Verification of DAG Structures in Cooperative Belief Network Based Multi-agent Systems
|
|
- Rosamond Gibbs
- 6 years ago
- Views:
Transcription
1 Vrton o AG Struturs n Cooprtv B Ntwor Bs Mut-nt Systs Y. Xn prtnt o Coputr Sn, Unvrsty o Rn Rn, Sstwn, Cn S4S 0A, yxn@s.urn. Astrt Mutpy ston Bysn ntwors (MSBNs) prov rwor or prost rsonn n opx sn usr ornt syst s w s n ooprtv ut-nt strut ntrprtton syst. urn t onstruton or yn orton o MSBN, utot vrton o t yty o t ovr strutur s sr. W nown orts or yty tst ssu ntrz stor o t strutur to tst. W suss wy ntrz tst s unsr n propos strut ort tt vrs t yty trou ooprton on sunts/nts. T ort os not rqur nt to rv ts ntrn strutur n tus supports onstruton o MSBN ro sunts ut y rnt vnors. Introuton Mutpy ston Bysn ntwors (MSBNs) s n xtnson o Bysn ntwors (BNs) [, 4, ]. A MSBN onssts o st o ntrrt Bysn sunts tt otvy n BN [, ]. E sunt srs non-pty st o vrs wt t st on otr sunt. Sunts r ornz nto yprtr strutur su tt prost nrn n pror ornty n our n strut son. T ourty provs nrn ny n opx sn usr ornt syst [0]. It so ows MSBNs to xtn nto ornt rwor or prost rsonn n ooprtv ut-nt strut ntrprtton systs [8]. T strutur o BN s rt y rp (AG). T ovr strutur o MSBN, t oposton o sunt struturs, s so AG. To nsur t orrt oposton, utot vrton o yty o t opos strutur s sr. Atou orts or tstn yty s on topoo sortn r w nown, s [6] or xp, ty ssu ntrz stor o t rp to tst. W nyz so sn onsrtons tt ntrz tst unsr. W tn propos strut ort n trs o st o strut oprtons or tstn yty o t opos strutur trou ooprton o sunts/nts. T tory n pptons o MSBNs r ry rvw n Ston. T onpts nssry to t rst o t ppr r ory n. W suss n Ston rsons
2 wy strut vrton o yty s prrr. It s sown n Ston 4 tt so ovous soutons to strut vrton o not sov t pro. T rp-tort ounton or t propos ort s rv n Ston 5 n t ort s prsnt n Ston 6 wt proo o ts orrtnss. Its opxty s nyz n Ston 7. Ovrvw o MSBNs In ts ston, w ry ovrvw t tory o MSBNs n tr pptons. Mor ts on MSBNs n oun n [, 0, 7, 8]. ABNS s trpt (N,,P) wr N s st o vrs, s AG wos nos r y nts o N, n P s ont proty struton (p) ovr N. W s N t on o S, t strutur o S n P t struton or p o S. A MSBN M s oton o Bysn sunts tt totr n BN. Ts sunts r rqur to stsy rtn ontons tt prt t onstruton o strut nrn orts. On o ts ontons rqurs tt nos sr y rnt sunts or -spst, s n ow. Lt G =(N,E )( =, ) two rps. W s rr to t rp G =(N N,E E ) s t unon o G n G, not y G = G G. nton (-spst) Lt =(N,E )( =, ) two AGs su tt = s AG. T ntrston I = N N s -spst twn n or vry A I wt ts prnts π n, tr π N or π N. E no n -spst s -spno. T -spst onpt s syntt onton. Snty, t n sown tt wn pr o sunts r sot ro M, tr -spst rnrs t ontony npnnt. Tror, -spst provs sp syntt ru to tt npnnt spton o snty orrt sunts. Fur (t) sows t tr AGs ( =,, ) o MSBN or noss o tr nurousur sss, Mn nrv son (Mn), Crp tunn synro (Cts) n Pxus uppr trun son (Pxut). T -spst twn pr o AGs s {Mn, Cts, P xut}. In nr, -spsts twn rnt prs o AGs o M y rnt. Just s t strutur o BN s AG, t strutur o MSBN s utpy ston AG (MSAG) wt yprtr ornzton, or spy yprtr MSAG n s oows: nton (Hyprtr MSAG) A yprtr MSAG =, wr s onnt AG, s AG tt s ut y t oown prour: Strt wt n pty rp (no no). Rursvy AG,yprno, to t xstn MSAG = sut to t onstrnts: [-spst] For (<), t ntrston I = N N s -spst wn t two AGs r sot. T xp s tn ro rton o PAINULIM [0] o or ustrton.
