Multi-Prefix Trie: a New Data Structure for Designing Dynamic Router-Tables

Transcription

1 1 Multi-Prix Tri: Nw Dt Strutur or Dsigning Dynmi Routr-Tls Sun-Yun Hsih Mmr, IEEE, Yi-Ling Hung, n Ying-Chi Yng Astrt IP looup ts th sp o n inoming pt n th tim rquir to trmin whih output port th pt shoul snt to; hn, it plys n importnt rol in th sign o routr-tls. In this ppr, w propos nw t strutur, ll multi-prix tri, or us in signing ynmi routrtls. On y tur o our t strutur is tht h no n stor mor thn on prix, whih rus th numr o mmory sss. Whn prorming looup, th strutur n srh mor prixs in on no n my in th longst mthing prix in n intrnl no rthr thn on l. Morovr, whn upting th routr-tl, it os not n to ronstrut th tl. As y-prout, th propos t strutur minimizs th tim rquir or ynmi routr-tl oprtions, inluing looup, insrtion, n ltion, n lso rus th numr o mmory sss. W rport th rsults o xprimnts onut to ompr th propos t strutur with othr struturs using th nhmr IPv4 prix ts AS4637 with 219,581 prixs. Inx Trms Clsslss intr omin routing, ynmi routr tls, IP rss looup, longst mthing prix, multi-prix tri. I. INTRODUCTION Th IP rsss o th lssul routing protool r otn ully utiliz, ut Clsslss Intr Domin Routing (CIDR) [] provis wy to llvit th prolm. Th routing ntris o CIDR r pirs o (p/l, o), whr p = p 0 p 1...p l 1 *is prix orm o inry its; l is th lngth o p, whih is t most 32 its or IPv4 [19] n 128 its or IPv6 [8]; n o is n output port intiir. Th prix lngth o CIDR is vril; hn it n provi mor IP rsss thn lssul routing. An intrnt routr lssiis inoming pts into lows s on inormtion ontin in th pt hrs n tl o (lssiition) ruls ll th routr-tl (quivlntly, rul-tl), whih ontins pirs o (p/l, o). Th ruls r us to in th st mthing prix n trmin th output port th pt shoul snt to. Currntly, th st nown prix mthing shm in th Intrnt protool is th so-ll longst prix mthing shm, whih omprs ll th prixs in th routing tl to th stintion rss it-y-it, n thn ins th longst on mong th st o mthing prixs. Whn pt with stintion rss Th uthors r with th Dprtmnt o Computr Sin n Inormtion Enginring, Ntionl Chng Kung Univrsity, No. 1, Univrsity Ro, Tinn 7, TAIWAN. E-mil rsss: hsihsy@mil.nu.u.tw (Corrsponing uthor: S. Y. Hsih); p @mil.nu.u.tw (Y. L. Hung); p @mil.nu.u.tw (Y. C. Yng). Th lssul routing protool hs our rss lsss: A, B, n C hv 8-it, 16-it, 24-it ntwor-rsss, rsptivly, n D is or multist. rrivs, w n in th pir o (p/l, o) in th routr tl i p with lngth l mths, n thn sn th pt to th output o. Howvr, this mthing shm uss mjor ottln in th routr. Clrly, th prormn o th longst prix mthing lgorithm plys n importnt rol in routing vis. Sin stti routr-tls rquir prohiitiv upt tim to ronstrut tl, thy r only suitl or routing nvironmnts without insrtion n ltion; thror, w onntrt on ynmi routr-tls to support rl-tim instnt upts. Intrst rrs my rr to [21] or mor inormtion out stti routr-tls, n to [25] or urthr tils out stti n ynmi routr-tls. In gnrl, our min tors t th prormn o routr-tl. 1) Looup sp: Th mount o Intrnt tri is so lrg tht th sp o trmining whih output port n inoming pt shoul orwr to is ritil. 2) Sp rquirmnt: Th sp rquirmnt must smll so tht th strutur o th routr-tl n stor in th mmory. 3) Slility: Th strutur o th orthoming IPv6 routing protool n ppli to th IPv6 routr. Th routr-tl must hv th slility to l with routing ntris or oth IPv4 n IPv6. 4) Upt sp: Sin vriility is y tur o th Intrnt, routr-tl ntris usully vry suh tht upts (insrtion n ltion) our rquntly. Eh oprtion must prorm iintly without ronstruting th routr tl. Dsigning routr-tls s on irnt t struturs hs riv grt l o ttntion in rnt yrs [2] [5], [7], [], [15] [18], [20], [22] [24], [26] [29], [31], [33], [34]. On lin o rsrh ouss on signing ynmi routrtls tht mnipult prixs using ontiguous intrvls [4], [5], [15], [26], [28], [29], [34]. For instn, whn th lngth o th IP rss quls 5, th intrvl rprsnting th prix p= is [0, 1] = [0, 15]. An intrvl iltr [u, v] mths th stintion rss D i u D v. Consquntly, th longst mthing prix prolm rus to ining th shortst intrvl ontining th stintion rss. Anothr lin o rsrh tris to in hrwr solutions. To this n, Srng Dhrmpurir t l. [9] mploy Bloom iltrs or mthing th longst prix. Jing t l. [13] propos n SRAM-s prlll multi-piplin rhittur or trit IP looup. Sitng Soh t l. [32] utiliz lxiogrphi orr prixs to ru th o-lin onstrution tim o th Full Expnsion Comprssion (FEC) lgorithm [6] n th Comprss Nxt-Hop Arry/Co Wor Arry (CNHA/CWA) [12]. Digitl Ojt Inntiir Authoriz.09/TC lins us limit to: Ntionl Chng Kung Univrsity //$26. Downlo on 20 Jun IEEE 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

