Survey and Taxonomy of IP Address Lookup Algorithms

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1 Survy n Txonomy of IP Arss Lookup Algorithms Migul Á. Ruiz-Sánhz, 2 3 Ernst W. Birsk, 4 Wli Dbbous 2 Jnury 5, 2 2 INRIA Sophi Antipolis Frn. 3 Univrsi Autónom Mtropolitn Iztplp Cmpus Mxio City. 4 Institut Euréom Sophi Antipolis Frn. Abstrt Du to th rpi growth of trffi in th Intrnt, bkbon links of svrl Gigbit/s r ommonly ploy. To hnl Gigbit/s trffi rts, th bkbon routrs must b bl to forwr millions of pkts pr son on h of thir ports. Fst IP rss lookup in th routrs, whih uss th pkts stintion rss to trmin for h pkt th nxt hop, is thrfor ruil to hiv th pkt forwring rts rquir. IP rss lookup is iffiult bus it rquirs longst mthing prfix srh. In th lst oupl of yrs, vrious lgorithms for high prformn IP rss lookup hv bn propos. W prsnt survy of stt-of-th rt IP rss lookup lgorithms n ompr thir prformn in trms of lookup sp, slbility, n upt ovrh. Introution Th primry rol of routrs is to forwr pkts towrs thir finl stintion. To this purpos, routr must i for h inoming pkt whr to sn it nxt. Mor xtly, th forwring ision onsists in fining th rss of th nxt-hop routr s wll s th grss port through whih th pkt shoul b snt. This forwring informtion is stor in forwring tbl tht th routr omputs bs on th informtion gthr by routing protools. To onsult th forwring tbl, th routr uss th pkt s stintion rss s ky; this oprtion is ll rss lookup. On th forwring informtion is rtriv, th routr n trnsfr th pkt from th inoming link to th pproprit outgoing link, in pross ll swithing. Th xponntil growth of th Intrnt hs strss its routing systm. Whil th t rts of links hv kpt p with th inrsing trffi, it hs bn iffiult for th pkt prossing pity of routrs to kp up with ths inrs t rts. Spifilly, th rss lookup oprtion is mjor bottlnk in th forwring prformn of toy s routrs. This ppr prsnts survy of th ltst lgorithms for ffiint IP rss lookup. W strt by tring th volution of th IP rssing rhittur. Th rssing rhittur is This rsrh ws support in prt by CS Tlom.

2 of funmntl importn to th routing rhittur n rviwing it will hlp us to unrstn th rss lookup problm.. Th Clssful Arssing Shm In IP vrsion 4, IP rsss r 32 bit long n, whn brokn up into 4 groups of 8 bits, r normlly rprsnt s four iml numbrs sprt by ots. For xmpl, th rss orrspons in th ott-iml nottion to On of th funmntl objtivs of th Intrnt Protool is to intronnt ntworks; so routing on ntwork bsis ws nturl hoi (rthr thn routing on host bsis). Thus, th IP rss shm initilly us simpl two-lvl hirrhy, with ntworks t th top lvl n hosts t th bottom lvl. This hirrhy is rflt in th ft tht n IP rss onsists of two prts, ntwork prt n host prt. Th ntwork prt intifis th ntwork to whih host is tth n thus ll hosts tth to th sm ntwork gr in th ntwork prt of thir IP rsss. Sin th ntwork prt orrspons to th first bits of th IP rss it is ll th rss prfix. W will writ prfixs s bit strings of up to 32 bits in IPv4 follow by *. For xmpl, th prfix * rprsnts ll th 2 6 rsss tht bgin with th bit pttrn. Altrntivly, prfixs n b init using th ott-iml nottion, so th sm prfix n b writtn s 3.86/6, whr th numbr ftr th slsh inits th lngth of th prfix. With two-lvl hirrhy, IP routrs forwr pkts bs only on th ntwork prt, until pkts rh th stintion ntwork. As rsult, forwring tbl only n to stor singl ntry to forwr pkts to ll th hosts tth to th sm ntwork. This thniqu is ll rss ggrgtion n llows using prfixs to rprsnt group of rsss. Eh ntry in forwring tbl ontins prfix, s n b sn in Tbl. So, fining th forwring informtion rquirs to srh for th prfix in th forwring tbl tht mths th orrsponing bits of th stintion rss. Dstintion Arss Nxt-hop Output Prfix intrf / / / / / Tbl : A forwring tbl Th rssing rhittur spifis how th llotion of rsss is prform, tht is it fins how to prtition th totl IP rss sp of 2 32 rsss. Spifilly, how mny ntwork rsss will b llow n of wht siz h of thm shoul b. Whn th Intrnt rssing ws initilly sign, rthr simpl rss llotion shm ws fin, whih is known toy s th lssful rssing shm. Bsilly, thr iffrnt sizs of ntworks wr fin in this shm, intifi by lss nm: lss A, B, n C (s figur ). Siz of ntworks ws trmin by th numbr of bits us to rprsnt th ntwork prt n th host prt. Thus ntworks of lss A, B or C onsist in n 8, 6 or 24-bit ntwork prt n orrsponing 24, 6 or 8-bit host prt. 2

3 Clss A 7 24 Ntwork Host Clss B 4 Ntwork 6 Host Clss C 2 8 Ntwork Host Figur : Clssful Arsss With this shm thr wr vry fw lss A ntworks n thir rssing sp rprsnt 5% of th totl IPv4 rss sp (2 3 rsss out of totl of 2 32 ). Thr wr 6,384 (2 4 ) lss B ntworks with mximum of 65,534 hosts pr ntwork n 2,97,52 (2 2 ) lss C ntworks with up to 256 hosts. This llotion shm work wll in th rly ys of th Intrnt. Howvr, th ontinuous growth of th numbr of hosts n ntworks hv m pprnt two problms with th lssful rssing rhittur. First, with only thr iffrnt ntwork sizs to hoos, th rss sp ws not us ffiintly n th IP rss sp ws gtting xhust vry rpily, vn though only smll frtion of th rsss llot wr tully in us. Son, lthough th stt informtion stor in th forwring tbls i not grow in proportion to th numbr of hosts, it still grw in proportion to th numbr of ntworks. This ws spilly importnt in th bkbon routrs, whih must mintin n ntry in th forwring tbl for vry llot ntwork rss. As rsult, th forwring tbls in th bkbon routrs wr growing vry rpily. Th growth of th forwring tbls rsult in highr lookup tims n highr mmory rquirmnts in th routrs n thrtn to impt thir forwring pity..2 Th CIDR Arssing Shm To llow for mor ffiint us of th IP rss sp n to slow own th growth of th bkbon forwring tbls, nw shm ll Clsslss Intr-omin Routing or CIDR ws introu. Rmmbr, tht in th lssful rss shm, only 3 iffrnt prfix lngths r llow: 8,6 n 24 orrsponing to th lsss A, B n C, rsptivly (s figur ). CIDR mks mor ffiint us of th IP rss sp by llowing finr grnulrity in th prfix lngths. With CIDR, prfixs n b of rbitrry lngth rthr thn onstrining thm to b 8, 6 or 24 bits long. To rss th problm of forwring tbl xplosion, CIDR llows rss ggrgtion t svrl lvls. Th i is tht th llotion of rsss hs topologil signifin. Thn, w n rursivly ggrgt rsss t vrious points within th hirrhy of th Intrnt s topology. As rsult, bkbon routrs mintin forwring informtion not t th ntwork lvl but t th lvl of rbitrry ggrgts of ntworks. Thus, rursiv rss ggrgtion rus th numbr of ntris in th forwring tbl of bkbon routrs. To unrstn how this works, onsir th ntworks rprsnt by th ntwork numbrs from /24 through /24 (s figurs 2 n 3). Suppos tht in routr ll ths ntwork rsss r rhbl through th sm srvi provir. From th binry rprsnttion w n s tht th lftmost 2 bits of ll th rsss in this rng r th sm ( ). Thus, w n ggrgt ths 6 ntworks into on suprntwork rprsnt by th 2-bit prfix, whih in iml nottion givs /2. Not tht initing th prfix lngth is nssry in iml nottion, bus th sm vlu my b ssoit to pr- 3

