Constructing Minimal Spanning/Steiner Trees. with Bounded Path Length. Department of EE-Systems

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1 Construting Miniml pnning/tinr Trs with Boun Pth Lngth Iksoo Pyo Jwon Oh n Mssou Prm JF1-61 Dprtmnt o EE-ystms Intl Corporttion Univrsity o outhrn Cliorni Hillsoro, OR 9714 Los Angls, CA Astrt This ppr prsnts n xt lgorithm n two huristis or solving th Boun pth lngth Miniml pnning Tr (BMT) prolm. Th xt lgorithm whih is s on itrtiv ngtiv-sum-xhng(s) hs polynomil sp omplxity n is hn mor prtil thn th mtho prsnt y Gow. Th rst huristi mtho (BKRU) is s on th lssil Kruskl MT onstrution. For ny givn vlu o prmtr, th lgorithm onstruts routing tr with th longst intronntion pth lngth t most (1 + ) R, n mpirilly with ost t most 1.19 tims ost(bmt ) whr R is th lngth o th irt pth rom th sour to th rthst sink n BMT is th optiml oun pth lngth MT. Th son huristi omins BKRU n ngtiv-sum-xhng(s) o pth to improv rsults. Extnsions o ths thniqus to th oun pth lngth Miniml tinr Trs, using th Elmor ly mol r prsnt s wll. Empiril rsults monstrt th tivnss o ths lgorithms on lrg nhmrk st. 1 Introution In th sign o high-prormn VLI systms, iruit sp n powr onsumption r importnt onsirtions. Routing optimiztion plys n importnt rol in hiving optiml iruit sp n miniml powr onsumption. In, ritil pth ly is untion o mximum intronntion pth lngth whil powr onsumption is untion o th totl intronntion lngth. A linr RC mol (whr intronntion ly twn sour n sink is proportionl to th wir lngth twn th two trminls) is otn us s simpl pproximtion or intronntion ly. First, w lso us wir lngth to pproximt intronntion ly uring th onstrution o routing trs. Ltr, w xtn this ly mol to mor urt RC ly mol. A routing tr us in synhronous systm hs n input, ll th rivr or sour, tht sns signls to h sink. Critil pth ly is n s th mximum ly rom th sour to ny sink. Th ritil pth ly o th hortst Pth Tr (PT) is minimum(in PT, h sink is onnt to th sour y th shortst possil pth.), ut PT hs xssiv routing ost n powr issiption s th powr onsum y th rivr hs linr rltion with This rsrhws un in prt y RC unr ontrt No. 94-DJ-559 n y NF NYI unr ontrt No. M/P th routing pitn. Miniml pnning Tr (MT) hs miniml routing ost, ut my ontin vry long sour-tosink pths whih gr th prormn. Alprt t l. [8] show how to tr th vrg sour-to-sink pth lngth or lowr totl routing ost y using linr omining ost untion onsisting o th sour-to-sink pth lngth n th wight o th g to uring th tr onstrution. In this ppr, w prsnt lgorithms or onstruting Boun pth lngth Miniml pnning Tr (BMT). Th routing tr hivs oun pth lngth, tht is, th lngth o th pth rom th sour to h trminl is oun. uh oun pth lngth tr provis goo initil topology or signrs to just or minimizing ritil pths using mor urt RC ly mol. Also, th tr hs smll routing ost whih is importnt rom r n powr onsumption viwpoints. Lt R th lngth o irt pth rom th sour to th rthst sink n non-ngtiv usr-spi prmtr. Our mtho onstruts spnning tr with rius t most (1+) R y using n nlogu o th lssil Kruskl MT onstrution [1]. W will show