2 2 Numrous tri-s routr-tl shms hv lso n propos [3], [18], [20], [22] [24], [29] [31], ut thy sur rom th ollowing shortomings. I. Aitionl mmory sp is rquir us som nos in ths struturs o not ontin ny routing inormtion [18], [20], [22] [24], [31]. II. Eh no stors xtly on prix, so ssing no n only mth on prix. Thus, routr-tl oprtions, suh s looup, insrtion, n ltion t grt l o tim [3], [18], [20], [31]. III. Looup oprtions in ths struturs must ompr th prixs stor in th nos rom th root to l, vn i th longst mthing prix is in som intrnl no. This will inrs th tim rquir or looup oprtions [3], [18], [20], [22] [24], [29] [31]. IV. Th routr-tls in [18], [20] r stti, so thy nnot upt in timly mnnr. Motivt y th ov osrvtions, w propos nw t strutur ll multi-prix tri or signing ynmi routr-tls. In th propos t strutur, ll th nos p th orrsponing routing inormtion; hn h no n stor mor thn on prix, whih rus th numr o mmory sss rquir or routr-tl oprtions. Morovr, multi-prix tri n rturn th longst mthing prix immitly whn it is oun in som intrnl no. This is quit irnt rom th othr t struturs, whih must go to l in orr to rturn th isovr longst mthing prix. Th ov vntgs ru th tim rquir or routr-tl oprtions. In ition, s on th multi-prix tri, w propos nothr t strutur, ll th inx multiprix tri, whih omins th inx tl with th multiprix tri to ru th hight o th tri n xpit routrtl oprtions. Not tht th two propos t struturs n ppli to oth IPv4 n IPv6. Th rminr o this ppr is orgniz s ollows: Stion II introus th propos multi-prix tri. Th ynmi routr-tl oprtions or th propos t struturs r sri in Stion III. W isuss th inx multi-prix tri n its oprtions in Stion IV. Th rsults o xprimnts onut to ompr th propos t struturs with othr t struturs r rport in Stion V. W thn summriz our inings in Stion VI. II. DESIGNING DYNAMIC ROUTER-TABLES USING A MULTI-PREFIX TRIE In this stion, w propos our t strutur, ll multiprix tri, or signing ynmi routr-tls. Th propos strutur utilizs th Prix-Tr () in in [3] s n uxiliry su-strutur. Bor sriing th t strutur, w in som trms us throughout th ppr. For prix p = p 0 p 1...p l 1 *, lt p = p 0 p 1...p i * or 0 i l 2 su-prix o p. Th lngth o prix p, not y ln(p), is th numr o non- symols; or xmpl, ln(p 0 p 1...p l 1 *)=l. Th lvl o no v in root tr, not y lvl(v), is th numr o gs on th pth rom th root to v. I th lst g on th pth rom th root o th tr is (y, x), thn y is th prnt o x, n x is hil o y. A. Prix-trs Eh no in prix-tr ontins xtly on prix, so th siz o th prix-tr is qul to th siz o th routing tl. As shown in Figur 1, h no ontins prix n two pointrs pointing to sussiv tr nos. Th lngth o th prix must grtr thn or qul to th lvl whr th prix is lot. Thus, in th igur, th lvl o th no ontining 1* is 1 n ln(1*)=1. For onvnin, w us LOOKUP, INSERT n DELETE, to rprsnt th rsptiv lgorithms or looup, insrtion, n ltion oprtions in prix-trs in O(W ) tim, whr W is th lngth o th IP rss. Th LOOKUP lgorithm n in th longst mthing prix [3]. Fig. 1. Prix * 1 1* * 1* A prix-tr * * 0 1 * 0 1 1* 0 1 B. Multi-prix tris A tri is root tr t strutur. A -stri Multi- Prix Tri (-M), whr is th stri whih is positiv intgr, ontins two typs o nos, primry no (p-no) n sonry no (s-no), whih possss th ollowing proprtis: P1. Eh p-no v ontins th ollowing ils: ) 0 t m is th numr o prixs stor in v, whr m = O(). ) Th t prixs, not y p 1 (v),p 2 (v),...,p t (v), r stor in non-inrsing orr with ln(p 1 (v)) ln(p 2 (v)) ln(p t (v)). ) port(p i (v)), th output port o p i (v). ) s pointr(v), pointr points to prix-tr ompos o s-nos, whih stor prixs o lngth t lst lvl(v), ut lss thn (lvl(v)+1). For onvnin, th tr init y s pointr(v) is ll th o v. ) Th ontnt o p-no v n rprsnt simply s (t, p 1 (v),p 2 (v),...,p t (v),s pointr(v)). P2. Th stri is th numr o its us y p-no to trmin whih rnh to t. A p-no whos stri is hs 2 hilrn orrsponing to th 2 possil Th s-nos in o p-no v stor prixs o lngth t lst lvl(v), ut lss thn (lvl(v)+1). This n viw s lssiition o prixs oring to thir lngths. Thr ws nothr prix prtitioning thniqu propos in [15] in whih th prixs in h prtition my rprsnt using ynmi routr-tl t strutur. Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

3 3 vlus or th us its. For s o prsnttion, w us hil 0 (v), hil 1 (v),..., hil 2 1(v) to rprsnt 2 hilrn orrsponing to 2 possil vlus rom } {{...0 } to }{{...1 }. Thus, i = 2, thr will our hilrn, hil 0 (v), hil 1 (v), hil 2 (v), n hil 3 (v), orrsponing to,,, n, rsptivly. P3. A p-no with m prixs is si to ull; othrwis, it is non-ull. Anintrnl p-no is ull p-no tht hs hilrn, n n xtrnl p-no is p-no without ny hilrn. Not tht n xtrnl p-no my non-ull. P4. Lt u n v two onsutiv p-nos on pth in T. I thr r two prixs p i (u) n p j (v) suh tht p j (v) is su-prix o p i (u), thn lvl(u) lvl(v). P5. Eh s-no w hs th ollowing ils: ) p(w), th prix stor in w. ) port(p(w)), th output port o th prix stor in w. ) lt(w), pointr initing th lt s-no o w i it xists; othrwis, th pointr is st s null. ) right(w), pointr initing th right s-no o w i it xists; othrwis, th pointr is st s null. A p-no is sri s mpty i it os not ontin ny prix. Figur 2 illustrts 2-M in whih,,,,, n r p-nos. Assum tht m =5; hn, vry p-no ontins t most iv prixs. Th intrnl p-nos n r ull, whil th xtrnl p-nos,,, n r non-ull. Th thr ott pointrs rprsnt s pointrs, whih point to thr s ontining prixs 0, *, n, rsptivly. Not tht, sin prix 0 in is su-prix in, w hv lvl() =0< 1=lvl(). Fig. 2. Prix 1* * 1 0 * 1 0 * 1*,,,,0 0 A 2-M * * 0,1,1,0,* 1 For -M T, th hight o T, not y h(t ), is mx{lvl(v) v is no (p-no or s-no) in T }. To trvrs pth in T rom th root ( p-no) to l, th urrnt p-no rquirs its to rnh out to th nxt p- no until it rrivs t th l. Hn, th longst pth rom th root to n xtrnl p-no is oun y W. Morovr, th s pointr o n xtrnl p-no my point to prixtr whos hight is oun y. Thror, th ollowing lmm hols. Lmm 1: Lt W th lngth o th IP rss n lt T -M. Thn, h(t ) W +. I w hv string x o lngth l n string y o lngth n, th ontntion o x n y, writtn xy, is th string otin y ppning y to th n o x, sinx 1 x l y 1 y n. Th ollowing proprtis r riv rom th struturl hrtriztion o -M. Lmm 2: Lt u p-no in -M T. Thn, th prixs in u hv th ommon su-prix o lngth t, whr t is th lngth o th pth rom th root o T to u. Proo: Lt P = v 0,v 1,...,v t, whr v t = u, th pth rom th root, sy v 0,tou. Thn, th ontntion o th inis (inry strings o lngth ) o th rnhs (v i,v i+1 ) or ll 0 i t 1 orms th ommon su-prix o th prixs in u. Sin th inx o h rnh is inry string o lngth n th pth P hs lngth t, th lngth o th ommon su-prix quls t. Lmm 3: Lt u p-no in -M T. I th prixtr o u xists, thn th prixs in hv th ommon su-prix o lngth t, whr t is th lngth o th pth rom th root o T to u. Proo: Aoring to proprtis P1, P2, n P5 o th - M, th rsult n shown y using mtho similr to tht utiliz to show Lmm 2. III. DYNAMIC ROUTER-TABLE OPERATIONS A. Crting n mpty -M To uil -M T, w irst us th ollowing lgorithm to rt n mpty root no n thn ll n insrtion lgorithm (sri in Stion III-B) to insrt nw prix. Both lgorithms us n uxiliry prour ll ALLOCATE P- NODE, whih llots th mmory sp to us y nw p-no in O(1) tim. Algorithm M CREATE(T ) 1: v := ALLOCATE P-NODE() 2: root(t ) := v B. Insrtion oprtion In this sustion, w prsnt th insrtion lgorithm or -M. Th ollowing inition is usul in our lgorithms. Dinition 1: Lt S = {(p 0 p 1...p l 1 *,i,j) p t {0, 1} or 0 t l 1 n 0 i j l 1}. Din th untion GET : S Z + s GET(p 0 p 1...p l 1,i,j)= j r=i p r2 j r. For xmpl, GET(, 0, 3) = () 2 =4. To insrt prix p = p 0 p 1...p l 1 * into -M, w strt rom th root n mov ownwrs until w in p-no to insrt p. Suppos tht v is th urrnt p-no visit uring this pross. I ln(p) < (lvl(v)+1), thn p will insrt into th o v. Othrwis, i ln(p) (lvl(v) +1), w trmin whthr or not no v is ull n xut th ollowing oprtion. Assum tht th urrnt ontnt o v is (t, p 1 (v),p 2 (v),...,p t (v),s pointr). I v is ull (i.., t = m) n ln(p m (v)) < ln(p), w rpl p m (v) with p, n thn insrt p m (v) into hil j (v), whr j=get(p m (v), Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