4 fixs of iffrnt lngths, for instn /2 ( *) is iffrnt from /22 ( *) /24 * /24 * /24 * /2 * Figur 2: Prfix ggrgtion / / /24 Totl IPv4 Arss Sp 2³²- Figur 3: Prfix Rngs Whil grt l of ggrgtion n b hiv if rsss r rfully ssign, in som situtions, fw ntworks n intrfr with th pross of ggrgtion. For xmpl, suppos now tht ustomr owing th ntwork /24 hngs its srvi provir n os not wnt to rnumbr its ntwork. Now, ll th ntworks from /24 through /24 n b rh through th sm srvi provir, xpt for th ntwork /24 (s figur 3). W nnot prform ggrgtion s bfor, n inst of only on ntry, 6 ntris n to b stor in th forwring tbl. On solution tht n b us in this sitution is ggrgting in spit of th xption ntworks n itionlly storing ntris for th xption ntworks. In our xmpl, this will rsult in only two ntris in th forwring tbl: /2 n /24, s figur 4 n tbl. Not howvr, tht now som rsss will mth both ntris bus prfixs ovrlp. In orr to lwys mk th orrt forwring ision, routrs n to o mor thn to srh for prfix tht mths. Sin xptions in th ggrgtions my xist, routr must fin th most spifi mth, n th most spifi mth is th longst mthing prfix. In summry, th rss lookup problm in routrs rquirs to srh th forwring tbl for th longst prfix tht mths th stintion rss of pkt / /2 Totl IPv4 Arss Sp 2³²- Ths rsss mth both prfixs Figur 4: Exption prfix 4

5 .3 Diffiulty of th Longst Mthing Prfix Srh In th lssful rssing rhittur, th lngth of th prfixs ws o in th most signifint bits of n IP rss (s figur ), n th rss lookup ws rltivly simpl oprtion: Prfixs in th forwring tbl wr orgniz in thr sprt tbls, on for h of th thr llow lngths. Th lookup oprtion mount to fin n xt prfix mth in th pproprit tbl. Th srh for n xt mth oul b prform using stnr lgorithms bs on hshing or binry srh. Whil CIDR llows to ru th siz of th forwring tbls, th rss lookup problm now boms mor omplx. With CIDR, th stintion prfixs in th forwring tbls hv rbitrry lngths n o not orrspon ny mor to th ntwork prt sin thy r th rsult of n rbitrry numbr of ntwork ggrgtions. Thrfor, whn using CIDR, th srh in forwring tbl nnot b prform ny longr by xt mthing bus th lngth of th prfix nnot b riv from th rss itslf. As rsult, trmining th longst mthing prfix involvs not only to ompr th bit pttrn itslf but lso to fin th pproprit lngth. Thrfor, w tlk bout srhing in two imnsions, th vlu imnsion n th lngth imnsion. Th srh mthos w will rviw try to ru th srh sp t h stp in both of ths imnsions. In wht follows w will us N to not th numbr of prfixs in forwring tbl n W to init th mximum lngth of prfixs, whih is typilly lso th lngth of th IP rsss..4 Rquirmnts on Arss Lookup Algorithms It is importnt to brifly rsum th hrtristis of th toy s routing nvironmnt to riv th qut rquirmnts n mtris for th rss lookup lgorithms tht w will survy. As w hv sn, using rss prfixs is simpl mtho to rprsnt groups of ontiguous rsss. Arss prfixs llow ggrgtion of forwring informtion n hn support th growth of th Intrnt. Figur 5 shows th growth of typil bkbon routr tbl. W n obsrv thr phss of tbl growth: Bfor th introution of CIDR growth ws xponntil (prtly visibl in rly 994). From th mi of 994 to th mi of 998, growth slow own n is nrly linr. From th mi of 998 up to now growth is gin xponntil. Sin th numbr of ntris in routr tbls still grows, it is importnt tht srh mthos ru rstilly th srh sp t h stp. Algorithms must b slbl with rspt to th numbr of prfixs. Anothr hrtristi of th routing nvironmnt is tht forwring tbl ns to b upt ynmilly to rflt rout hngs. In ft, instbilitis in th bkbon routing protools n hng firly frquntly th ntris in forwring tbl. Lbovitz [8] foun tht bkbon routrs my riv bursts of rout hngs t rts xing svrl hunr prfix upts pr son. H lso foun tht, in vrg, rout hngs our on hunr tims pr son. Thus, upt oprtions must b prform in ms or lss. Th prfix lngth istribution in th forwring tbls n b us s mtri of th qulity of th Intrnt hirrhy n rss ggrgtion. Shortr prfixs rprsnt grtr gr of ggrgtion. Thus, rs in th vrg prfix lngth woul init improv ggrgtion n hirrhy in th Intrnt. In figur 6 w n s tht th historil lss C with its 24-bit prfix lngth still omints th numbr of ntris in th forwring tbl (not tht sl is logrithmi). A rnt stuy shows tht th numbr of xptions in th rss ggrgtion is growing. Mor prisly, Huston [7] foun tht urrntly 4 % of th ntris of typil bkbon forwring tbl r prfix xptions. 5

6 Tlstr Routr 2 Dmbr, 2 8 Numbr of ntris Jn-94 Jn-95 Jn-96 Jn-97 Jn-98 Jn-99 Jn- Jn- Figur 5: Tbl growth of typil bkbon routr Tlstr Routr 2 Dmbr, 2 Numbr of ntris (log sl) Prfix lngth Figur 6: Prfix lngth istribution of typil bkbon routr 6

7 2 Clssil Solution 2. Binry Tri A nturl wy to rprsnt prfixs is using tri. A tri is tr-bs t strutur llowing th orgniztion of prfixs on igitl bsis by using th bits of prfixs to irt th brnhing. Figur 7 shows binry tri (h no hs t most two hilrn) rprsnting st of prfixs of forwring tbl. Prfixs * b * * * * f * g * h * i * f g h i b Figur 7: Binry tri for st of prfixs. In tri, no on lvl l rprsnts th st of ll rsss tht bgin with th squn of l bits onsisting of th string of bits lbling th pth from th root to tht no. For xmpl, no in figur 7 is t lvl 3 n rprsnts ll rsss bginning with th squn. Th nos tht orrspon to prfixs r shown in rk olor n ths nos will ontin th forwring informtion or pointr to th forwring informtion. Not lso tht prfixs r not only lot t lvs but lso t som intrnl nos. This sitution riss bus of xptions in th ggrgtion pross. For xmpl, in figur 7 th prfixs b n rprsnt xptions to prfix. Figur 8 illustrts this sitution bttr. Th tri shows th totl rss sp, ssuming 5-bit long rsss. Eh lf rprsnts on possibl rss. W n s tht rss sps ovr by prfixs b n ovrlp with th rss sp ovr by prfix. Thus, prfixs b n rprsnt xptions to prfix n rfr to spifi subintrvls of th rss intrvl ovr by prfix. In th tri in figur 7, this is rflt by th ft tht prfixs b n r snnts of prfix, or in othr wors, prfix is itslf prfix ofb n. As rsult, som rsss will mth svrl prfixs. For xmpl, rsss bginning with will mth both, prfix n prfix. Nvrthlss, prfix must b prfrr bus it is mor spifi (longst mth rul). Tris llow in strightforwr wy to fin th longst prfix tht mths givn stintion rss. Th srh in tri is gui by th bits of th stintion rss. At h no, th srh pros to th lft or to th right oring to th squntil insption of th rss bits. Whil trvrsing th tri, vry tim w visit no mrk s prfix (i.., rk no) w rmmbr this prfix s th longst mth foun so fr. Th srh ns, whn thr is no mor brnh to tk n th longst or bst mthing prfix will b th lst prfix rmmbr. For instn, if w srh th bst mthing prfix (BMP) for n rss bginning with th bit pttrn w strt t th root in figur 7. Sin th first bit of th rss is w mov to th right, to th no mrk with prfix n w rmmbr s th BMP foun so fr. Thn w mov to th lft sin th son rss bit is, this tim th no is not mrk s prfix, so is still th BMP foun so fr. Nxt th thir rss bit is but t this point thr is no brnh lbl, so srh ns n th lst rmmbr BMP 7