th sm mtho n xtn to Elmor ly mol n to oun pth lngth tinr tr. Furthrmor, th tr ost is mpirilly osrv to t most 1.19 o tht o n optiml BMT. W nxt sri n xt lgorithm u to Gow [5] whih prous n optiml BMT with xponntil tim n sp omplxity. Thn, w propos nw xt lgorithm whih rquirs xponntil tim ut hs polynomil sp omplxity. This mtho onstruts n optiml tr y ngtiv-sum-xhng(s) on n initil sil solution. W lso propos nothr huristi whih rsolvs th omplxity prolms o th xt lgorithm n prous ttr vrg rsults thn th Kruskl s mtho. Bkgroun On Mhttn (L 1 mtri) or n Eulin (L mtri) pln, lt G =(V;E) (jvj=n)ntwork whr V is st o rnomly istriut trminl pins ll sinks with istinguish pin ll th sour(s), n E is th st o gs onnting V. BMT sks to onnt ll nos o V in G y st o gs in E o miniml totl lngth with oun pth lngth rom th sour to ny sink. This prolm is known to NP-omplt [7]. W propos novl lgorithm - tht is, Boun pth lngth Kruskl (BKRU) - or solving this prolm huristilly. A tr gnrt y our BKRU mtho is ll Boun pth ED&TC /96 $ IEEE

2 lngth Kruskl miniml spnning Tr (BKT). Cong t l. [] propos two huristis or solving th BMT prolm. In th rst mtho o Cong t l., i.. th Boun Prim (BPRIM) lgorithm, vn though th mpiril rsults r promising, th worst-s prormn rtio is unoun whr prormn rtio is n s ost(bprim)/ost(mt) (s Tl n Figur 5). In th son mtho o Cong t l., i.. th Boun Rius, Boun Cost (BRBC) lgorithm, th worst-s prormn rtio is oun. Howvr, BRBC mtho uss minimum pth (shortst pth) rom th sour to sink whnvr th sour-to-sink pth lngth violts th lngth oun ost(; sink) uring th pth rst tr trvrsl. Hn, it my introu unnssry routing ost. Thir nhmrk rsults [] show outworst-s prormn rtio o :66 n n vrg prormn rtio o 1:57. 3 A Huristi: BKRU Bor sriing our pproh, w giv som nitions. Th sum o ll g wights o T is th ost o th tr, ost(t ). Th shortst pth istn twn u n v in grph G is ist G(u; v). Th shortst pth istn twn u n v in tr T is ist T (u; v). Th rius o no v G is rius G(v) (i.. mxist G(v; u)g, 8 u V ). imilrly, th rius o no v T is rius T (v) (i.. mxist T (v; u)g, 8 u V ). Th prtil tr whih ontins no v is rprsnt y t v. nots th sour. BKRU lgorithm solvs th BMT prolm y solving th ollowing prolm: Givn th routing grph G(V;E) in L 1 or L sp, n miniml ost routing tr BKT with rius BKT () (1 + ) R. Th lssil Kruskl lgorithm s n g (u; v) in G to MT, or quivlntly, mrgs two prtil trs t u n t v y th g (u; v) i: (1) (u; v) is th lst wight g mong th vill gs n () t u 6= tv. For (1), ll th gs r sort in nonrsing orr. For (), isjoint st on V is implmnt. Thr oprtions on th st r MAKE ET, FIND ET n UNION, th mnings o whih r sl-xplntory. Mrging two prtil trs is on y th UNION oprtion ollow y th Mrg routin to isuss ltr, whil onition () is sily tst y th FIND ET oprtion. BKRU lgorithm s on mor onition s ollows: (3) th mrg tr stiss th pth lngth oun (1 + ) R rom th sour to th rthst sink. Lt t M th mrg tr, i.., t M = t u [ tv [ (u; v). Two ss r possil: (3-) I t u ontins th sour, thn th ollowing onition shoul stis: ist t u(; u)+ist G(u; v)+rius t v (v) (1 + ) R in nos in t u lry stisy