4 4 lvl(v), (lvl(v) +1) 1). Th ontnt o v is urthr upt s (t, p 1 (v),p 2 (v),...,p m 1 (v),p,s pointr). Howvr, i v is ull n ln(p m (v)) ln(p), w insrt p into hil j (v), whr j=get(p, lvl(v), (lvl(v) + 1) 1). Othrwis, v is non-ull, i.., t < m, so w just insrt p into v. Now, th ontnt o v is upt s (t +1,p 1 (v),p 2 (v),...,p t (v),p,s pointr). Th lgorithm is til low. Algorithm M INSERT(p, v, lvl) 1: i v is null thn 2: v :=ALLOCATE P-NODE() 3: i IN (ln(p), lvl) thn / p shoul insrt into th o v / 4: u := ALLOCATE S-NODE() / llot th storg or nw s-no / 5: INSERT(p, u, s pointr(v)) / rll tht INSERT is th insrtion lgorithm or prix-trs / 6: ls i Is Full(v) thn 7: i ln(p m (v)) < ln(p) thn / lngth o th lst prix in v<lngth o p / 8: rpl p m (v) with p in v 9: rrng th prixs in v in non-inrsing orr o thir lngth : r := GET(p m (v), lvl, (lvl +1) 1) : v := hil r (v) 12: M INSERT(p m (v), v, lvl +1) 13: ls 14: r := GET(p, lvl, (lvl +1) 1) 15: v := hil r (v) 16: M INSERT(p, v, lvl +1) 17: ls 18: insrt p into v 19: t(v) := t(v)+1 / inrs th numr o th prixs in v / 20: rturn Th initil ll to insrt prix p is M INSERT(p, root, 0). Nxt, w sri how th ov lgorithms wor (s lso Figur 3). Exmpl 1: To insrt th prix into 2-M with m = 5, s shown in Figur 3(), n not rpl with ny prix in p-no (us ln(0)=6> 3 = ln()) or insrt into th o (us ln()=3 > (lvl() + 1) = 2(0 + 1)). W otin th irst two its () 2 vi GET(,0,1) n thn go to hil 1 (), i.., p-no. Sin IN (ln(),1) is TRUE, Algorithm IN (l, lvl) 1: i l< (lvl +1) thn 2: rturn TRUE 3: ls 4: rturn FALSE Algorithm IS FULL(v) 1: i v is ull thn 2: rturn TRUE 3: ls 4: rturn FALSE is insrt into th o (s Figur 3()). Lt us onsir insrting nothr prix. Sin th lngth o th lst prix 0 in is smllr thn ln()=7, 0 is rpl with s shown in Figur 3(). Nxt, w insrt 0 into p-no. Sin p-no is not ull, w insrt th prix into (s Figur 3()). Th ollowing lmms r usul or monstrting th orrtnss o th M INSERT lgorithm. Lmm 4: Atr xuting Algorithm M INSERT to insrt th prixs, th lngth o h prix in p-no v will lrgr thn tht o h prix in th o v. Proo: Lt p n ritrry prix in v n lt p n ritrry prix in th o v. Suppos, y ontrition, tht ln(p) ln(p ). Thn, oring to Lins 3 5 o th lgorithm n Algorithm IN, w hv ln(p ) < (lvl(v)+1). Hn, ln(p) ln(p ) < (lvl(v)+1), whih mns p is in th o v. This ontrits th ssumption tht p is in v. Consir p-no v in -M T. Lt root(t ) th root o T. Any p-no u on th uniqu pth rom th root o T to v is ll n nstor o v. Iu is n nstor o v, thn v is snnt o u. (Evry no is oth n nstor n snnt o itsl.) I u is n nstor o v n u v, thn u is propr nstor o v, n v is propr snnt o u. Th snnt sutr root t v, not y T (v), is th tr inu y snnts o v, root t v. Lmm 5: Lt x n y two prixs tht r insrt using Algorithm M INSERT, whr y is su-prix o x. Ix n y r oth in p-nos, thn x n y r in th sm p-no; othrwis, th lvl o th p-no ontining x is smllr thn th lvl o th p-no ontining y. Proo: W hv th ollowing snrios. Cs 1: x is insrt or y. Lt u th urrnt p- no ontining x. Aoring to th lgorithm, ll th propr nstors o u must ull. Morovr, whn y is insrt into p-no using M INSERT(y, root, 0), y must ollow th ownwr pth rom th root n nountr p-no u us ln(y) <ln(x). Thr r thr ss. Cs 1.1: u is ull n ln(p m (u)) < ln(y). Aoring to Lins 6 12 o th lgorithm, y will lso insrt into u y rpling p m (u). Hn, x n y r now in th sm p-no n th rsult hols. Cs 1.2: u is ull n ln(p m (u)) ln(y). Aoring to Lins o th lgorithm n y th ssumption tht y is in p-no, y will insrt into p-no whos lvl is lrgr thn tht o u. Hn, th rsult hols. Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

5 5 1*,,,,0 1*,,,,0 * 0,1,1,0,* * 0,1,1,0,* 0 * 1 0 * 1 () () 1*,,,, 1*,,,, * 0,1,1,0,* 0 * 0,1,1,0,* 0 * 1 0 * 1 () () Fig. 3. Illustrtion o how Algorithm M INSERT wors. () Th originl 2-M. () 2-M tr insrting prix. () To insrt prix, th lst prix in no is rpl with prix. () Th 2-M tr insrting prix. Cs 1.3:u is non-ull. Aoring to Lins 17 19, y will lso insrt into u. Hn, x n y r now in th sm p-no n th rsult hols. Cs 2: x is insrt tr y. Lt v th urrnt no ontining y. Aoring to th lgorithm, ll th propr nstors o v must ull n th lngth o h prix stor in ny propr nstor o v must t lst ln(y). Whn x is insrt using M INSERT(x, root, 0), ithr x is rpl y som prix in propr nstor o v, or it ollows th ownwr pth rom th root until it nountrs p-no v. In th ormr s, x will in p-no whos lvl is smllr thn lvl(v); n in th lttr s, w hv th ollowing thr snrios. Cs 2.1: v is ull n ln(p m (v)) < ln(x). Aoring to Lins 6 12 o th lgorithm, x will insrt into v y rpling p m (v). I y = p m (v), thn y will insrt into propr snnt o v. Thus, th rsult hols. Othrwis, i y p m (v), thn x n y will insrt into th sm p-no. Cs 2.2:v is ull n ln(p m (v)) ln(x). This ls to ln(y) ln(p m (v)) ln(x), whih ontrits th t tht ln(y) < ln(x) us y is su-prix o x. Cs 2.3:v is non-ull. Aoring to Lins 17 19, x will lso insrt into v. Hn, x n y r now in th sm p-no n th rsult hols. Thorm 1: Algorithm M INSERT(p, root, 0) n orrtly insrt prix p into -M T. Proo: Th orrtnss ollows rom Lmms 4 5 n th orrtnss o Algorithm INSERT. Thorm 2: Algorithm M INSERT(p, root, 0) n implmnt to run in O(W ) tim. Proo: During th xution o th lgorithm, w irst h whthr th givn prix p shoul insrt into th o som p-no v, oring to Lins 3 4. This stp n on in O(1) tim. I th onition hols, it ts O(W ) tim to insrt prix into th oring to Lin 5. Othrwis, w n h whthr or not th urrnt p-no v is ull in O(1) tim. I v is ull n ln(p m (v)) <ln(p), w rpl p m (v) with p n insrt p into th propr position in v n p m (v) into hil j (v) with j=get(p m (v), lvl(v), (lvl(v)+1) 1), oring to Lins Sin m = O(), th ov oprtion ts O() tim without onsiring th rursiv ll. I v is ull n ln(p m (v)) ln(p), w insrt p into hil j (v) with j=get(p, lvl(v), (lvl(v)+1) 1), oring to Lins o th lgorithm. Without onsiring th rursiv ll, this ts O() tim. Othrwis, i v is non-ull, th lgorithm only ns O() tim to insrt p into v. Sin th numr o th p-nos rom th root to n xtrnl p-no in - M is O( W ) n insrting prix into o p-no Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