3 Pxut pns Mn wr oup Cts pn Pxut ps Mn prt Cts p Pxut Mn Cts p up {Mn,Cts, Pxut} {Mn,Cts,Pxut} Fur : Lt: T AGs o n xp MSBN, wr -spno s t y ott r. M: T yprtr ornzton o t AGs n t t. Rt: A nr yprtr MSAG (unrt to t t). [Lo ovrn] Tr xsts (<) su tt, or (<; ), w v I N. For n rtrry osn su, I s t yprn twn yprnos n, n n r s to nt. Not tt yprtr MSAG s tr wr no s yprno s n ov n n s yprn. T AGs n Fur (t) n ornz nto t trv yprtr MSAG n Fur (), wr yprno s y AG n yprn s y -spst. Fur (rt) pts nr yprtr MSAG. Atou AGs o MSBN sou ornz nto yprtr, AG y utpy onnt (or tn on pt xst twn pr o nos),..,. Morovr, tr n utp pts twn pr o nos n rnt AGs n yprtr MSAG. For nstn, utp pts r or twn p n p tr n r unon. T o ovrn onton nsurs tt or ny unrt y ross two nt AGs, ot o ts two pts r trou t orrsponn -spst. Totr wt t -spst onton, ty nsur tt n yprtr strutur M, yprn rnrs t two prts o M tt t onnts ontony npnnt. An ntutv ustton o ts strutur s vn n [9]. A MSBN s n s oows. Rrs r rrr to [] or or ts. nton A MSBN M s trpt (N,, P). N = N s t tot unvrs wr N s st o vrs. = ( yprtr MSAG) s t strutur wr nos o AG r y nts o N. P = P (N )/ P (I ) s t ont proty struton (p). E P (N ) s proty struton ovr N su tt wnvr n r nt n, t rnztons o P (N ) n P (N ) onto t -spst I r nt. E P (I ) s su rn struton ovr yprn o. E trpt S =(N,,P ) s sunt o M. Wtout onuson, w s sy tt two sunts S n S r nt n r nt. A MSBN n us s rwor or prost rsonn n sn usr ornt syst n r pro on. T sn usr ps tt vn n qurs r rstrt to on suon t t. Usn MSBN s ost n suons o t pro on r oosy oup (t sz o -spst s rsony s rtv
4 to t sz o t suon) n vn n qurs r ous on on suon or pro o t or stn to rnt suon. For xp n Fur (rt), t usr y ous ttnton on sunt S wos strutur s. Atr svr ps o vn r ntr n qurs r ssu to ts sunt, t usr y st ttnton to t sunt S. T nrn oprtons o MSBNs w tn propt vn ro S to S n tn to S. T usr n tn ntr vn on vrs ontn n S. It n sown tt wt su rstrt propton urn ttnton st, t nswrs to qurs otn n S r wys onsstnt to vn uut n t ntr MSBN. Coputton opxty, owvr, s ru y not vn to upt ny sunts not on t yprpt ro t urrnt sunt to t nxt trt sunt. Appton ons o sn-usr MSBNs nu noss o ntur systs [0] n o-s noss o rt systs [5]. Snsors Bysn Sunt son Mr Atutor Snstvty Anyzr Rsonr Countor strut Go Strutur Vrr Fur : Mn oponnts o n nt n MSBN-s ut-nt syst. MSBNs n xtn nto rwor or prost rsonn n ooprtv ut-nt strut ntrprtton systs. E nt os ts prt prsptv o r pro on (Sunt n Fur ), sss o vn sour (Snsors n Fur ), ounts wt otr nts nrqunty (Countor), rsons wt t o vn n t o vn (Rsonr), n nswrs qurs (Rsonr) or ts tons (son Mr/Atutor). It n sown [8] tt nts r ooprtv (vs s-ntrst), n pr o nt nts r ontony npnnt vn tr sr vrs n v oon nt on t sr vrs, tn ont syst s w n w s nt to nt s wtn ts suon n suppnt to t nt s outs t suon. Evn tou utp nts y qur vn synronousy n pr (opr wt t sn usr ornt syst wr vn s wys ntr nto t urrnt sunt o ous), t ounton oprtons o MSBNs nsur tt t nswrs to qurs ro nt r onsstnt wt vn qur n t ntr syst tr ounton. Sn ounton s nrqunt, t oprtons so nsur tt twn two sussv ountons, t nswrs to qurs or nt r onsstnt wt o vn tr so r n r onsstnt wt vn tr n t ntr syst up to t st ounton. Tror, MSBN n rtrz s on o untony urt, ooprtv strut systs []. Potnt pptons nu son support to ooprtv un Atou MSBNs r not rrn rty, t rprsntton ors us s sp s o MSBNs. For xp, t st o nput nos I, output no O, o no M, n uy no [5], w ors n ntr twn r v n owr v n t rry, s -spst []. T opost ont tr [5] orrspons to t yprtr []. T wy n w nrn s pror n t opost on tr orrspons to t oprton StAttnton []. 4