th uppr oun onstrint, this onition nsurs tht nos in t v will lso stisy th uppr oun onstrint tr th mrg. Th s whr t v ontins th sour is similr. (3-) I nithr t u nor t v ontins th sour, thn thr must no x t M suh tht: ist G(; x)+rius tm (x) (1 + ) R This onition nsurs tht ll th nos in th mrg tr t M n onnt to th sour without violting th uppr oun pth lngth onstrint y hving t lst irt pth rom th sour to no x. Tht is, th xistn o suh noxgurnts tht ll nos in t M n stisy th uppr oun onstrint. I no suh no xists in t M, thn (u; v) shoul rjt s thr is no wy to stisy th upproun onstrint or ll th nos in t M.W n now giv two importnt nitions. Dnition 3.1 A sil no: I thr xists no x in t M suh tht istg(,x) + riustm (x) (1 + ) R, thn noxissil no in t M. Dnition 3. A sil g: I g (u; v) stiss onitions () n (3), thn it is sil g. Fsil gs n sly to th spnning tr unr onstrution. BKRU mintins th rius o h no in th prtil tr it longs to, n th pth lngths twn vry pir o nos within th prtil tr thy long to. Lt th rry D[V,V] ontin inormtion out th Mnhttn or th Eulin istns twn vry pir o nos, i.. D[x; y] = ist G(x; y). Lt th rry P[V,V] th pth lngth twn vry pir o nos in th routing tr, i.. P[x; y] =ist T (x; y). Also lt th vtor r[v] th rii o nos in th tr thy long to. Initilly, th rry P n th vtor r r initiliz to zro. As th tr grows, P n r r upt y th Mrg routin givn low: // Mrg two sutrs tu n tv y g (u; v) Algorithm Mrg(u; v) 1 or h x tu n y tv o P[x; y] =P[y; x] =P[x; u] +D[u; v] +P[v; y] 3 n or 4 or h x tu o 5 r[x] = mx(r[x], P[x; i], 8 i tv) 6 n or 7 or h y tv o 8 r[y] = mx(r[y], P[i; y], 8 i tu) 9 n or Figur 1 shows n xmpl o how Mrg routin works. Th two prtil trs r mrg y th g (; ). Th lthn si tr is t n th righthn si tr is t. Bor th mrging tks pl, ll o th non-zro lmnts (xpt th igonl lmnts) in mtrix P n th vtor rwr omput rom prvious mrgings. Not tht lmnts o r r th mximum o h rowop. Th Mrg routin lvs thos non-zro lmnts unhng n upts P[x; y] only whn x n y r in irnt prtil trs. For xmpl, P[; ] n omput y P[; ] =P[; ] + D[; ] +P[; ]. On th P mtrix is upt y lin 1-3 in th lgorithm, nw rius r[x] n oun y tking th mximum mong th ol rius (ol r[x]) n th P[x; y]s or ll y t v.for xmpl, nw r[] n oun y tking th mximum mong ol r[], P[; ], P[; ]g, whih is9,

3 4 3 5 Bor Mrg P = Atr Mrg P = r = r = Figur 1: Exmpl o Mrging Two Prtil Trs 11, 13g. o th nw r[] is 13. W n sily s tht th tim omplxity o Mrg is O(V ). in w n nw rius o no x in th mrg tr to tst th siility ox, it sms tht mrging is n or th siility tst is prorm. Howvr, w n n th nw rius o ny no without n tul mrging. Using th sm nottion s or, suppos x longs to t u. Thn it n sily sn tht nw rius o x = mx r[x], P[x; u] +D[u; v] +r[v]g whr r n P vlus r r rom th rrys or th mrg. Th s whr x longs to t v is similr. With this, siility tst or no n on in O(1). o th onition (3-) n tst in O(V ). W lso not tht onition (3-) n tst in O(1). Th omplt BKRU lgorithm is summriz in th ollowing: Algorithm BKRU(G) 1 or h vrtx x V o MAKE ET(x) 3 r[x] =0 4 n or 5 or vry pir o vrtis x; y V o 6 P[x; y] =0 7 n or 8 sort th g st