6 6 ts O(W ) tim, th totl tim omplxity o th lgorithm is O( ( W )+W)=O(W). C. Looup oprtion Th stps o our srh lgorithm, ll LOOKUP, r s ollows. Givn th stintion rss DA, w srh - M strting rom th root in top-own mnnr. Whn p-no v is visit, w irst srh th prixs o v strting rom p 1 (v) to trmin whthr th DA mths som prix in v. I suh prix xists, thn it is th longst mthing prix oring to Proprtis P1() n P4. Othrwis, w srh th orrsponing prix-tr n trmin whthr it ontins mthing prix. I mthing prix is oun, w stor it n pro with urthr itrtions until n xtrnl p-no is visit. Th lgorithm is til low. Algorithm M LOOKUP(DA,v,lvl) / nxt hop is us to ror th output port o th urrnt ttr mthing prix, n ult rout is us to ror th ult output port. / 1: lvl := 0 2: nxt hop := ult rout 3: whil v null o 4: i thr is prix in v tht mths DA thn 5: in th longst prix p i (v) tht mths DA 6: rturn port(p i (v)) 7: ls 8: nxt hop := LOOKUP(DA, s pointr(v)) 9: r := GET(DA, lvl, (lvl +1) 1) : v := hil r (v) : lvl := lvl +1 12: rturn nxt hop Th initil ll is M LOOKUP(DA, root, 0), whr root is th root o th givn -M. W us n xmpl to xplin how th lgorithm wors (s lso Figur 2). Exmpl 2: I DA=, w gin srhing th root n in tht mths this rss. Hn, th longst mthing prix is. I DA =, w gin srhing th root, ut in tht no prix mths. Bs on s pointr(), w go to th orrsponing n in mthing prix. W thn stor th output port numr o, n t th irst two its o DA to go to hil 0 () (i.., p-no ). No prix in mths, ut th orrsponing hs prix 0 tht mths. Sin p-no hs no hil, 0 is thus th longst mthing prix. Thorm 3: Algorithm M LOOKUP(DA, root, 0) n orrtly in th longst mthing prix or DA i suh prix xists. Proo: To vriy tht th lgorithm wors orrtly, w us th ollowing loop invrint : At th strt o h itrtion o th whil loop in Lins 3, th output port o ithr th longst mthing prix or th urrnt st nown mthing prix is oun. W n to show tht 1) this invrint is tru prior to th irst loop itrtion; 2) h itrtion o th loop mintins th invrint; n 3) th invrint provis usul proprty to monstrt th orrtnss whn th loop trmints. Initiliztion: Th ntir -M is not srh prior to th irst itrtion o th loop; nxt hop is st s ult rout n th loop invrint hols trivilly. Mintnn:To h whthr h itrtion mintins th loop invrint, w mth DA with th prixs p 1 (v),p 2 (v),...,p t (v) stor in th urrnt p-no v. First, w onsir th sitution whr DA mths p i (v) or som 1 i t s on th ollowing two ss. Cs 1. nxt hop ult rout. Hr, th DA must mth som prix p in o p-no u, whr lvl(u) < lvl(v). Aoring to Proprty P1() o th -M, lvl(u) ln(p ) < (lvl(u)+1). Not tht ln(p i (v)) lvl(v). Hn, ln(p ) < ln(p i (v)) y ln(p ) < (lvl(u)+1) lvl(v) ln(p i (v)). Morovr, i thr xists nothr mthing prix p in p-no w or in th o w with lvl(w) > lvl(v), thn, oring to Lmm 4 n Proprtis P1() n P4 o th -M, ln(p ) < ln(p i (v)). Bs on th ov osrvtions n th t tht ln(p i (v)) ln(p i+1 (v)) ln(p t (v)), p i (v) is slt s th longst mthing prix n Lin 6 o th lgorithm will rturn th orrt output port. Cs 2. nxt hop = ult rout. Hr, p i (v) is th irst mthing prix. I thr is nothr mthing prix p in p-no w or in th o w with lvl(w) > lvl(v), thn, oring to Lmm 4 n Proprtis P1() n P4 o th -M, ln(p ) < ln(p i (v)). Morovr, us ln(p i (v)) ln(p i+1 (v)) ln(p t (v)), p i (v) is slt s th longst mthing prix n Lin 6 o th lgorithm will rturn th orrt output port. Nxt, w onsir th sitution whr th DA os not mth ny prix in p-no. I th DA mths prix in th o v, thn Lin 8 o th lgorithm will rturn th output port o th longst mthing prix mong th prixs in th. Clrly, th isovr mthing prix is urrntly th st on. Howvr, i th DA os not mth ny prix in th o v, thn in Lins 9, w gt th nxt its to trmin whih hil o v shoul visit nxt. Rnwing v n inrmnting lvl r-stlishs th loop invrint or th nxt itrtion. Trmintion: At th trmintion, v is null. By th loop invrint, h tion o rnwing nxt hop will p th output port o th urrnt st mthing prix. Sin thr r no othr nos to visit, Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