8 Prfixs * b * * * * f * g * h * i * f g h i b f f g g h h i i Figur 8: Arss sp (prfix ) is th longst mthing prfix. In ft, wht w r oing is squntil prfix srh by lngth, trying t h stp to fin bttr mth. W bgin by looking in th st of lngth- prfixs, whih r lot t th first lvl in th tri, thn in th st of lngth-2, lot t th son lvl, n so on. Morovr, using tri hs th vntg tht whil stpping through th tri, th srh sp is ru hirrhilly. At h stp, th st of potntil prfixs is ru n srh ns whn this st is ru to on. Upt oprtions r lso strightforwr to implmnt in binry tris. Insrting prfix bgins by oing srh. Whn rriving t no with no brnh to tk, w n insrt th nssry nos. Dlting prfix strts gin by srh, unmrking th no s prfix n, if nssry lting unus nos, i.. lv nos not mrk s prfixs. Not finlly tht sin th bit strings of prfixs r rprsnt by th strutur of th tri, th nos mrk s prfixs o not n to stor th bit strings thmslvs. 2.2 Pth-omprss Tris Whil binry tris llow th rprsnttion of rbitrry lngth prfixs thy hv th hrtristi tht long squns of on-hil nos my xist (s prfix b in figur 7). Sin ths bits n to b inspt, vn though no tul brnhing ision is m, srh tim n b longr thn nssry for som ss. Also, on-hil nos onsum itionl mmory. In n ttmpt to improv tim n sp prformn, thniqu ll pth-omprssion n b us. Pth-omprssion onsists in ollpsing on-wy brnh nos. Whn on-wy brnh nos r rmov from tri, itionl informtion must b kpt in rmining nos, so tht srh oprtion n b prform orrtly. Thr r mny wys to xploit th pth-omprssion thniqu; prhps th simplst to xplin is illustrt in figur 9, orrsponing to th binry tri in figur 7. Not tht th two nos pring b now hv bn rmov. Not lso tht sin prfix ws lot t on-hil no, it hs bn mov to th nrst snnt not bing on-hil no. Sin in pth to b omprss svrl on-hil nos my ontin prfixs, in gnrl, list of prfixs must b mintin in som of th nos. Bus on-wy brnh nos r now rmov, w n jump irtly to th bit whr signifint ision is to b m, bypssing th bit insption of som bits. As rsult, bit numbr fil must b kpt now to init whih bit is th nxt bit to inspt. In figur 9 ths bit numbrs r shown nxt to th nos. Morovr, th bit strings of prfixs must b xpliitly stor. A srh in this kin of pth-omprss tris is s follows: Th lgorithm prforms, s usul, snt in th tri unr th guin of th rss bits; but this tim, only inspting bit positions init 8

9 by th bit-numbr fil in th nos trvrs. Whn no mrk s prfix is nountr, omprison with th tul prfix vlu is prform. This is nssry sin uring th snt in th tri w my skip som bits. If mth is foun, w pro trvrsing th tri n kp th prfix s th BMP so fr. Srh ns whn lf is nountr or mismth is foun. As usul th BMP will b th lst mthing prfix nountr. For instn, if w look for th BMP of n rss bginning with th bit pttrn in th pth omprss tri shown in figur 9, w pro s follows: W strt t th root no n sin its bit numbr is w inspt th first bit of th rss. Th first bit is so w go to th lft. Sin th no is mrk s prfix w ompr th prfix with th orrsponing prt of th rss (). Sin thy mth w pro n kp s th BMP so fr. Sin th no s bit numbr is 3 w skip th son bit of th rss n inspt th thir on. This bit is so w go to th lft. Agin w hk whthr th prfix b mths th orrsponing prt of th rss (). Sin thy o not mth, srh stops n th lst rmmbr BMP (prfix ) is th orrt BMP. Pth-omprssion ws first propos in shm ll PATRICIA [], but this shm os not support longst prfix mthing. Sklowr propos shm with moifitions for longst prfix mthing in [3]. In ft, this vrint ws originlly sign not only to support prfixs but mor gnrl non-ontiguous msks. Sin this ftur ws rlly nvr us, urrnt implmnttions iffr somhow from th Sklowr s originl shm. For xmpl, th BSD vrsion of th pth-omprss tri (rfrr to s BSD tri) is ssntilly th sm s w hv just srib. Th bsi iffrn is tht in th BSD shm, th tri is first trvrs without hking th prfixs t intrnl nos. On t lf, th trvrs pth is bktrk in srh of th longst mthing prfix. At h no with prfix, or list of prfixs, omprison is prform to hk for mth. Srh ns whn mth is foun. Comprison oprtions r not m on th ownwr pth in th hop tht not mny xption prfixs xist. Not tht with this shm, in th worst s, th pth is ompltly trvrs two tims. In th s of th originl Sklowr s shm th bktrk phs lso ns to o rursiv snts of th tri bus non-ontiguous msks r llow. Prfixs * b * * * * f * g * h * i * b f g h i Figur 9: A pth-omprss tri Until rntly, th longst mthing prfix problm hs bn rss by using t struturs bs on pth-omprss tris, lik th BSD tri. Pth-omprssion mks muh sns whn th binry tri is sprsly popult. But whn th numbr of prfixs inrss n th tri gts nsr, using pth omprssion hs littl bnfit. Morovr, th prinipl isvntg of pth-omprss tris, s wll s binry tris in gnrl, is tht srh ns to o mny mmory sss, in th worst s 32 for IPv4 rsss. For xmpl, for typil bkbon routr [8] with 473 prfixs, th BSD vrsion for pth-omprss tri rts 9334 nos. Th mximl hight is 26, whil th vrg hight is lmost 2. For th sm prfixs, simpl binry tri (with on-hil nos) hs mximl hight of 3 n n vrg hight of lmost 22. As w n s, th hights of both tris r vry similr n th BSD tri my prform itionl omprison oprtions whn bktrking is n. 9