E in nonrsing orr o wights 9 or h g (u; v) in th sort g list o 10 i FIND ET(u) 6= FIND ET(v) thn 11 i ithr onition (3-) or (3-) is stis thn 1 UNION(u; v) 13 Mrg(u; v) 14 output th g (u; v) 15 n i 16 n i 17 n or Th ominting omplxity o BKRU is on lin 11 n 13. in lin 11 is xut E tims n lin 13 is xut V 1 tims, th omplxity o BKRU is oun y O(EV + V V )=O(V 3 ). Hr, w xplin BKRU lgorithm with simpl xmpl. uppos w hv sour n thr sinks s shown in () o Figur. I th uppr oun pth lngth is st to (1 + ) R = 8, BKRU works in th orr o (), (), n () n prous BKT with totl ost 8 whih is optiml. Th slt lightst g - o () stiss ov thr onitions sin is th sil no. o th g - is sil g. Th nxt lightst g - o () stiss () () () () Figur : BKRU Exmpl ov thr onitions sin is th sil no. Finlly, g - is inlu sin is th sil no. 3.1 Extnsion o BKRU to us th Elmor Dly Mol Th BKRU lgorithm n xtn to th Elmor ly mol so tht th pth lngth rom th sour to ny sink is rpl with th signl propgtion ly. To nsur th xistn o solution, th sour shoul l to supply vry lrg mount o urrnt, i.. it must hv vry smll rivr rsistn so tht PT n solution. R is st to th longst -sink lys o PT. For two nos u n v, th ly rom u to v is not simply proportionl to th pth lngth twn u n v, ut lso pnnt on th tr topology. o th mtho in BKRU or omputing th rius o no os not work. Th nw rii r in BKRU lgorithm must ompltly romput tr tmporrily mrging th two sutrs. Th rius r[v] onovis th longst ly rom v to ny sink in th mrg tr root t v, n n oun in linr tim. At th sm tim, th totl pitn, whih is sum o th wir n lo pitns in th mrg tr (not y C(mrg tr)) is lso omput. Th siility tsts (3-) n (3-) r thn rstt s: (3-) 0 r[sour] (1 + ) R in th mrg tr (3-) 0 thr xists no x in th mrg tr suh tht R s istg(; x) (C s istg(; x)=+c(mrg tr)) + r[x] < (1 + ) R, whr R s n C s not th sht rsistn n pitn o th wir, rsptivly. (3-) 0 tks O(V ) whil (3-) 0 tks O(V ). As rsult o ths moitions to BKRU, th siility tst omints th totl omplxity o th lgorithm, whos omplxity oms O(EV ). 3. Construting Boun Pth Lngth tinr Trs: BKT Boun Pth Lngth tinr Trs n onstrut on hnnl intrstion grph or on Hnn's gri grph [9] using moi BKRU. A spnning tr tht spns ll th sinks n th sour on ths routing grphs oms tinr tr. W ll this vrition o Boun Kruskl mtho Boun Kruskl Tinr (BKT). Initilly, th istns twn vry pir o sinks on th routing grph r omput n stor in hp. Ths istns r nlogous to th g wights in BKRU. Thn w xtrt th smllst istn rom th hp n hk its siilty. I it is sil, th pth in th routing grph tht hivs

4 our our our Initil olution Tr g () () () our our T-xhng T-xhng T-xhng T-xhng pth 1 () i h g i h g () Figur 3: Exmpl o tinr BKRU (BKT) this istn is oun n to th tinr tr unr onstrution. I thr r multipl suh pths, w hoos only L-shp pths (no zigzg pths). Also, mong th two possil L-shp pths, whoos th pth whos ornr is losst to th sour. Th nos tht li on th pth whih wr just to th tinr tr, r trt s nw sinks. Nxt, th istns twn th nw sinks n ll othr sinks whih r not in th urrnt mrg tr r omput n stor in th hp. Th nxt itrtion piks th smllst istn rom th hp. This