7 7 Lin 12 o th lgorithm will rturn th output port o th longst mthing prix i suh prix xists. Thorm 4: Algorithm M LOOKUP(DA, root, 0) n implmnt to run in O( W 2 ) tim. Proo: For h prix p = p 0 p 1...p l 1 *, w us n intrvl I p =[, ] to rprsnt p, whr = l 1 r=0 p r2 W r 1 n = l 1 r=0 p r2 W r 1 + W 1 r=l 2 W r 1. For xmpl, whn W =8, I 0 = [32, 63]. Not tht th DA mths prix [, ] i th DA s iml vlu is twn n. Hn, hing whthr DA mths prix will t O(1) tim. In h itrtion o th whil loop, w n h whthr DA mths som prix in th urrnt p-no in O() tim. Sin th numr o th p-nos rom th root to n xtrnl p-no in -M is O( W ) n loouping prix in ts O(W ) tim, th ovrll tim omplxity o th lgorithm is O(( + W ) W ) = O( W 2 ). D. Dltion oprtion To lt prix p = p 0 p 1...p l 1 *, w strt rom th root, ollow th ownwr pth, n xut th ollowing oprtion. Suppos tht v is th p-no ing visit urrntly. First, w utiliz Algorithm IN to h whthr p is in th o v. Ip is in th, w simply lt p n rls th storg llotion o th orrsponing s-no ontining p. Howvr, i p is not in th o v, thr r two possil snrios pning on whthr p is in v or not. W irst onsir th s whr p is in v. Lt (t, p 1 (v),p 2 (v),...,p t (v),s pointr(v)) th urrnt ontnt o v suh tht p i (v) = p or som 1 i t. I v is n xtrnl p-no, w rmov p rom v n upt th ontnt o v s (t 1,p 1 (v),p 2 (v),...,p i 1 (v),p i+1 (v),...,p t (v),s pointr(v)). Not tht whn t 1 = 0 n v hs no, i.., s pointr(v)=null, w only n to rls th storg llotion o v. Howvr, i v is n intrnl p-no, w lt p irst. Lt hil r (v) or som 0 r 2 1 th hil ontining th longst prix, sy y, mong th prixs stor in th hilrn o v. W insrt y into v n upt th ontnt o v s (t, p 1 (v),p 2 (v),...,p i 1 (v),p i+1 (v),...,p t (v),y, s pointr(v)). By xuting th ov oprtions, w rursivly ll th ltion lgorithm to lt y rom hil r (v). For th sitution whr p is not in v, w gt th sir its rom p to rnh out rom som hil o v, n rursivly ll th ltion lgorithm. Th omplt lgorithm is prsnt in Algorithm M DELETE(p, v, lvl). Th initil ll or lting prix p is M DELETE(p, root, 0). Exmpl 3: Th stps o th M DELETE lgorithm r s ollows. First, w lt th prix * rom th 2-M shown in Figur 4(). Sin * is not in p-no or in th o, w gt th irst two its () 2 n go to hil 1 () =. Howvr, * is in th o, so w lt it rom th (s Figur 4()). Nxt, w lt prix 0 rom th 2-M shown in Figur 4(). Sin 0 is not in p-no or in th o, w gt th irst two its () 2 n go to Algorithm M DELETE(p, v, lvl) / This lgorithm uss two uxiliry prours, FREE S- NODE n FREE P-NODE, whih r th storg llotion o n s-no n p-no, rsptivly, in O(1) tim. / 1: i v is null thn 2: output p is not oun 3: i IN (ln(p), lvl) thn 4: DELETE(p, s pointr(v)) 5: FREE S-NODE 6: ls i p is in v thn 7: lt p rom v 8: i v is n xtrnl p-no thn 9: t(v) := t(v) 1 / rs th numr o th prixs in v / : i t(v) =0n s pointr(v)=null thn : FREE P-NODE(v) 12: ls 13: in prix y in hil r (v) s.t. ln(y) = mx{ln(p) p hil i (v) or 0 i 2 1} 14: insrt y s th lst prix in v 15: v := hil r (v) 16: M DELETE(y, v, lvl +1) 17: ls 18: r := GET(p, lvl, (lvl +1) 1) 19: v := hil r (v) 20: M DELETE(p, v, lvl +1) 21: rturn hil 3 () =. W in tht 0 is in p-no, sow lt it (s Figur 4()). As shown in Figur 4(), 0 is rpl with prix 1 in p-no, n 1 is lt rom. As p-no is now mpty, w rls its storg llotion. Finlly, w onsir to lt rom p-no, s shown in Figur 4(). Th intrmit 2- M is shown in Figur 4(). W slt th longst prix mong th prixs stor in th hilrn o. Th lngth o th longst prix quls 6, ut thr r mny prixs o lngth 6 stor in th hilrn o. Without loss o gnrlity, w ssum tht 0 in p-no is slt. W thn insrt it s th lst prix in n lt it rom (s Figur 4()). Not tht sin th o v still xists tr lting 0, w o n not to rls th storg llotion o. Th ollowing lmms r usul or monstrting th orrtnss o th ltion lgorithm. Lmm 6: Atr using Algorithm M DELETE (z, root, 0) to lt prix z rom p-no v, h prix in v will still longr thn ny prix in th o v. Proo: Aoring to Lins o th lgorithm, th longst prix in som hil o v my mov to v. Lt x n ritrry prix in th o v n lt y th prix mov up to v. Sin ln(x) < (lvl(v)+1) n (lvl(v)+ 1) ln(y), ln(x) <ln(y). Lmm 7: Lt x n y two prixs in p-no n lt y su-prix o x. Thn, tr using Algorithm M DELETE(z, root, 0) to lt prix z / {x, y}, on Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

8 8 1*,,,, 1*,,,, 0 * 0,1,1,0,* 0 * 0,1,1,0,* 0 * () () 1*,,,, 1*,,,, 0 * 1,1,0, 0 * 1,1,1,0,* () () 1*,,, 1*,,,,0 0 * 1,1,1,0,* * 1,1,1,0,* 0 0 () () Fig. 4. Th oprtions o th M DELETE lgorithm. () Th originl 2-M. () 2-M tr lting prix *. () Prix 0 is lt rom p-no. () Prix 1 is mov to no n th storg llotion o is rls. () Prix is lt rom p-no. () Th longst prix, 0, in th hilrn o is mov to no suh tht p-no is now mpty. o th ollowing two onitions will hol: (1) x n y will stor in th sm p-no suh tht x will ppr or y; or (2) th lvl o th p-no ontining x will smllr thn th lvl o th p-no ontining y. Proo: Lt u n v two p-nos ontining x n y, rsptivly, or xuting Algorithm M DELETE(z, root, 0). W hv th ollowing snrio. Cs 1: u = v. Sin ln(y) <ln(x), x pprs or y in u. Aoring to th lgorithm, thr r two su-ss. Cs 1.1:Th irst prix p 1 (u) is mov up to th prnt o u.ix = p 1 (u), thn x is mov to th prnt o u tr th lgorithm hs n xut; thus, Conition (2) hols. Howvr, i x p 1 (u), thn x n y will still in th sm p-no tr th lgorithm hs n xut. Hn, Conition (1) hols. Cs 1.2:Th irst prix p 1 (u) is not mov up to th prnt o u. In this su-s, x n y r still in u n rtin thir rltiv orr. Hn, Conition (1) hols. Cs 2: u v. By Lmm 5, lvl(u) < lvl(v). Atr xuting th lgorithm, ithr lvl(u) < lvl(v) still hols or lvl(u)+1 = lvl(v) n y is mov up to u. In th lttr s, sin ln(x) >ln(y), Conition (1) hols oring to Lin 14 o th lgorithm. Thorm 5: Algorithm M DELETE(p, root, 0) n or- Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