10 3 Nw IP Lookup Algorithms W hv sn tht th iffiulty with th longst prfix mthing oprtion is its ul imnsion: lngth n vlu. Th nw shms for fst IP lookups iffr in th imnsion to srh n whthr this srh is linr or binry srh. In th following, w prsnt lssifition of th iffrnt IP lookup shms. 3. Txonomy of IP Lookup Algorithms Srh on vlus pprohs: Squntil srh on vlus is th simplst mtho to fin th BMP. Th t strutur n is just n rry with unorr prfixs. Th srh lgorithm is vry simpl. It gos through ll th ntris ompring th prfix with th orrsponing bits of givn rss. Whn mth is foun, w kp th longst mth so fr n ontinu. At th n, th lst prfix rmmbr is th BMP. Th problm with this pproh is tht th srh sp is ru only by on prfix t h stp. Clrly th srh omplxity in tim for this shm is funtion of th numbr of prfixs O(N), n hn th shm is not slbl. With th srh on vlu pproh, w gt ri of th lngth imnsion bus of th xhustiv srh. It is lr tht binry srh on vlus woul b bttr, n w will s in stion 6 how this n b on. Srh on lngths pprohs: Anothr possibility is to bs th srh on th lngth imnsion n to us linr srh or binry srh. Two possibl wys of orgnizing th prfixs for srh on lngths xist. In ft w hv lry sn linr srh on lngths, whih is prform on tri. Tris llow t stp i to hk th prfixs of lngth i. Morovr, prfixs in tri r orgniz in suh wy tht stpping through th tri rus th st of possibl prfixs. As w will s in stion 4, on optimiztion to this shm onsists in using multibit tris. Multibit tris still o linr srh on lngths, but inspt svrl bits simultnously t h stp. Th othr possibl wy of orgnizing th prfixs tht llows srh on lngths is to us iffrnt tbl for h possibl lngth. Thn, linr srh on lngths n b m by oing t h stp srh on prtiulr tbl using hshing, for instn. W will s in stion 5 how Wlvogl t l. [7] us hsh tbls to o binry srh on lngths. In ition to th lgorithm-t strutur spt, vrious pprohs us iffrnt thniqus suh s trnsformtion of th prfix st, omprssion of runnt informtion to ru th mmory rquirmnts, pplition of optimiztion thniqus, n xploittion of th mmory hirrhy in omputrs. W introu h of ths spts brifly in th following substion n thn isuss th nw lookup shms in til oring to th lgorithm-t strutur spt in th nxt stions. 3.2 Auxiliry Thniqus Prfix trnsformtion: Forwring informtion is spifi with prfixs tht rprsnt rngs of rsss. Although th st of prfixs to us is usully trmin by th informtion gthr by th routing protools, th sm forwring informtion n b xprss with iffrnt sts of prfixs. Vrious trnsformtions r possibl oring to spil ns, but on of th most ommon prfix trnsformtion thniqus is prfix xpnsion. Expning prfix mns trnsforming on prfix into svrl longr n mor spifi prfixs tht ovr th sm rng of rsss. As n xmpl, th rng of rsss ovr by prfix * n lso b spifi with th two prfixs *, *; or lso, with th four prfixs: *, *, *, *. Ifwo prfix xpnsion ppropritly, w n gt st of prfixs tht hs fwr iffrnt lngths, whih n b us to mk fstr srh, s w will show ltr.

11 W hv sn tht prfixs n ovrlp (s figur 4). In tri, whn two prfixs ovrlp, on of thm is itslf prfix of th othr, s figurs 7 n 8. Sin prfixs rprsnt intrvls of ontiguous rsss, whn two prfixs ovrlp this mns tht on intrvl of rsss ontins nothr intrvl of rsss, s figur 4 n 8. In ft, tht is why n rss n b mth to svrl prfixs. If svrl prfixs mth, th longst prfix mth rul is us in orr to fin th most spifi forwring informtion. On wy to voi th us of th longst prfix mth rul n to still fin th most spifi forwring informtion is to trnsform givn st of prfixs into st of isjoint prfixs. Disjoint prfixs o not ovrlp n thus no rss prfix is itslf prfix of nothr on. A tri rprsnting st of isjoint prfixs will hv prfixs t th lvs but not t intrnl nos. To obtin isjoint-prfix binry tri, w simply lvs to nos tht hv only on hil. Ths nw lvs r nw prfixs tht inhrit th forwring informtion of th losst nstor mrk s prfix. Finlly, intrnl nos mrk s prfixs r unmrk. For xmpl, figur shows th isjoint-prfix binry tri tht orrspons to th tri in figur 7. Prfixs, 2, 3 hv inhrit th forwring informtion of th originl prfix, whih now hs bn supprss. Prfix hs bn obtin in similr wy. Sin prfixs t intrnl nos r xpn or push own to th lvs of th tri, this thniqu hs bn ll lf pushing by Srinivsn t l. [4]. Figur shows th isjoint intrvls of rsss tht orrspon to th isjoint-prfix binry tri of figur. Prfixs * b * * * * f * g * h * i * ¹ ¹ b ² ³ f g h i Figur : Disjoint-prfix binry tri Prfixs * b * * * * f * g * h * i * ¹ ¹ ³ f g h i ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ b ² ³ ³ ¹ ¹ ¹ ¹ f f g g h h i i Disjoint intrvls of rsss Figur : Expn isjoint-prfix binry tri Comprssion thniqus: Dt omprssion tris to rmov runny from th noing. Th i to us omprssion oms from th ft tht xpning th prfixs inrss informtion runny. Comprssion

12 shoul b on in suh wy tht mmory onsumption is rs n tht rtriving th informtion from th omprss strutur n b on sily n with minimum numbr of mmory sss. Run-lngth noing is vry simpl omprssion thniqu tht rpls onsutiv ourrns of givn symbol with only on ourrn plus ount of how mny tims tht symbol ours. This thniqu is wll pt to our problm bus prfixs rprsnt intrvls of ontiguous rsss tht hv th sm forwring informtion. Applition of optimiztion thniqus: Thr is mor thn on wy to trnsform th st of prfixs. Optimiztion llows to fin som onstrints n to fin th right st of prfixs stisfying thos onstrints. Normlly w wnt to minimiz th mount of mmory onsum. Mmory hirrhy in omputrs: On of th hrtristis of toy s omputrs is th iffrn in sp btwn prossor n mmory n lso btwn mmoris of iffrnt hirrhis (h, RAM, isk). Rtriving informtion from mmory is xpnsiv, so smll t struturs r sirbl bus thy mk it mor likly tht th forwring tbl fits into th fstr h mmory. Furthrmor, th numbr of mmory sss must b minimiz to mk srh fstr. Nw lgorithms to th longst prfix mthing problm us on or svrl of th spts just outlin. W will survy th iffrnt lgorithms by lssifying thm oring to th lgorithm-t strutur spt n w will isuss othr spts s wll. It is worth to mntion tht orgnizing th prfixs in iffrnt wys llows for iffrnt troffs btwn th srh ost n th upt ost, s wll s mmory onsumption. W isuss ths troffs whn w xplin th iffrnt shms. W now prsnt in til som of th most ffiint lgorithms for IP rss lookup. 4 Srh on Prfix Lngths using Multibit Tris 4. Bsi Shm Binry tris provi n sy wy to hnl rbitrry lngth prfixs. Lookup n upt oprtions r strightforwr. Nvrthlss, th srh in binry tri n b rthr slow bus w inspt on bit t tim n in th worst s 32 mmory sss r n for n IPv4 rss. On wy to spup th srh oprtion is to inspt not just on bit tim but svrl bits simultnously. For instn, if w inspt 4 bits t tim w woul n only 8 mmory sss in th worst s for n IPv4 rss. Th numbr of bits to b inspt pr stp is ll stri n n b onstnt or vribl. A tri strutur tht llows th insption of bits in stris of svrl bits is ll multibit tri. Thus, multibit tri is tri whr h no hs 2 k hilrn, whr k is th stri. Sin multibit tris llow to trvrs th t strutur in stris of svrl bits t tim, thy nnot support rbitrry prfix lngths. To us givn multibit tri, th prfix st must b trnsform into n quivlnt st with th prfix lngths llow by th nw strutur. For instn, multibit tri orrsponing to our xmpl from figur 7 is shown in figur 2. W s tht first stri of two bits is us, so prfixs of lngth on r not llow, n w n to xpn prfixs n to prou four quivlnt prfixs of lngth two. In th sm figur it is shown how prfix hs bn xpn to lngth 4. Not tht th hight of th tri hs rs n so th numbr of mmory sss whn oing srh. Figur 3 shows iffrnt multibit tri for our xmpl. W n s gin tht prfixs n hv bn xpn but now to lngth thr. Howvr, two of th prfixs prou by xpnsion lry xist (prfixs n ). W must prsrv th forwring informtion of prfixs n sin thir forwring informtion is mor spifi thn th on of th xpn prfix. Thus, xpnsion of prfixs n finlly rsults in six prfixs n not ight. In gnrl, whn n 2