ontinus until vry sink is ovr. I thr r m sinks (oth givn n sinks) in th tinr tr, th omplxity o BKRU is O(V m ). In th worst s, m is o O(V ). Howvr, in prti, m is not lrg. In our nhmrk iruits, m ws usully no mor thn 10 tims o V. In mny VLI signs, spilly in stnr ll signs, th sink lotions r rgulr. o thr r not so mny Hnn points. Ths ts nl us to run BKT on lrg nhmrks s wll. Figur 3 shows n xmpl o this lgorithm on Hnn gri grph. In th Figur, () shows th givn sour n our sink lotions. Th ott lins n thir intrstion points r th gs n nos o th Hnn gri grph. Initilly, th istns twn th 5 points (our,,,, ) r omput n stor in th hp. From th hp, th shortst istn (,) is xtrt. Assum tht it is sil. Thn th pth pth(; ) shown in () is to th tinr tr. W thn hv two nw sinks n. Th istns rom, to our,, r omput n stor in th hp. Th nxt shortst istn in th hp is (,) n it is sil, so pth(;) is to th tinr tr in (). Th nxt shortst istn pth(; ), howvr, is not sil, so it is rjt. Th nxt shortst n sil istn is pth(our; g), so it is in (). Finlly, pth(i; ) is inlu in (). 4 Gow's Ext Mtho: BMT G Lt us sri n optiml lgorithm or th Boun pth lngth Miniml pnning Tr (BMT) prolm. This optiml lgorithm is opt rom [5], lthough our implmnttion is somwht irnt. Gow's lgorithm prous ll spnning trs in orr o inrsing ost with tim omplxity oo(kelog (1+E=V ) V ) j pth Figur 4: BKEX Ngtiv-sum-Exhng rh Tr n sp omplxity oo(k) whr K is th totl numr o spnning trs gnrt 1. W riy sri his lgorithm, omitting mny tils. Intrst rrs my rr to [5]. Lt T spnning tr o G. A T-xhng is pir o gs (, ) whr T, G T n T [ is spnning tr. Th wight o xhng (, ) iswight() wight(). Th g pir (, ) whih hivs th minimum wight oxhng is miniml T-xhng. Not tht i T is miniml spnning tr, thr is no ngtiv wight T- xhng. I T is miniml spnning tr n (, ) is miniml T -xhng, thn T [ is spnning tr with th nxt smllst ost. This is th sis o th lgorithm. W trmint Gow's lgorithm whn th gnrt spnning tr stiss th uppr oun. Th mjor shortoming o Gow's lgorithm is th sp omplxity. Totl numr o spnning trs in omplt grph is V V [6]. This mks Gow's lgorithm imprtil vn or s w s 10 nos. W hv n l to somwht ru th sp n tim omplxitis y liminting som gs s prprossing to th Gow's lgorithm tht nnot l to solution tr. Using this thniqu, whv us Gow's lgorithm on trs with s mny s 15 sinks. In prtil CMO iruit, gt usully rivs lss thn 10 gts. o this lgorithm n us in most prtil ss. 5 Yt Anothr Ext Mtho n Huristi: BKEX n BKH Boun Kruskl EXhng (BKEX) is post-prossing lgorithm tht strts rom n initil solution tr n rus th routing ost towr th optiml. I th initil tr is not n optiml solution, BKEX ns g xhng(s) suh tht routing ost is ru. W ll suh xhng(s) ngtiv-sum-xhng(s). Dnition 5.1 Ngtiv-sum-xhng(s): A squn o T-xhng(s) whr th sum o th wight(s) o xhng(s) is ngtiv. BKEX strts rom ny solution tr, ns ngtiv-sumxhng(s), onvrts th solution tr to nw solution tr y xhnging gs, n itrts until no mor possil xhng(s) r oun. Lt's not th srh tr in Figur 4 s. Eh no in rprsnts spnning tr. 1 W liv this is th orrt tim omplxity inst o Gow's lim o O(KE(E;V )).