9 9 rtly lt prix p rom -M. Proo: Th orrtnss ollows irtly rom th lgorithm n Lmms 6 n 7. Thorm 6: Algorithm M DELETE(p, root, 0) n implmnt to run in O( 2 W ) tim. Proo: During th xution o th lgorithm, w irst h whthr p is in th o th urrnt p-no v (Lins 3 5) in O(1) tim. I p is in th, it ts O(W ) tim to lt it. Othrwis, w h whthr p is in v n lt it in O() tim. I p is in v, thn th orrsponing oprtions (without rgr to rursiv ll) sri in Lins 6 16 o th lgorithm n implmnt in O(2 ) tim us w in th longst prix mong th prixs in O(2 ) hilrn o v. Othrwis, p is not in v. Th orrsponing oprtions (without rgr to rursiv ll) sri in Lins o th lgorithm n implmnt in O() tim. Thror, th totl omplxity is O(2 ( W ) + W )=O( 2 W ). IV. INDEX MULTI-PREFIX TRIE W now prsnt nw t strutur ll th -stri Inx Multipl Prix Tr (-IM or short), whih is s on th -M. Th -IM prtitions -M into svrl smllr -Ms to ru its hight. This i hs n us in mny routrs [14]. Th prtition -Ms r mrg through th inx tl t[ }{{...0 }, }{{...1 },, } {{...1 } ]. α α α Givn ix lngth α, th inx tl hs 2 α ntris suh tht th ntry t[ α 1 ] ontins pointr, whih inits th -M tht stors th prixs with th ommon suprix α 1 (s Figur 5). To xut th routr-tl storg rquirmnt. Th lgorithms or looup n ltion wor similrly. Dinition 2: Lt S = {(p 0 p 1...p l 1 *,α) p i {0, 1} or 0 i l 1 n 0 l < α}. Din th untion TAKE PREFIX ENDPOINT : S Z + s TAKE PREFIX ENDPOINT(p 0 p 1...p l 1,α) = l 1 r=0 p r2 α r 1 + α 1 r=l 2α r 1. For xmpl, TAKE PREFIX ENDPOINT(1, 8) = (11) 2 = 163. Th lgorithms or th looup, insrtion, n ltion oprtions or prix p = p 0 p 1...p l 1 * on th -IM r til low. Algorithm IM LOOKUP(α, p, DA) 1: s := GET(p, 0,α 1) 2: M LOOKUP(DA, t[s] root, 0) Algorithm IM INSERT(α, p) 1: s := GET(p, 0,α 1) 2: i ln(p) α thn 3: i (t[s] root) is null thn 4: M CREATE(t[s]) 5: M INSERT(p, t[s] root, 0) 6: ls 7: := TAKE PREFIX ENDPOINT(p, α) 8: or i := s to o 9: M INSERT(p, t[i] root, 0) Fig. 5. A -IM Inx Tl -M -M -M Algorithm IM DELETE(α, p) 1: s := GET(p, 0,α 1) 2: i ln(p) α thn 3: M DELETE(p, t[s] root, 0) 4: ls 5: := TAKE PREFIX ENDPOINT(p, α) 6: or i := s to o 7: M DELETE(p, t[i] root, 0) oprtions (looup, insrtion, n ltion) in -IM, w irst mth th inx tl n thn xut th oprtions in th orrsponing -M. For xmpl, i w insrt prix p into -IM, w irst gt th α its o p to trmin th inx vlu. I ln(p) α, w n insrt p into th orrsponing rry. Howvr, i ln(p) <α, thn p orrspons to mor thn on inx vlu. For instn, ssum tht p =1 n α = 8 (i.., th irst ight its o prix rprsnt th inx vlu). Bus 1 ontins,,, n 11, th prix p =1 must insrt into th -Ms init y t[], t[], t[], n t[11], rsptivly. Sin storing uplit prixs is iniint, w must hoos n α vlu tht n ru th V. EXPERIMENTAL RESULTS W onut xprimnts on th nhmr IPv4 prix ts AS4637, t Novmr 21, 27 [1]. It ontins 219,581 prixs. W implmnt th -M n -IM or vrious, s wll s or othr t struturs (or omprison with -M n -IM) using C++. All th xprimnts wr prorm on 3.40GHz PC with 512MB mmory. Figur 6 shows th totl prix-popultion y th prix lngth. In Stion V-A, w slt propr n m or -M n - IM, n nlyz thir prormn. Thn, in Stion V-B, w ompr th propos t struturs with othr t struturs. Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

10 @ 9 TABLE I AVERAGE LOOKUP TIME (# CLOCK CYCLES) OF -MS WITH DIFFERENT AND m m \ -M 1-M 2-M 3-M 4-M 5-M TABLE II AVERAGE UPDATE TIME (# CLOCK CYCLES) OF -MS WITH DIFFERENT AND m m \ -M 1-M 2-M 3-M 4-M 5-M ? 6 ; : >= ;< 6 9 : & ' ) * +, - ) / IM. As w n s in Tl I, it n hiv th st vrg looup tim whn = 2 n m = For vrg upt tim shown in Tl II, 4-M hs ttr prormn. Morovr, it n hiv th st vrg upt tim whn =4n m =3. Sin (vrg upt tim on =4n m =2 +1) (vrg upt tim on =4 n m = 3) = = 53 is smll; hn, w slt m = 2 +1 or ompring -M with othr t struturs in Stion V-B. On th othr hn, th -IM hs similr rsults. Th tls o vrg looup tim n vrg upt tim or -IM with irnt n m r shown t Fig. 6. Prix-popultion in AS4637 A. Slting n m or M n IM Sin th numr m = O() or prixs in th p-no o M my ts prormn, w n to slt propr vlus pproprit or n m. Th xprimntl rsults or th vrg looup tim n vrg upt tim or - M with irnt n m s on AS4637 r shown in Tl I n Tl II, rsptivly. W insrt n lt 5,0 prixs in AS4637 to omput th vrg upt tim. Sin th lngth o h prix in AS4637 is t lst 8, w st α =8to voi uplit storg o th sm prix in W slt m =2+1 or 1 5 to ompr th storg rquirmnt y th xprimnt. Th xprimntl rsults or th numr o nos, th hight, n th storg rquirmnts or uiling -M n -IM s on AS4637 r shown in Tl III. Not tht or only on -M, th mximum hight n vrg hight r th sm. On th othr hn, sin on -IM T my ontin svrl -Ms, th mximum hight o T is in s mx{h(t ) T is -M ontin in T }; n th vrg hight o T is th vrg hight o ll th -Ms in T. As shown in Tl III, whn inrss, th hight rss, ut mor storg is rquir. Th rson is tht lrgr n stor mor prixs in p- no, whih implis tht mor storg is rquir. To hiv ttr prormn, w n to onsir th hight, oprting tim, n storg rquirmnt whn slting. As shown in Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

11 1-M 2-M 3-M 4-M 5-M 1-IM 2-IM 3-IM 4-IM 5-IM TABLE III REQUIREMENTS FOR BUILDING -MS AND -IMS BASED ON AS4637 # Nos Mx Hight Avg. Hight Storg (KB) # p-nos # s-nos Tl III, 2-M rquirs lss storg thn 4-M. From th ov osrvtions, w onlu tht th looup prormn o 2-M is ttr, n its vrg upt tim is just littl highr thn tht o 4-M. Thror, w ompr 2-M with th othr t struturs in Stion V-B. W hv similr osrvtions on -IM n slt 2-IM with m =2 +1 to ompr it with th othr t struturs in th nxt stion. B. Comprison with othr t struturs Tl IV omprs th numr o nos, th numr o ummy nos, th hight, n th storg rquirmnt whn implmnting th AS4637 routr-tl with th ollowing struturs: inry tri, n LC-tri [18], moii LC-tri [20], prix-tr [3], Dynmi Tr BitMp () [30], Fix-Stri Tri (FST) [22], 2-M, n 2- IM. For th multi-it tri s strutur, w opt th -FST, ix-stri tri with th lvl. W ppli th lgorithm propos y Shni n Kim [22] to omput th ost n th st stri o -FST or 3 7. In our xprimnt, 7-FST is suprior thn th othr -FSTs (3 6) with rspt to th worst-s looup tim, numr o nos, numr o ummy nos, storg n upt tim. Morovr, th vrg looup tim is omptitiv to othr hois o. Thror, w slt 7-FST to ompr with th propos t struturs. Not tht th inry tri, th LC-tri, th moii LC-tri,, n 7-FST ontin ummy nos, whih inrss th storg ost sustntilly. Morovr, th inry tri, th LC-tri, th moii LC-tri, th prix-tr,, n 7-FST hv mor nos thn our t struturs. Th storg rquirmnt o th LC-tri n th moii LC-tri n to stor inormtion out th nxt hop, th prix, s wll s sip n rnh oprtions in h no; hn, thir storg rquirmnts r grtr thn thos o th othr struturs. Spiilly, th hight (oth th mximum hight n th vrg hight) o 2-M (2-IM) is lss thn tht o th othr struturs xpt or n 7- FST, whih rus th numr o mmory sss rquir or routr-tl oprtions. Although th hight o (7- FST) is smllr thn our t struturs, th looup tim o our t struturs is omptitiv to n 7-FST in th xprimnt. Sin h mmory ss rquirs grt l o tim, th numr o mmory sss ts th oprting tim. Tl V, Figur 7, n Figur 8 ompr th prormn o th looup oprtions o th irnt t struturs. Both 2-M n 2-IM rquir wr mmory sss thn th othr struturs xpt or n 7-FST, whih implis tht th vrg looup tim lso shortr thn th ompr struturs. On y rson is tht our t struturs n rturn th longst mthing prix whn it is oun in n intrnl no, without going to l to rturn. Although th (rsptivly, 7-FST) rquirs wr mmory sss thn 2- M (rsptivly, 2-M n 2-IM), 2-M n 2-IM r str in trms o looup tim. Th rsons r s ollows. (1) Whn no v is visit in th, n uxiliry untion is ll to trmin whthr thr is mthing prix mths th stintion rss. I suh prix xists, thn it will stor s th urrntly st mthing prix. Morovr, th orrsponing nxt hop is lso pt. (2) Whn no is visit in th 7-FST, th orrsponing stri n to Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