13 xpn prfix ollis with n xisting longr prfix, forwring informtion of th xisting prfix must b prsrv to rspt th longst mthing rul. Prfixs * b * * * * f * g * h * i * f g h i b Figur 2: A vribl stri multibit tri Prfixs * b * * * * f * g * h * i * b f f g g h h i i Figur 3: A fix stri multibit tri Srhing in multibit tri is ssntilly th sm s in binry tri. To fin th BMP of givn rss onsists in sussivly looking for longr prfixs tht mth. Th multibit tri is trvrs n h tim prfix is foun t no, it is rmmbr s th nw BMP sn so fr. At th n, th lst BMP foun is th orrt BMP for th givn rss. Multibit tris still o linr srh on lngths s o binry tris, but th srh is fstr bus th tri is trvrs using lrgr stris. In multibit tri, if ll nos t th sm lvl hv th sm stri siz w sy tht it is fix stri, othrwis it is vribl stri. W n hoos multibit tris with fix stris or vribl stris. Fix stris r simplr to implmnt thn vribl stris but in gnrl wst mor mmory. Figur 3 is n xmpl of fix stri multibit tri, whil figur 2 shows vribl stri multibit tri. 4.2 Choi of Stris Choosing th stris rquirs to mk troff btwn srh sp n mmory onsumption. In th xtrm s, w oul mk tri with singl lvl, tht is on-lvl tri with 32 bit stri for IPv4. Srh woul tk in this s just on ss but w woul n hug mount of mmory to stor 2 32 ntris. 3

14 On nturl wy to hoos stris n ontrol th mmory onsumption is to lt th strutur of th binry tri trmin this hoi. For xmpl, if w look t figur 7, w n obsrv tht th subtri hving s root th right hil of no is full subtri of two lvls ( full binry subtri is subtri whr h lvl hs th mximum numbr of nos). W n rpl this full binry subtri with on-lvl multibit subtri. Th stri of th multibit subtri is simply th numbr of lvls of th substitut full binry subtri, two in our xmpl. In ft, this trnsformtion hs bn lry m in figur 2. This trnsformtion is strightforwr, but s it is th only trnsformtion w n o in figur 7, it hs limit bnfit. W will s ltr how to rpl, in ontroll wy, binry subtris tht r not nssry full subtris. Hight of th multibit tri will b ru whil ontrolling mmory onsumption. W will s lso, how optimiztion thniqus n b us to hoos th stris. 4.3 Upting Multibit Tris Siz of stris lso trmins upt tim bouns. A multibit tri n b viw s tr of on-lvl subtris. For instn, in figur 3 w hv on subtri t th first lvl n thr subtris t th son lvl. Whn w o prfix xpnsion in subtri, wht w tully o is omput for h no of th subtri its lol BMP. Th BMP is lol bus it is omput from subst of th totl of prfixs. For instn, in th subtri t th first lvl w r only onrn to fin for h no th BMP mong th prfixs,,,. In th lftmost subtri t th son lvl th BMP for h no will b slt from th only prfix b. In th son subtri t th son lvl, th BMP is slt for h no mong th prfixs f,g, n th rightmost subtri is onrn only with prfixs h,i. Som nos my b mpty initing tht thr r no BMP for ths nos, mong th prfixs orrsponing to this subtri. As rsult, multibit tris ivi th problm of fining th BMP into smll problms in whih lol BMPs r slt mong subst of prfixs. Hn, whn looking for th BMP of givn rss w trvrs th tr n rmmbr th lst lol BMP s w go through it. It is worth to not tht th BMPs omput t h subtri r inpnnt of th BMPs omput t othr subtris. Th vntg of this shm is tht insrting or lting prfix ns only to upt on of th subtris. Prfix upt is ompltly lol. In prtiulr if th prfix is or will b stor in subtri with stri of k bits, th upt ns to moify t most 2 k; nos ( prfix popults t most th hlf of th nos in subtri). Thus, hoosing pproprit stri vlus llows to boun th upt tim. Lol BMPs llow inrmntl upts but rquir tht intrnl nos, bsis lvs, stor prfixs n thus mmory onsumption is inrmnt. As w know, w n voi prfixs t intrnl nos if w us st of isjoint prfixs. W n obtin multibit tri with isjoint prfixs if w xpn prfixs t intrnl nos of th multibit tri own to its lvs (lf pushing). Figur 4 shows th rsult of this pross whn ppli to th multibit tri in figur 3. Nvrthlss not tht now, in th gnrl s, prfix n b thortilly xpn to svrl subtris t ll lvls. Clrly with this pproh, th BMPs omput t h subtri r not ny mor lol n thus upts will suffr of longr worst s tims. As w n s, multibit tri with svrl lvls llows, by vrying th stri k, n intrsting troff btwn srh tim, mmory onsumption, n upt tim. Th lngth of th pth n b ontroll to ru th srh tim. Choosing lrgr stris will mk fstr srhs but mor mmory will b n n upts will rquir to moify mor ntris bus of xpnsion. As w hv sn inrmntl upts r possibl with multibit tris, if w o not us lf pushing. Howvr, insrting n lting oprtions r slightly mor omplit thn with binry tris bus of th prfix trnsformtion. Insrting on prfix mns fining th pproprit subtri, oing n xpnsion n insrting h of th rsulting prfixs. Dlting is still mor omplit bus it mns lting th xpn prfixs 4