5 usrs n unrtn ons n trousootn opx syst y utp now s susysts [8]. Wy strut Vrton? As n n Ston, t strutur o MSBN s yprtr MSAG w sou AG. Autot vrton o yty o ts strutur s sr n t onstruton o r MSBNs. Aorts tt tst wtr rt rp s AG s on topoo sortn r w nown, s or xp [6]. Ts orts, owvr, ssu ntr rprsntton o t rp strutur to tst. A ntr rprsntton o AGs n MSBN s not sr or t st two rsons. Frst, t onstruton o ut-nt MSBN rqurs ony t now o t untonty o sunt n t ntr (-spst) twn sunts (BNs). Known t ntrn strutur o sunt s not nssry. Tror, sunt y vop y n npnnt vnor wo y not wn to sos t strutur ts. T ssupton o ntr rprsntton o AGs w nt t possty o ooprtn nts ut y su vnors. Sony, MSBN n potnty yn. Tt s, sunts y on or v t MSBN s t syst s untonn. It s sr to vry t orrtnss o t strutur o t syst wnvr t r sunts n. It s so sr tt t vrton os not rqur t ounton o AGs to ntr oton or os not pn upon sn nt to ntn rpostory o AGs n t urrnt syst. In ts ppr, w propos strut ort or vrton o t yty o MSBN strutur. urn t vrton pross, nt ony provs nswrs to nt sunts on qustons rrn -spnos, n t os not rv ts ntrn strutur yon tt. 4 Issus n strut Vrton R ro nton tt yprtr MSAG s ut ro st o AGs sut to t -spst n o ovrn ontons. Howvr rt rp ut ro st o AGs sut to ts two ontons y st ontn rt ys. W s rr to t rsutnt rp s yprtr AG unon sn t y not quy s yprtr MSAG. Fur : A y AG unon. 5
6 Fur sows two AGs n wt tr -spst n {, }. I w unon t two AGs, t ry stss t o ovrn onton. Howvr, t unon ontns t rt y (,,,,, ) n tus s not AG. T ov y n tt w unon t pr o AGs n tst t yty. Atou t prws vrton y tt so rt ys, prws yty n yprtr AG unon os not urnt t o yty. n o Fur 4: Tr AGs w r prws y ut wos unon s y. Consr t tr AGs n Fur 4. T unon o n s y n so s t unon o n. Howvr, wn t tr AGs r unon, rt y {,,,, n,,,,, } s or. Cry, strut vrton o yty rqurs ooprton yon prs. 5 Vrton y Mrn Nos In ts ston, w sow tt yty o rt rp n vr y rn root n nos rursvy. On t s sts, w n r non--spnos oy n r -spnos y ooprton s prsnt n t nxt ston. A no s root r onnt to t s rt wy ro t. A no s r onnt to t s rt towr t. A no x s r x n rs onnt to x r nor ro urtr vrton pross. T oown two propostons sow tt rn o root/ nos os not n yty. Proposton 4 Lt G rt rp n x tr root or n G. Tn t yty o G rns tr x s r. Proo: I G s y, tn rn x nnot rt rt y n G. Suppos G s y. Tn tr xsts non-pty st O o rt ys n G. Ix s root, t os not v ny non r. I x s, t os not v ny outon r. Tror, x nnot prtpt n ny ys n O, w ps tt non o t ys n O w n tr x s r. On root or s r, otr nos y o roots or vs. Hn rn roots n vs n pror rursvy w prsrvn t yty. 6