5 Th root o is th initil solution. A hil no is gnrt y T-xhng rom its prnt no. Th gs o r ll with th wight ot-xhng. BKEX srhs ngtiv-sum-xhng(s) in pth rst srh mnnr. Not tht on n rh ny spnning tr rom th root y sris o t most V 1 T -xhngs. o srhing own to th pth o V 1 lwys ns th optiml solution. Howvr, in most ss, BKEX ns n optiml solution in muh smllr pth. BKEX kps trk o th sum o T -xhng wights rom th root to th urrnt no uring th pth rst srh. I th sum t rtin no is ngtiv, nw tr with lss ost is onstrut t tht no n w stor th nw tr s minimum ost tr. Whnvr ttr solution is oun uring th srh, this nw tr is put on th root o n nw srh gins. in th numr o possil T -xhngs in tr T is O(EV ), no in hs O(EV ) hilrn. o hs O(E n V n ) nos whr n is th pth o. For h no in, BKEX ns to hk i th urrnt spnning tr is sil, whih tks O(V ). o th tim omplxity o BKEX is O(E n V n+1 ). This is highr tim omplxity thn Gow's, ut sp omplxity is only O(E). Th initil solution signintly ts th prormn o BKEX. Whn BKEX strts rom vry goo initil solution (suh s BKT), th tul srh sp is muh smllr thn E n V n. In our xprimntl rsults show tht BKEX is muh str thn Gow's mtho. Bsis, BKEX ns th solution whn Gow's lgorithm ils or lrgr nhmrks u to its xponntil sp omplxity. W tst BKEX with,750 rnomly gnrt nhmrks. Th numr o sinks o ths nhmrks r twn 5 n 15. Th vlu hs rng rom 0.0 to 1.0. BKEX rhs optiml solutions o %, % n % with pth two, thr n our rsptivly. Only on nhmrk ws lt unoptiml with pth v n it ws solv y pth six. W implmnt nothr huristi mtho BKH whih limits th pth o th srh tr y two. It n shown tht BKT is lol optimum with rspt to singl T -xhng. To otin ttr lol optimum thn BKT, t lst oul T -xhngs r n. Thus BKH is propos to n lol optimum with rspt to two T -xhngs. Th omplxity o BKH is O(E V 3 ). in this omplxity is rltivly high, w oun tht BKH is niil whn V is lss thn 300 (s Tl 1). 6 Exprimntl Rsults W implmnt BKRU, BMT G, BKEX, BKH, n BKT lgorithms in C on HPPA n UN worksttions in th UNIX nvironmnt. W us thr sts o nhmrks: (1) th sink plmnts or MCNC Primry1 n Primry nhmrks us in [3]; n () th sink plmnts or th v nhmrks r1-r5 us in [4]; n (3) v sts o 5 to 15 sinks n 50 rnom tst ss or h st. W on mor no s th sour to th r* n primry* nhmrks us thy i not om with sour. All th rsults r omput in Mnhttn mtri. BKRU BKH pr. pth pr. pth pr. r. rtio rtio pu rtio rtio pu % pr k k 9.37 pr k k k 0.18 r k k r k k k.58 r k k 1.71 r k k k k 0.00 r k k k k k 0.00 pr. rtio (Tr) = ost(tr) = ost(mt) pth rtio (Tr) = longst pth(tr) = longst pth(pt) pr. ruttion = (1 BKH/BKRU) 100 CPU tim is msur in sons. BKH limits CPU tim to out 1 hours. GABOW, BKEX n BPRIM r imprtil to gnrt outputs. Tl 1: BKRU n BKH rsults or lrg nhmrks BPRIM BKT ε= 0.0 MT n BKT ε= ε = 0.5 Cost = Cost = Cost = BKT ε = 0.0 Cost = Figur 5: Exmpl whr th prormn rtio o BRPIM is not oun or ny As xplin in [], BPRIM n gnrt solution whos ost is not oun or ny. BPRIM gnrts 4: ost(mt) n 3:3 ost(bkt) or th nhmrk shown in Figur 5 whn =0. A omprison o BKRU n BKH ovr MT is givn in Tl 1 or nhmrks (1), (). Th rsults show tht th prormn rtio o BKT ovr MT is t most For (3) nhmrks, th omprison o BPRIM, BRBC, BKRU, BKH, BMT G n BKT in trms o routing ost is shown in Tl. Th nhmrk rsults show out vrg prormn rtios o 1.8, 1.45, 1.0, 1.0 n 1.03 or BPRIM, BKRU, BKH, BMT G n BKT rsptivly in th worst s. In th s o 15 points with = 0:1, th vrg ost rutions r 18.19%, 10.9%, 10.6% n 6.5% ovr BPRIM or BKT, BMT G (BKEX), BKH n BKRU rsptivly. BKRU mtho ors ontinuous, smooth tr-o twn th ompting rquirmnts o longst pth lngth n totl wir lngth in trms o.