15 Prfixs * b * * * * f * g * h * i * b f f g g h h i i Figur 4: A isjoint-prfix multibit tri n mor importntly upting th ntris with th nxt bst mthing prfix. Th problm is tht originl prfixs r not tully stor in th tri. To s this bttr, suppos w insrt prfixs *, * n * in th multibit tri in figur 3. Clrly prfix will isppr n if ltr w lt prfix *, for instn, thr will b no wy to fin th nw BMP () for no. Thus, upt oprtions n n itionl strutur for mnging originl prfixs. 4.4 Multibit Tris in Hrwr Th bsi shm of Gupt t l. [6] uss 2 lvl multibit tri with fix stris similr to th on shown in figur 4. Howvr, th first lvl orrspons to stri of 24 bits n th son lvl to stri of 8 bits. On ky obsrvtion in this shm is tht in typil bkbon routr, most of th ntris hv prfixs with lngth 24-bits or lss (s figur 6 with logrithmi sl on th y xis). As rsult, using first stri of 24 bits llows to fin th BMP in on mmory ss for th mjority of th ss. Also sin fw prfixshv lngth longr thn 24, thr will b only smll numbr of subtris t th son lvl. In orr to sv mmory, intrnl nos r not llow to stor prfixs. Hn, shoul prfix orrspon to n intrnl no it will b xpn to th son lvl (lf pushing). This pross rsults in multibit tri with isjoint xpn prfixs similr to th on illustrt in figur 4, for th xmpl in figur 3. Th first lvl of th multibit tri hs 2 24 nos n is implmnt s tbl with th sm numbr of ntris. An ntry in th first lvl ontins ithr th forwring informtion or pointr to th orrsponing subtri t th son lvl. Entris in th first tbl n two byts to stor pointr hn mmory bnk of 32 Mbyts is us to stor 2 24 ntris. Atully th pointrs us 5 bits bus th first bit of n ntry inits if th informtion stor is th forwring informtion or pointr to son lvl subtri. Th numbr of subtris t th son lvl pns on th numbr of prfixs longr thn 24 bits. In th worst s h of ths prfixs will n iffrnt subtri t th son lvl. Sin th stri for th son lvl is 8 bits, subtri t th son lvl hs 2 8 =256 lvs. Th son lvl subtris r stor in son mmory bnk. Th siz for this son mmory bnk pns on th xpt worst s prfix lngth istribution. In th MEst tbl [8]w xmin in August 6 999, only 96 prfixs wr longr thn 24 bits. For xmpl for mmory bnk of 2 2 ntris of byt h, tht is mmory bnk of Mbyt, th sign supports mximum of 2 2 =496 subtris t th son lvl. In figur 5 w n s how th oing of stintion rss is on to fin th orrsponing forwring informtion. Th first 24 bits of th stintion rss r us to inx into th first mmory bnk (first lvl of th multibit tri). If th first bit of th ntry is, th ntry ontins th forwring informtion, othrwis th forwring informtion must b look up in th son mmory bnk (son lvl of th multibit 5

16 tri). In tht s, w ontnt th lst 8 bits of th stintion rss with th pointr just foun in th first tbl. Th rsult is us s n inx to lookup in th son mmory bnk th forwring informtion. First Mmory bnk Dstintion rss Forwring informtion / ntris Son Mmory bnk 23 3 / ntris Figur 5: Gupt t l. hrwr shm Th vntg of this simpl shm is tht th lookup rquirs mximum of two mmory sss. Morovr, sin it is hrwr pproh, th mmory sss n b piplin or prllliz. As rsult th lookup oprtion tks prtilly on mmory ss tim. Nvrthlss, sin th first stri is of 24 bits n lf pushing is us, upts my tk long tim for som ss. 4.5 Multibit tris with Pth-omprssion Thniqu Nilsson t l. [] rursivly trnsform binry tri with prfixs into multibit tri: Strting t th root, w rpl th lrgst full binry subtri with orrsponing on-lvl multibit subtri. This pross is rpt rursivly with th hilrn of th multibit subtri obtin. Aitionlly, on-hil pths r omprss. Sin w rpl t h stp binry subtri of svrl lvls with multibit tri of on lvl, th pross n b viw s omprssion of th lvls of th originl binry tri. LC (lvl-omprss) tri is th nm givn by Nilsson to ths multibit tris. Nvrthlss, ltting th strutur of th binry tri stritly trmin th hoi of stris os not llow to ontrol th hight of th rsulting multibit tri. On wy to furthr ru th hight of th multibit tri, is to lt th strutur of th tri only gui n not trmin th hoi of stris. In othr wors, w will rpl nrly full binry subtris with multibit subtri, tht is binry subtris whr only fw nos r missing. Nilsson proposs to rpl nrly full binry subtri with multibit subtri of stri k if th nrly full binry subtri hs suffiint frtion of th 2 k nos t lvl k, whr suffiint frtion of nos is fin by using singl prmtr ll fill ftor x, with < x. For instn, in figur 7, if th fill ftor is.5, th frtion of nos t th fourth lvl is not nough to hoos stri of 4. Sin only 5 of th 6 possibl nos, r prsnt. Inst, thr r nough nos t th thir lvl (5 of th 8 possibl nos) for multibit subtri of stri 3. In orr to sv mmory sp, ll th nos of th LC tri r stor in singl rry. First th root, thn ll th nos t th son lvl, thn nos t thir lvl, t. Morovr, intrnl nos r not llow to stor prfixs. Inst, h lf hs linr list with prfixs, in s th pth to th lf shoul hv on or svrl prfixs (lss spifi prfixs). As rsult, srh in n LC tri pros s follows: Th LC tri is trvrs lik in th bsi multibit tri. Nvrthlss, sin pth omprssion is us, xpliit omprison must b prform whn rriving t lf. In s of mismth, srh in th list of prfixs must b prform (lss spifi prfixs, i.. prfixs in intrnl nos in th originl binry tri). 6

17 Sin th LC tri is implmnt using singl rry of onsutiv mmory lotions n list of prfixs must b mintin t lvs, inrmntl upts r vry iffiult. 4.6 Multibit Tris n Optimiztion Thniqus On sy wy to boun worst-s srh tims is by fining fix stris tht yil wll fin hight for th multibit tri. Nvrthlss th problm is tht in gnrl, mmory onsumption will b lrg, s stion 4.4. On th othr hn, w n minimiz th mmory onsumption by ltting th prfix istribution stritly trmin th hoi of stris. Unfortuntly, th hight of th rsulting multibit tri nnot b ontroll n pns xlusivly on th spifi prfix istribution. W sw in th lst stion tht Nilsson uss th fill ftor s prmtr to ontrol th influn of th prfix istribution in th stri hoi n so influns somhow th hight of th rsulting multibit tri. Sin th prfix istribution still guis th stri hoi, mmory onsumption is still ontroll. Nvrthlss, th us of th fill ftor is simply rsonbl huristi n mor importntly it os not llow to gurnt worst-s hight. Srinivsn t l. [4] us ynmi progrmming to trmin, for givn prfix istribution, th optiml stris tht minimiz th mmory onsumption n gurnt worst-s numbr of mmory sss. Th uthors giv mtho to fin th optiml stris for th two typs of multibit tris: fix stri n vribl stri. Anothr wy of minimiz th lookup tim is by tking into ount, on on hn th hirrhil strutur of th mmory in systm, n on th othr th probbility istribution of th usg of prfixs (whih is trffi pnnt). Chung t l. [] giv mthos to minimiz th vrg lookup tim pr prfix for this s. Thy suppos systm hving thr typs of hirrhil mmoris with iffrnt ss tims n sizs. Using optimiztion thniqus mks sns if th ntris of th forwring tbl o not hng t ll or hng vry littl, but this is rrly th s for bkbon routrs. Insrting n lting prfixs grs th improvmnt u to optimiztion n rbuiling th strutur my b nssry. 4.7 Multibit Tris n Comprssion Doing xpnsion rts svrl prfixs tht ll inhrit th forwring informtion of th originl prfix. Thus, if w us multibit tris with lrg stris, w will hv grt numbr of ontiguous nos with th sm BMP. W n us this ft n omprss th runnt informtion, whih will llow to sv mmory n to mk th srh oprtion fstr bus of th smll hight of th tri. On xmpl of this pproh is th Full xpnsion/comprssion shm propos by Crsnzi t l. [2]. W will illustrt thir mtho with smll xmpl whr w o mximl xpnsion supposing 5-bit rsss n using two lvl multibit tri. Th first lvl uss stri of 2 bits n th son lvl stri of 3 bits, s it is shown in figur 6. Th i is to omprss h of th subtris t th son lvl. In figur 7 w n s how th lvs of h son lvl subtris hv bn pl in vrtil fshion. Eh olumn orrspons to on of th son lvl subtris. Th gol is to omprss th rpt ourrns of th BMPs. Nvrthlss, th omprssion is on in suh wy tht t h stp th numbr of omprss symbols is th sm for h olumn. With this strtgy th omprssion is not optiml for ll olumns but sin th omprssion is m in synhroniz wy for ll th olumns, ssing ny of th omprss subtris n b m with on ommon itionl tbl of pointrs, s it is shown in figur 7. To fin th BMP of givn rss w trvrs th first lvl of th multibit tri s usul, tht is th first two bits of th rss r us to 7