7 Nxt, w sow tt rt rp s y, vry no n t w r y rursv pptons o Proposton 4. On t otr n, t s y, t st tr nos w t unr. Proposton 5 Lt G rt rp. G s y t s pty tr rursv rn o roots n vs. Proo: Wtout osn nrty, w ssu tt G s t st two nos n s onnt. Suppos G s y. Tn G s t st on root n on. Aorn to Proposton 4, tr o t r r, t rsutnt rp s st y n s nw roots n vs. Sn G s nt nur o nos, tr rursv rn o roots n vs, vntuy G w v no unr nos. Nxt, suppos G s y. Tn G s t st on rt y θ onsstn o t st tr nos. For no x n θ, t s ntr root nor n tus nnot r s su. Mrn o ny nos outs θ nnot turn x nto root or. Hn non o t nos n θ n r y rursv rn o roots n vs. Sn G s nt nur o nos, tr rursv rn o roots n vs, vntuy tr w no roots or vs to r n G w nos (t st tr) n θ r unr. W now onsr yprtr AG unon G wos nos r ss nto -spnos n non--spnos. Sn G s onnt rt rp, Proposton 5 n pp to tst ts yty. Howvr, tou non--spno roots n vs n ronz oy wtn sunt, -spno roots n vs n ony ronz trou ooprton on sunts. For xp, t no ( non--spno) n Fur 4 s ot n (pprn r oy) n n t AG unon (n r oy). On t otr n, ( -spno) s n (pprn r oy), root n (pprn r oy), ut non-root/non- n t AG unon (n not r oy). Morovr, rn o -spno roots n vs y turn so non--spnos nto nw roots or vs. T oown proposton sows tt rursv n trnt rn o non--spno roots/vs n -spno roots/vs s sunt to tst t yty o yprtr AG unon. Corory 6 Lt G yprtr AG unon. Lt G t rp rsutn ro rursv n trnt rn o non--spno roots/vs n -spno roots/vs n G unt no or nos n r. Tn G s y G s pty. Proo: Aorn to Proposton 5, G s y, t roun o rursv rn, tr so non--spno roots/vs or so -spno roots/vs n r, unt G s pty. I G s y, t roun o rursv rn, tr so non--spno roots/vs or so -spno roots/vs n r, unt ony nos n rt ys n G r t unr (t st tr). T trnt rn o -spnos n non--spnos s nssry. Otrws, t rn y t prtury vn G s y. To ustrt t nssty o trnt rn, onsr t y AG unon n Fur 5. Wtout usn trnt rn, w v ony two optons: () rursv rn o 7
8 non--spno roots/vs oow y rursv rn o -spno roots/vs, or () rursv rn o -spno roots/vs oow y rursv rn o non--spno roots/vs. Usn opton (), non--spnos (root) n () n w r n t rst st. Ntr o t non--spnos n n n r t ts st. In t son st, t -spnos (now root) n (now ) n r. T rn trnts wt n unr. Usn opton (), no -spno n r n t rst st sn ntr nor s root or. In t son st, non--spnos (root) n () n w r. T rn trnts wt,, n unr. Usn trnt rn y strtn wt non--spnos, n w r n t rst st. In t son st, -spnos n w r. In t tr st, non--spnos n w r. Now t rsutnt rp s pty. Atrnt rn y strtn wt -spnos vs t s rsut. Fur 5: A y AG unon. Not tt t orory os vn wn G s n rtrry AG unon (not sut to t two ontons n nton ), n -spnos r rp y sr nos. Su nrty s not n or our purpos. Corory 6 ors t ss or strut vrton ort w w prsnt n Ston 6. 6 Cooprtv Vrton As onstrt n Stons 4 n 5, n orr to vry t yty o yprtr AG unon, nts ust ooprt. Sn ooprton rqurs ounton w nurs ovr, t s sr to spy t ts or ooprton s u s poss. Aorn to Corory 6, non--spno roots/vs n yprtr AG unon n r sprty n rursvy. W n prprossn oprton to r ts nos or ooprton strts. Lt AG n yprtr AG unon G rtrry osn. I w trt ts AG s t root o t yprtr n rt t yprns o t yprtr wy ro t, tn t yprtr s onvrt nto rt tr. For vn AG, w n tn rr to nt AG s ts or ts prnt n t nor sns. Oprton 7 (PrPross) Wn PrPross s n AG, t oown r pror: 8