12 12 TABLE IV STRUCTURE COMPARISON OF DIFFERENT DATA STRUCTURES # Nos # p-nos # s-nos # Dummy Nos Mx. Hight Avg. Hight Storg (KB) Binry Tri LC-Tri Moii LC-Tri Prix Tr FST M IM TABLE V LOOKUP TIME AND THE NUMBERS OF MEMORY ACCESSES FOR DIFFERENT DATA STRUCTURES Tim (# Clo Cyls ) # Mmory Asss Avg. Worst Avg. Worst Binry tri LC-tri Moii LC-tri Prix-tr FST M IM Avg. Looup Tim Worst-s Looup Tim Tim (# Clo Cyls) Binry Tri LC-Tri Moii LC-Tri Prix-Tr 7-FST 2-M 2-IM Tim (# Clo Cyls) Binry Tri LC-Tri Moii LC-Tri Prix-Tr 7-FST 2-M 2-IM () () Fig. 7. Th vrg looup tim n worst-s looup tim or irnt t struturs omput in orr to go to th nxt no. Ths oprtions t longr tim. Tl VI, Figur 9, n Figur show th prormn omprison o upting 5,0 prixs in AS4637 with irnt ynmi t struturs. Sin th LCtri, moii LC-tri, n FST r stti, th whol tl must ronstrut whn insrting or lting prix; hn, th ov thr t struturs r xlu rom th omprison. As shown in th tl n igurs, oth 2-M n 2-IM rquir shortr upt tim. Sin th inry tri my insrt or lt longst prix, th mmory ss is 32 or IPv4 (128 or IPv6). Thror, th upt tim o th inry tri is lrgr thn our t struturs. Morovr, th upt tim n th numr o mmory sss or th inry tri n th prix-tr r lmost th sm us, in oth struturs, th insrtion n ltion o mny routing ntris in AS4637 otn ollow ownwr pths to lvs in orr to xut th oprtions. Not tht th upt tim n th numr o mmory sss or th inry tri n th prix-tr r lrgr thn 2-M n 2-IM. Although th numr o mmory sss o is lss thn 2-M n 2-IM, th upt tim o is lrgr thn 2-M n 2-IM. Th rson is tht ns mor omplx oprtions to prorm llotion whn lting prix. VI. CONCLUDING REMARKS W hv propos two nw t struturs, M n IM, or ynmi routr-tl sign. Sin th struturs o not ontin ummy nos, thy rquir lss storg n thy r not s high s othr trs. In ition, us o th lowr hight, thy rquir wr mmory sss or routr- Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

13 13 Avg. # Mmory Asss or Looup Worst-s # Mmory Asss or Looup # Mmory Asss Binry Tri LC-Tri Moii LC-Tri Prix-Tr 7-FST 2-M 2-IM # Mmory Asss Binry Tri LC-Tri Moii LC-Tri Prix-Tr 7-FST 2-M 2-IM () () Fig. 8. Th vrg numr mmory sss n th worst-s numr o mmory sss or irnt t struturs TABLE VI THE UPDATE TIME AND THE NUMBER OF MEMORY ACCESSES FOR DIFFERENT DATA STRUCTURES Tim (# Clo Cyls) # Mmory Asss Avg. Worst Avg. Worst Binry tri Prix-tr M IM Avg. Upt Tim Worst-s Upt Tim Tim (# Clo Cyls) Binry Tri Prix-Tr 2-M 2-IM Tim (# Clo Cyls) Binry Tri Prix-Tr 2-M 2-IM () () Fig. 9. Th vrg upt tim n th worst-s upt tim or irnt t struturs tl oprtions. Th simultion rsults rvl tht our t struturs r suprior to thos tri-s t struturs in rspt o storg, IP looup tim, n upt tim. Tl VII summrizs th struturl proprtis o irnt tri-s t struturs. Th inry tri, th (moii) LCtri n ontin ummy nos, so it is nssry to go to l to in th longst mthing prix. Our propos t struturs o not sur rom this limittion. Our utur wor ttmpts to ru th numr o mmory sss n th mmory rquirmnt o th propos t struturs whil prsrving st upting. ACKNOWLEDGMENT Th uthors r ply ppritiv or th ommnts n suggstions givn y th itor n th nonymous rrs. REFERENCES [1] BGP Tl otin rom [2] A. Bsu n G. Nrlir, Fst inrmntl upts or piplin orwring ngins, IEEE/ACM Trnstion on Ntworing, vol. 13, no. 3, pp , Jun 25. [3] M. Brgr, IP looup with low mmory rquirmnt n st upt, in Proings o IEEE High Prormn Swithing n Routing, pp , Jun 23. [4] Y. K. Chng, Simpl n st IP looups using inomil spnning trs, Computr Communitions, vol. 28, no. 5, pp , Mr. 25 [5] Y. K. Chng n Y. C. Lin, Dynmi sgmnt trs or rngs n prixs, IEEE Trnstions on Computrs, vol. 56, no. 6, pp , Jun 27 [6] P. Crsnzi, L. Drini, n R. Grossi, IP rss looup m st n simpl, in Proings o Svnth Ann. Europn Symp. Algorithms, Thnil Rport TR-99-, Univ. o Pis, [7] M. Dgrmr, A. Broni, S. Crlsson, n S. Pin, Smll orwring tls or st routing looups, in Proings o ACM SIGCOMM, pp. 3 14, [8] S. Dring n R. Hinn, Intrnt protool vrsion 6 (IPv6) spiition, RFC 1883, D [9] S. Dhrmpurir, P. Krishnmurthy n D. E. Tylor, Longst prix mthing using loom iltrs, IEEE/ACM Trnstions on Ntworing, vol. 14, no. 2, pp , Apr. 26 [] W. Doringr, G. Krjoth, n M. Nsshi, Routing on longst- Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

