrg o h mximum-low vlu > 0 o Prolm 4, provi h l C l 1 or ny u C. Proo: Immi rom Lmm 1 n 2. B. Corrn o h Lx-Conrol Algorihm W ir prn wo propri h ini how o pinpoin h riil link ( Sion IV-A) in iriu hion. Propry 1: In mximum-low oluion, i link l L li on minimum u, hn i i riil. Proo: Th k o o link l L i l = l l /. Sin l i ix n i h mximum low, h k o l n only r y ruing l. I link l li on minimum u, i i ur (i.., low o link l qul i link piy). W n hn rgr h ruion o l h r in h piy o link l. Thi rul in h r in h piy o h minimum u n, y h mx-low-min-u horm, h r in h mximum low. Thu, h minimiz wor- k o =1/ inr. By iniion, link l i riil. Propry 2: For vry link l L ir rom no u o no v, whr u, v N, i no v i no rhl rom no u in h riul nwork G ( Sion IV-A or i iniion), hn link l li on minimum u. Proo: L S h o no rhl rom no u in G n T =N S. By umpion, w hv u S n v T. W no h link l L rri low rom u o v (ohrwi, v i rhl rom u in G ) n h low origin rom our, o i rhl rom u in G. I ollow h S. Similrly, h low rriving v will vnully rh, o v i rhl rom in G. Thi impli h T (i S in, v i rhl rom u vi in G ). Morovr, in h no in T r no rhl rom h no in S, hr r no link ir rom S o T in G, o h link rom S o T in G r ur n hy mu rprn minimum u. Sin l i on o h link ir rom S o T, l li on h minimum u. B on Propri 1 n 2, h no u N n invok ny lgorihm h n hk h onniviy o grph (.g., h rh-ir rh) on G. Thi hk i u o rmin whhr i nighor in G r rhl in G. I no, h orrponing link l L(u) wn no u n i nighor in G r lying on minimum u n hn r riil. Thi nl h iniiion o ll riil link in iriu hion. Now, w ormlly prov h h Lx-Conrol lgorihm onvrg o h lxiogrphilly opiml oluion ( Prolm 5 n 6 in Sion II). Lmm 3: In h Lx-Conrol lgorihm, i link i rmin o riil in lxiogrphi irion, i rmin riil in uqun lxiogrphi irion. Proo: Conir h link h r oun o riil. By Propry 1, hy li on om minimum u. L C hi minimum u. From lin 9-10 o Algorihm 2, h piy o h u C i pii l C l, whr i h low vlu rhing h ink. By low onrvion, w hv l C l =1, n hu h piy o C i pii. In h nx

14 14 lxiogrphi irion, u o low onrvion, h low ro C i h nwly ompu mximum low whih lo qul h pii piy o C. By h mx-low min-u horm, C i ill minimum u n hn h unrlying link rmin riil. Rmrk: Lmm 3 impli h h k o o vry riil link rmin unhng in uqun lxiogrphi irion. Lmm 4: Bor h Lx-Conrol lgorihm n, vry lxiogrphi irion in nw riil link. Morovr, mong h non-riil link h r inii o riil, l on o hm h h k o 1/, whr i h mximum low rurn rom h prviou xuion o h Boun-Conrol lgorihm. Proo: Suppo h lgorihm pro o nw lxiogrphi irion. Thi impli h i l hn h uiinly lrg vlu U (u o lin 3 o Algorihm 2), whr i h mximum low ompu rom h prviou xuion o h Boun-Conrol lgorihm. By h mx-low min-u horm, qul h piy o om minimum u, y C, n hi piy i qul o l C min(1/ l, l ). To hiv < U, w mu hv minimum u C in whih l on link l h piy qul o 1/ l in o l o h i oun wy rom U (ohrwi, = U i h mximum low n h lgorihm rmin). Thi link l i prviouly non-riil (ohrwi, i piy i pii y l u o lin 10 o Algorihm 2) n i now inii o riil (in i li on minimum u). Furhrmor, i k o i givn y 1/. Rmrk: Lmm 4 impli h l on nwly inii riil link xhii h minimiz wor- k o ompu rom h l xuion o h Boun-Conrol lgorihm. Lmm 5: Wihin h Lx-Conrol lgorihm, h mximum low ompu in h xuion o h Boun-Conrol lgorihm i rily inring. Proo: From Lmm 4, h mximum low, y, ompu in n xuion o h Boun-Conrol lgorihm i givn y l C min(1/ l, l ), whr C no minimum u h inlu non-riil link l. Noi h C i no minimum u in h prviou xuion o h Boun-Conrol lgorihm, or link l woul hv lry n inii riil. Thu, C h grr piy. By h mx-low min-u horm, h ompu mximum low om grr, n i hu rily inring in h xuion o h Boun-Conrol lgorihm. Thorm 2: Th Lx-Conrol lgorihm onvrg o h lxiogrphilly opiml oluion. Proo: By Lmm 3 n 4, h lxiogrphi irion o h Lx-Conrol lgorihm inii wo yp o riil link: h lry po on (i ny) n h nwly po on. By Lmm 3, h k o o h lry inii riil link rmin h m. Mnwhil, y h iniion o riil link n Lmm 4, h nw riil link hv hir k o minimiz uj o h ompu minimiz wor- k o h i xhii y l on nw riil link. Thu, h Lx-Conrol lgorihm pproh h lxiogrphilly opiml oluion mor riil link r inii. By Lmm 5, h mximum low rurn rom h Boun- Conrol lgorihm i rily inring, o i vnully rh h vry lrg vlu U. In hi, or ny rmining nonriil link l, i k o i givn y l = l l /U, whih i ngligily mll (or imply rgr zro). Thu, h k o o ny rmining link r h opimiz vlu (whih r zro). A h k o o h riil link r minimiz (y h iniion o riil link), h Lx-Conrol lgorihm rmin wih h lxiogrphilly opiml oluion.