9 . rursvy rs non--spno root or.. s PrPross n AG. Atr PrPross s opt n G, nos t unr n AG r tr sot -spnos, or nos tt or rt pts n wt -spnos. Cooprton on AGs s n to urtr t vrton pross. Fur 6 sows t tr AGs n Fur 4 tr PrPross s ntt n ny o t. Ony rt pts r t n ts s. W w s sot -spnos n tr xp. n o Fur 6: T AG unon n Fur 4 tr o prprossn. Mr nos r sown s ry. To n out -spno x n r, AG uss t oprton CotFyIno to trn x s root or trou ooprton. T oprton psss trp (x, p, ) roun AGs w ontn x. T purpos s to ot t prnt/ norton or x, wr p s ount o t nur o AGs tt ontn prnts o x n s ount o t nur o AGs tt ontn rn o x. T r n t oown nton rrs to tr prnt AG or t nxt r v o oprton w ntt ts oprton. Oprton 8 (CotFyIno) Wn CotFyIno(x) s n AG, t oown r pror:. ors trp t 0 =(x, p 0, 0 ), wr p 0 = ontns n (unr) prnt o x n p 0 =0otrws, n 0 = ontns n (unr) o x n 0 =0 otrws.. I s no AG to w x s -spno, or p 0 = 0 =, tn rturns t 0 to r.. Otrws, s CotFyIno(x) n AG to w x s - spno. 4. Atr AG n s rturn tr trps (ssun AGs r ), t,t,...,t, rturns trp t =(x, p = x =0 p,= x =0 ) to r. On -spno x s trn to root (p = 0) or ( = 0), t oprton strutmr s us to r t n vry AG tt ontns t. 9
10 Oprton 9 (strutmr) Wn strutmr(x) s n AG, t oown r pror:. rs t no x.. rursvy rs ny non--spno root or.. I s ny nt AG to w x s -spno xpt r, tn s strutmr(x) n o t. Not tt n ts oprton t rn o -spno s trnt wt t rn o non--spnos s rqur y Corory 6. T oprton MrNo ons CotFyIno n strutmr to pror on roun o rn o -spnos. It rs -spno root/ own t yprtr, n trnts rn wt t rn o non--spno roots/vs n AG. Oprton 0 (MrNo) Wn MrNo s n AG, t oown r pror:. rturns s t s no AG, otrws ontnus.. For unr -spno x wt AG o, s CotFyIno(x) n ts. Wn t trp (x, p, ) s rturn to, s strutmr(x) n ts p =0or =0.. s MrNo n AG. 4. I ny AG rturns tru or strutmr(x) ws n, tn rturns tru to r. Otrws, rturns s (no no s r). T oprton MrA s wtr nos n yprtr AG unon v n r tr roots n vs v n rursvy r. Aorn to Corory 6, G s y tru s rturn. Oprton (MrA) Wn MrA s n AG, t oown r pror:. I tr xsts no n tt s not n r, tn rturns s.. Otrws, s no AG, t rturns tru. I s AGs, s MrA n AG.. I ny AG rturns s (wt unr nos), tn rturns s. Otrws, rturns tru. Fny, t top v oprton TstAyty ons t prvousy n oprtons to vry t yty o G. Oprton (TstAyty) Wn TstAyty s ntt n yprtr AG unon G, t oown r pror: 0
11 . A AG s rtrry osn s t root o t yprtr.. s PrPross n ts.. s MrNo n ts rpty unt s s rturn (no no s r n t st ). 4. s MrA n ts. I tru s rturn, tn TstAyty rturns y (G s y). Otrws, rturn y. T oown tor stss t orrtnss o t ort. Tor T oprton TstAyty orrty trns t yty o yprtr AG unon. Proo: Aorn to Corory 6, t s sunt to r non--spno roots/vs n - spno roots/vs rursvy n trnty. PrPross os t rst roun o rursv rn o non--spno roots/vs, n rpt MrNo prors t susqunt rursv n trnt rn. E MrNo nts -spno roots/vs y CotFyIno (tr p =0or = 0) n tn rs t s w s nw non-spno roots/vs y strutmr. By Corory 6, MrNo w not rturn s unt roots n vs r r. MrA tsts t AG unon s pty n w trn t yty orrty. W ustrt t prorn o TstAyty wt two xps, y AG unon n n y on. T rst s t AG unon pt n Fur 4. Suppos s st s t root. Atr PrPross t unon oos s Fur 6. Wn s MrNo n ts, t s n ts CotFyIno() w s tn propt to. rturns trp (,, 0) to. Susqunty, nrts t n trp (,, ) n trnts CotFyIno(). Sn ntr p nor s zro, strutmr() s not. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). s MrNo n. s n ts CotFyIno() w s tn propt to. rturns trp (,, 0) to. Susqunty, nrts t n trp (,, ) n trnts CotFyIno(). NostrutMr s. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). s MrNo n w rturns s sn s no AG. Ts uss to rturn s to w so rturns s n trnts MrNo. s MrA n ts n rturns s ty. TstAyty tn trnts wt y rturn.