6 nt BPRIM BRBC BKRU BKH BMT G BKT siz v mx mx v mx pu v mx pu v mx pu min v mx pu rnom tst ss wr gnrt or h point. CPU tim is th vrg o 50 rnom tst ss msur in sons. Minimum vlus r 1.008, 1.038, n or BPRIM, BKRU, BKH n BMT G rsptivly t = 0.0 o nt 1. Th othrs r BRBC is shown only with mximum vlus sin minimum n vrg vlus o BRBC r lwys wors thn thos o BPRIM. Tl : Th Rtio o th Routing Cost ovr MT MT BMT_G BKEX BKH BKRU PT Mximl pnning Tr Figur 6: Routing Cost Chrt From ths rsults, th vrious BMT mthos n orr y thir routing osts s shown in Figur 6. This hrt shows th vrg rltiv positions. Th rsult o Boun Kruskl tinr Tr (BKT) on nhmrk st (3) shows tht its ost is lowr thn ny othr spnning tr huristis. Th svings r 5% to 30% ovr othr huristis. Not tht th svings r vn grtr whn is los to zro. This is u to th t tht whn is los to zro, thr r mny irt sour-to-sink pths in th spnning tr solutions whil in th tinr solutions, ths irt pths r rpl y wr irt sour-to-sink pths. Although BKT prous lowr ost trs, w l tht spnning tr huristis r worthwhil us thy run muh str. 7 Conlusion W hv prsnt oun pth lngth miniml spnning/tinr tr shms whih n ontrol longst pth lngth n routing ost. Our mtho hivs smllr ost thn tht o BPRIM n BRBC. Futur rsrh inlus onsiring th ts o uring n wir sizing, xtning this work to lowr n uppr oun tinr Trs n prsrving plnrity uring th onstrution prour. Rrns [1] J. B. Kruskl, \On th shortst spnning sutr o grph n th trvling slsmn prolm," Proings o th Amrin Mthmtil oity, Vol. 7, pp , [] Jingshng Cong, A. Khng, G. Roins, M. rrzh, n C. K. Wong, \Provly Goo Prormn-Drivn Glol Routing," IEEE Trnstions on Computr Ai Dsign, Vol. 11, NO. 6, pp , Jun, 199. [3] M. A. B. Jkson, A. rinivsn, n E.. Kuh, \Clok routing or high-prormn ICs," 7th Dsign Automtion Conrn, pp , [4] R- Tsy, \Ext zro skw," Intrntionl Conrn on Computr-Ai Dsign, pp , [5] Hrol N. Gow, \Two lgorithms or gnrting wight spnning trs in orr," IAM J. Comput.,Vol. 6, No. 1, pp , Mrh [6] F. Hrry, \Grph Thory," Aison-Wsly, Msshustts, pp , [7] J. Ho, D. T. L, C. H. Chng, n C. K. Wong, \Boun imtr spnning trs n rlt prolms," Proings o ACM ymposium Computtionl Gomtry, pp. 76-8, [8] C.J. Alprt, T.C. Hu, J.H. Hung n A.B. Khng, \A irt omintion o th Prim n Dijkstr onstrutions or improv prormn-rivn glol routing," Intrntionl ymposium on Ciruit n ystms, pp , [9] M. Hnn, \On tinr's prolm with rtilinr istn," IAM Journl o Appli Mthmtis, pp , Mrh 1966.