18 hoos th orrt subtri t th son lvl. Thn, th lst thr bits of th rss r us to fin th pointr in th itionl tbl. With this pointr w n rily fin th BMP in th omprss subtri. For xmpl, srhing for th rss will gui us to th thir subtri (olumn) in th omprss strutur n using th pointr ontin in th ntry of th itionl tbl, w will fin s th bst mthing prfix. In th tul shm propos by Crsnzi prfixs r xpn to 32 bits. A multibit tri of two lvls is lso us but th stri of th first lvl n son lvl is 6 bits. It is worth to not tht vn though omprssion is on, th rsulting strutur is not smll nough to fit in th h mmory. Nvrthlss, bus of th wy to ss th informtion, srh tks lwys only thr mmory sss. Th rport mmory siz for typil bkbon routr tbl is.2 Mbyts. Prfixs * b * * * * f * g * h * i * b f f g g h h i i Figur 6: A two lvl full xpn multibit tri Prfixs * b * * * * f * g * h * i * b f f g h i Comprss tri lvs b f f g g h h i i Aitionl tbl Tri lvs Figur 7: Full xpnsion prlll omprssion shm Anothr shm tht ombins multibit tris with th omprssion i hs bn ubb th Lul lgorithm [3]. In this shm, multibit tri with fix stri lngths is us. Th stris r 6,8,8, for th first, son n thir lvl rsptivly, whih givs tri of hight 3. In orr to o n ffiint omprssion, th Lul shm must us st of isjoint prfixs. Hn th Lul shm first trnsforms th st of prfixs into isjoint-prfix st. Thn, th prfixs r xpn in orr to mt th stri onstrints of th multibit tri. Aitionlly, in orr to sv mmory, prfixs r not llow t intrnl nos of th multibit tri n thus lf pushing is us. Agin, th i is to omprss th prfix informtion in th subtris by supprssing th rpt ourrns of onsutiv BMPs. Nvrthlss, ontrry to th lst shm, h subtri is omprss inpnntly of th othrs. On subtri is omprss, lvr oing mhnism llows th ss to th bst mthing prfixs. Du to lk of sp w o not giv hr th tils of th oing mhnism. 8

19 Whil th tri hight in th Lul shm is 3, tully mor thn 3 mmory rfrns r n bus of th oing rquir to ss th omprss t strutur. Srhing t h lvl of th multibit tri ns, in gnrl, 4 mmory rfrns. This mns tht in th worst s 2 mmory rfrns r n for IPv4. Th vntg of th Lul shm, howvr, is tht ths rfrns r lmost lwys to th h mmory bus th whol t strutur is vry smll. For instn, for forwring tbl ontining prfixs th rport siz of th t strutur is 6 Kbyts. Shms using multibit tris n omprssion giv vry fst srh tims. Howvr omprssion n th lf pushing thniqu us o not llow inrmntl upts. Rbuiling th whol strutur is th only solution. A iffrnt shm using omprssion is th Full Tr Bit Mp by Ethrton [4]. Lf pushing is voi n so inrmntl upts r llow. 5 Binry Srh on Prfix Lngths Th problm with rbitrry prfix lngths is tht w o not know how mny bits of th stintion rss shoul b tkn into ount whn ompr with th prfix vlus. Tris llow squntil srh on th lngth imnsion: first w look in th st of prfixs of lngth, thn in th st of lngth 2 prfixs n so on. Morovr t h stp th srh sp is ru bus of th prfix orgniztion in th tri. Anothr pproh to squntil srh on lngths without using tri is by orgnizing th prfixs in iffrnt tbls oring to thir lngths. In this s, hshing thniqu n b us to srh in h of ths tbls. Sin w look for th longst mth, w bgin th srh in th tbl holing th longst prfixs n srh ns s soon s mth is foun in on of ths tbls. Nvrthlss, th numbr of tbls is qul to th numbr of iffrnt prfix lngths. If W is th rsss lngth, whih is 32 for IPv4, th tim omplxity of th srh oprtion is O(W ) ssuming prft hsh funtion, whih is th sm s for tri. In orr to ru th srh tim, binry srh on lngths ws propos by Wlvogl t l. [7]. In binry srh, w ru th srh sp in h stp by hlf. Whih hlf to ontinu th srh pns on th rsult of omprison. Howvr, n orring rltion ns to b stblish bfor bing bl to mk omprisons n pro th srh in irtion oring to th rsult. Comprisons r usully on using ky vlus. But our problm is iffrnt sin w o binry srh on lngths. W r rstrit to hk whthr t givn lngth, mth xists. Using mth to i wht to o nxt is possibl: if mth is foun, w n ru th srh sp to only longr lngths. Unfortuntly, if no mth is foun, w nnot b sur tht th srh shoul pro in th irtion of shortr lngths, bus th bst mthing prfix oul b of longr lngth s wll. Wlvogl t l. insrt xtr prfixs of qut lngth, ll mrkrs, to b sur tht, whn no mth is foun, th srh must pro nssrily in th irtion of shortr prfixs. To illustrt this pproh onsir th prfixs shown in th figur 8. In th tri w n obsrv th lvls t whih th prfixs r lot. At th right, binry srh tr shows th lvls or lngths tht r srh t h stp of th binry srh on lngths lgorithm. Not tht th tri is only shown to unrstn th rltionship btwn mrkrs n prfixs but th lgorithm os not us tri t strutur. Inst, for h lvl in th tri, hsh tbl is us to stor th prfixs. For xmpl, if w srh th BMP for th rss, w bgin by srhing th tbl orrsponing to lngth 4, mth will b foun bus of th prfix f, n th srh pros in th hlf of longr prfixs. Thn w srh t lngth 6, whr th mrkr * hs bn pl. Sin mth is foun, th srh pros to th lngth 7 n fins prfix k s th BMP. Not tht without th mrkr t lvl 6, th srh prour woul fil to fin prfix k s th BMP. In gnrl, for h prfix ntry sris of mrkrs r n to gui th srh. Sin binry 9