14 14 Avg. # Mmory Asss or Upt Worst-s # Mmory Asss or Upt #MmoryAsss Binry Tri Prix-Tr 2-M 2-IM #MmoryAsss Binry Tri Prix-Tr 2-M 2-IM () () Fig.. Th vrg numr o mmory sss n th worst-s numr o mmory sss or irnt t struturs TABLE VII SUMMARY OF PROPERTIES FOR DIFFERENT DATA STRUCTURES Dummy no Tl Hight Compr-to-L Btring Binry Tri Y ynmi W Y N LC-tri Y stti W Y Y Moii LC-Tri Y stti W Y N Prix-Tr N ynmi W Y N Y ynmi W Y N 7-FST Y stti Y N M N ynmi W + N N IM N ynmi W + N N Th strutur must ompr th prixs stor in th nos rom th root to l, vn i th longst mthing prix is in som intrnl no. Th prix o th l in this strutur os not mth th stintion rss, so it must tr rom th l to th uppr nos to in th longst mthing prix. mthing prixs, IEEE/ACM Trnstions on Ntworing, vol. 4, no. 1, pp , [] V. Fullr, T. Li, J. Yu n K. Vrhn, Clsslss intr-omin routing (CIDR): n rss ssignmnt n ggrgtion strtgy, RFC 1519, Sp [12] N. F. Hung n S. M. Zho, A novl IP-routing looup shm n hrwr rhittur or multigigit swithing routrs, IEEE Journl on Slt Ars in Communitions, vol. 17, no. 6, pp , Jun [13] W. Jing, Q. Wng, n V. Prsnn, Byon TCAMs: n SRAM-s prlll multi-piplin rhittur or trit IP looup, in Proings o IEEE INFOCOM, pp , Apr. 28. [14] B. Lmpson, V. Srinivsn, n G. Vrghs, IP looups using multiwy n multiolumn srh, in Proings o IEEE INFOCOM, pp , Apr [15] H. Lu, K. S. Kim n S. Shni, Prix n intrvl-prtition ynmi IP routr-tls, IEEE Trnstions on Computrs, vol. 54, no. 5, pp , My 25 [16] J. H. Mun, H. Lim n C. Yim, Binry srh on prix lngths or IP rss looup, IEEE Communitions Lttrs, vol., no. 6, pp , Jun 26 [17] S. Nilsson n G. Krlsson, Fst rss loo-up or Intrnt routrs, IEEE Bron Comm., pp. 22, [18] S. Nilsson n G. Krlsson, IP-rss looup using LC-tri, IEEE Journl on slt Ars in Communitions, vol. 17, no. 6, pp , Jun [19] J. Postl, Intrnt protool rp intrnt progrm protool spiition, RFC791, Sp [20] V. C. Rviumr, R. Mhptr, J. C. Liu, Moii LC-tri s iint routing looup, in Proings o th th IEEE Intrntionl Symp. on Moling, Anlysis n Simultion o Computr n Tlommunition Systms (MASCOTS), pp , Ot. 22. [21] M. A. Ruiz-Snhz, E. W. Birs, n W. Dous, Survy n txonomy o IP rss looup lgorithms, IEEE Ntwor, vol. 15, pp. 8 23, 20. [22] S. Shni n K. S. Kim, Eiint onstrution o ix-stri multiit tris or IP looup, in Proings o th 8th IEEE Worshop on Futur Trns o Distriut Computing Systms, pp , 20. [23] S. Shni n K. S. Kim, Eiint onstrution o vril-stri multiit tris or IP looup, in Proings o IEEE Symp. Applitions n th Intrnt (SAINT), pp , 22. [24] S. Shni n K. S. Kim, Eiint onstrution o multiit tris or IP looup, IEEE/ACM Trnstion on Ntworing, vol., no. 3, pp , 23. [25] S. Shni, K. S. Kim, n H. Lu, Dt struturs or on- imnsionl pt lssiition using most-spii-rul mthing, Intrntionl Journl o Fountions o Computr Sin, vol. 14, no. 3, pp , 23. [26] S. Shni n K. S. Kim, An O(log n) ynmi routr-tl sign, IEEE Trnstions on Computrs, vol. 53, no. 3, pp , Mr. 24. [27] S. Shni n K. S. Kim, Eiint onstrution o piplin multiittri routr-tls, IEEE Trnstions on Computrs, vol. 56, no. 1, pp , Jn. 27. [28] S. Shni n H. Lu, Enhn intrvl trs or ynmi IP routrtls, IEEE Trnstions on Computrs, vol. 53, no. 12, pp , D. 24 [29] S. Shni n H. Lu, A B-tr ynmi routr-tl sign, IEEE Trnstions on Computrs, vol. 54, no. 7, pp , July 25. [30] S. Shni n H. Lu, Dynmi tr itmp or IP looup n upt, in Proings o th Sixth Intrntionl Conrn on Ntworing (ICN 07), pp.79, 27. [31] K. Slowr, A tr-s pt routing tl or Brly unix, in Proings o Wintr Usnix Con, pp , [32] S. Soh, L. Hirynto n S. Ri, Eiint prix upts or IP routr using lxiogrphi orring n uptl rss st, IEEE Trnstions on Computrs, vol. 57, no. 1, pp , Jn. 28 [33] V. Srinivsn n G.Vrghs, Fstr IP looups using ontroll prix xpnsion, ACM Trnstions on Computr Systms, pp. 1 40, F [34] P. R. Wrh, S. Suri, n G. Vrghs, Multiwy rng tr: sll Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.

15 15 IP looup with st upts, Computr Ntwor, vol. 44, no. 3, pp , F. 24. PLACE PHOTO HERE Sun-Yun Hsih riv th PhD gr in omputr sin rom Ntionl Tiwn Univrsity, Tipi, Tiwn, in Jun H thn srv th ompulsory two-yr militry srvi. From August 20 to Jnury 22, h ws n ssistnt prossor t th Dprtmnt o Computr Sin n Inormtion Enginring, Ntionl Chi Nn Univrsity. In Frury 22, h join th Dprtmnt o Computr Sin n Inormtion Enginring, Ntionl Chng Kung Univrsity, n now h is ull prossor. H riv th 27 K. T. L Rsrh Awr, Prsint s Cittion Awr (Amrin Biogrphil Institut) in 27, th Enginring Prossor Awr o Chins Institut o Enginrs (Kohsiung Brnh) in 28, th Ntionl Sin Counil s Outstning Rsrh Awr in 29, n Distinguish Prossor Awr t Ntionl Chng Kung Univrsity in 20. His urrnt rsrh intrsts inlu sign n nlysis o lgorithms, ult-tolrnt omputing, ioinormtis, prlll n istriut omputing, n lgorithmi grph thory. PLACE PHOTO HERE Yi-Ling Hung riv th B.S. gr in Computr Sin n Inormtion Enginring rom Ntionl Chnghu Univrsity o Eution, Tiwn, in 26, n th M.S. gr in Computr Sin n Inormtion Enginring rom Ntionl Chng Kung Univrsity, Tiwn, in 28. In July 28, sh join th ntwor oprtions lortory o Chunghw Tlom Lortoris, n now sh is n ssoit rsrhr. Hr rsrh intrsts inlu IP looup n pt lssiition lgorithms, progrmml routrs, th sign n nlysis o sll srhing lgorithms n rhitturs, n gogrphi inormtion systm lgorithms. PLACE PHOTO HERE Ying-Chi Yng riv th B.S. gr in th Dprtmnt o Computr Sin n Enginring rom Ntionl Chung-Hsing Univrsity, Tiwn, in 28. Sh is urrntly woring towr th M.S. gr in omputr sin n inormtion nginring t Ntionl Chng Kung Univrsity, Tiwn. Hr rsrh intrsts inlu IP routing n srhing lgorithms. Authoriz lins us limit to: Ntionl Chng Kung Univrsity. Downlo on Jun 21,20 t 13:28:08 UTC rom IEEE Xplor. Rstritions pply.