20 Prfixs * b * * * * f * g * h * i * j * k * p * j M p Binry srh on lngths 2 3 f g h i 4 b 5 M 6 k 7 Figur 8: Binry srh on prfix lngths srh only hks mximum of log 2 W lvls, h ntry will gnrt mximum of log 2 W mrkrs. In ft, th numbr of mrkrs rquir will b muh smllr for two rsons: No mrkr will b insrt if th orrsponing prfix ntry lry xists (prfix f in figur 8), n singl mrkr n b us to gui th srh for svrl prfixs, s for xmpl prfixs n p whih us th sm mrkr t lvl 2. Howvr, for th vry sm rsons, th srh my b irt towrs longr prfixs lthough no longr prfix will mth. For xmpl, suppos w srh th BMP for rss. W bgin t lvl 4 n fin mth with th prfix f, so w pro to lngth 6, whr w fin gin mth with th mrkr, so w pro to lvl 7. Howvr, t lvl 7 no mth will b foun bus th mrkr hs gui us in th b irtion. Whil mrkrs provi vli hints in som ss, thy n misl in othr ss. To voi bktrking whn bing misl, Wlvogl uss promputtion of th BMP for h mrkr. In our xmpl, th mrkr t lvl 6 will hv f s th promput BMP. Thus, s w srh, w kp trk of th promput BMP so fr, n thn in s of filur w lwys hv th lst bst mthing prfix. Th mrkrs n th promput BMP vlus inrs th mmory rquir. Aitionlly, th upt oprtions bom iffiult bus of th svrl iffrnt vlus tht must b upt. 6 Prfix Rng Srh Srh on vlus only, to fin th longst mthing prfix, is possibl if w n gt ri of th lngth imnsion. On wy of oing this is by trnsforming th prfixs to uniqu lngth. As prfixs r of rbitrry lngths, w n to o full xpnsion, trnsforming ll prfixs to 32 bit lngth prfixs, in th s of IPv4. Whil binry srh on vlus oul b on now, this pproh ns hug mount of mmory. Fortuntly, it is not nssry to stor ll of th 2 32 ntris. As full xpnsion hs bn on, informtion runny xists. A prfix rprsnts n ggrgtion of ontiguous rsss, in othr wors prfix trmins wll fin rng of rsss. For xmpl, supposing 5-bit lngth rsss, prfix=*fins th rng of rsss [,5]. So, why not simpl stor th rng npoints inst of vry singl rss. Th BMP of th npoints is, in thory, th sm for ll th rsss in th intrvl. An srh of th BMP for givn rss, woul b ru to fin ny of th npoints of th orrsponing intrvl. For instn, th prssor, whih is th grtst npoint smllr thn or qul to givn rss. Th BMP problm woul b rily solv, bus fining th prssor of givn rss n b prform with lssil binry srh mtho. 2

21 Unfortuntly this pproh my not work bus prfix rngs my ovrlp, tht is prfix rngs my b inlu in othr prfix rngs, s figur 4. For xmpl, figur 9 shows th full xpnsion of prfixs ssuming 5-bit lngth rsss. Th sm figur shows th npoints of th iffrnt prfix rngs, in binry s wll s iml form. Thr, w n s tht th prssor of th rss vlu 9, for instn, is th npoint vlu 8; nvrthlss th BMP of th rss 9 is not th on ssoit to npoint 8 (b), but th on ssoit to npoint () inst. Clrly, th ft tht rng my b ontin in nothr rng os not llow this pproh to work. On solution is to voi intrvl ovrlp. In ft, by obsrving th npoints w n s tht ths vlus ivi th totl rss sp into isjoint bsi intrvls. In bsi intrvl, vry rss hs tully th sm BMP. Figur 9 shows th BMP for h bsi intrvl of our xmpl. Not tht for h bsi intrvl, its BMP is th BMP of th shortst prfix rng nlosing th bsi intrvl. Th BMP of givn rss n now b foun by using th npoints of th bsi intrvls. Nvrthlss, w n obsrv in figur 9 tht som bsi intrvls o not hv xpliit npoints (for xmpl I3 n I6). In ths ss, w n ssoit th bsi intrvl with th losr npoint to its lft. As rsult, som npoints n to b ssoit to two bsi intrvls n thus npoints must mintin in gnrl two BMPs, on for th intrvl thy blong to n on for th potntil nxt bsi intrvl. For instn, th npoint vlu 8 will b ssoit to bsi intrvls I 2 n I 3, n must mintin BMP b n. Prfixs * b * * * * f * g * h * i * f g h i b f f g g h h i i Prfix rngs b f g h i BMP Bsi intrvls b f g h i I I2 I3 I4 I5 I6 I7 I8 I9 I Figur 9: Binry rng srh Figur 2 shows th srh tr initing th stps of th binry srh lgorithm. Th lvs orrspon to th npoints, whih stor th two BMPs ( = n > ). For xmpl, if w srh th BMP for th rss (22) w bgin ompring th rss with th ky 26, s 22 is smllr thn 26 w tk th lft brnh in th srh tr. Thn, w ompr 22 with ky 6 n go to th right, thn t no 24 w go to th lft rriving t no 9 n finlly w go to th right n rriv t th lf with ky 9. Bus th rss (22) is grtr thn 9 th BMP is th vlu ssoit with >, tht is. As with tritionl binry srh, th implmnttion of this shm n b m by xpliitly builing 2

22 th binry srh tr. Morovr, inst of binry srh tr, multiwy srh tr n b us to ru th hight of th tr n thus mk th srh fstr. Th i is similr to th us of multibit tris inst of binry tris. In multiwy srh tr, intrnl nos hv k brnhs n k- kys, this is spilly ttrtiv if n ntir no fits into singl h lin bus srh in th no will b ngligibl ompr to norml mmory sss. As w hv prviously mntion, th BMP for h bsi intrvl ns to b promput by fining th shortst rng (longst prfix) nlosing th bsi intrvl. Th problm with this pproh, whih ws propos by Lmpson t l. [9], is tht insrting or lting singl prfix my rquir to romput th BMP for mny bsi intrvls. In gnrl, vry prfix rng spns svrl bsi intrvls. Th mor bsi intrvls < 26 >= < 2 < >= 6 >= >= >= >= < < < 24 >= < >= >= < < 28 < >= 29 3 >= >= < < >= = = = = = = = = = = = = = = b f> f g g h h i> i > > > > > > > > > > > f g g > h h i i Figur 2: Bsi Rng srh tr prfix rng ovr, th highr th numbr of BMPs to potntilly romput. In ft, in th worst s w woul n to upt th BMP for N bsi intrvls, N s usul bing th numbr of prfixs. This is th s whn ll th 2N npoints r ll iffrnt n on prfix ontins ll th othr prfixs. On i to ru th numbr of intrvls ovr by prfix rng is to us lrgr, yt still isjoint intrvls. Th lvs of th tr in figur 2 orrspon to bsi intrvls. A ruil obsrvtion is tht intrnl nos orrspon to intrvls tht r th union of bsi intrvls, s figur 2. Also, ll th nos t givn lvl form st of isjoint intrvls. For xmpl, t th son lvl th nos mrk 2, 24 n 28 orrspon to th intrvls [,5], [6,25] n [26,29] rsptivly. So why stor BMPs only t lvs. For instn, if w stor t th no mrk 2, in th son lvl, w will not n to stor t lvs n upt prformn woul b bttr. In othr wors, inst of omposing prfix rngs into bsi intrvls, w ompos prfix rngs into isjoint intrvls s lrgr s possibl. Figur 2 shows how prfixs n b stor using this i. Srh oprtion is lmost th sm xpt tht now ns to kp trk of th BMP nountr whn trvrsing th pth to th lvs. W n ompr th bsi shm to using lf pushing whil th nw mtho os not. Agin, w n s tht pushing informtion to lvs mks upt iffiult, bus th numbr of ntris to moify grows. Th multiwy rng tr pproh [6] prsnts n vlops this i to llow inrmntl upts. 7 Comprison n Msurmnts of Shms Eh of th shms prsnt hs its strngths n wknsss. In this stion, w ompr th iffrnt shms n isuss th importnt mtris to vlut ths shms. Th il shm woul b on with 22

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