12 n o Fur 7: An y AG unon. As notr xp, onsr t AG unon n Fur 7. It s nt to tt n Fur 4 xpt tt t r ro to s now rvrs. o () n n o () () n o () o n n o () Fur 8: Prorn o TstAyty n t AG unon o Fur 7. Suppos s st s t root. Fur 8 () sows t unon tr PrPross. Not tt nos n r sot n ut nnot r urn PrPross. PrPross s so n sn n sot -spno n on AG y st prtpt n rt y n otr AGs. It wou nonsstnt to r t n on AG n to p t unr n notr. Wn MrNo s rst n wtn TstAyty, t s CotFyIno() n ts n tn CotFyIno() s n t prvous xp.
13 s MrNo n. s n ts CotFyIno() w s tn propt to. rturns trp (, 0, 0) to. Susqunty, nrts t n trp (, 0, ) n trnts CotFyIno(). Sn p =0, s strutmr() n ts. It rs n, n tn s strutmr() n w rs s w. Fur 8 () sows t rsutnt unon. tn s n ts CotFyIno() w vntuy trnts sry s CotFyIno(). Fur 8 () sows t rsutnt unon. s MrNo n w rturns s. Sn strutmr ws n, t rturns tru to w rturns tru n trnts t rst o MrNo n TstAyty. Wn MrNo s n t son t wtn TstAyty, s n ts CotFyIno() w s tn propt to. rturns trp (, 0, 0) to. Susqunty, nrts t n trp (, 0, ) n trnts CotFyIno(). Sn p =0, s strutmr() n ts. Ts uss t rn o,, n n n. Fur 8 () sows t rsutnt unon. tn s n ts CotFyIno(). It vntuy trnts sry s CotFyIno() wt r n ot n. Fur 8 () sows t rsutnt unon. s MrNo n w tn s MrNo n. s no AG n rturns s, w uss to rturn s to. Sn strutmr ws n, t rturns tru n trnts t son o MrNo n TstAyty. Wn MrNo s n t tr t wtn TstAyty, t propts MrNo to n tn to. Evntuy, s s rturn. Wn MrA s n, t propts t oprton to t rst o t unon. Evntuy, tru s rturn. T oprton TstAyty trnts wt y rturn. 7 Copxty Anyss W not t xu nur o nos n AG y, t xu nur o nt nos o no n AG y t, t xu nur o nos n -spst y, t xu nur o AGs tt y ontn -spno y s, n t tot nur o AGs n t yprtr AG unon y n. To rursvy r non--spnos n AG, O(t ) nos n to. Hn PrPross s O(n t) nos. E CotFyIno (rtv to sn -spno) tsts -spno n O(s) AGs. To r -spno, O(s) AGs pror t rn, o w so s
14 O(t ) non--spnos. Hn t opxty o CotFyIno n strutmr or r -spno s O(s t). T opxty o MrNo ro TstAyty s tn O(n st). Sn t st on -spno w r or o MrNo, Mrno w O(n ) ts. Hn t opxty o MrNo s s O(n st). MrA s O(n) nos. Tror, t worst s opxty o TstAyty s O(n st). 8 susson In ts ppr, w prsnt n nt strut ort, or vrton o yty o t ovr strutur o MSBN. An portnt tur o t ort s tt t os not rqur sunt/nt n t syst to rv ts ntrn strutur. Fro t nton o CotFyIno, ry nt ony provs norton rrn wtr sr no s ny prnt or n t AG tt t nt s rspons or. Tror, t ort supports t onstruton o MSBNs onstrut ro utp oputton nts ut y utp vnors w provn utot vrton o orrtnss o t ovr strutur. Our strut ort s s on Corory 6 w n turn s s on Proposton 5. Proposton 5 xtns t o topoo sortn n tt t ttr s quvnt to rn ony root nos. Evn n ntrz tst, Proposton 5 ows or nt tst tn topoo sortn. Ts s us Proposton 5 ts on vr tw s ny r nos t rursv rn, vn ss nos to n susqunt rn prosss. Atou topoo sortn n so xtn nto strut ort, t vnt o TstAyty s or pronnt n ooprtv tst. Sn MrNo rs or nos n roun tn topoo sortn os, ny ss s o MrNo w (u or nt ooprton), w trnsts nto ruton n ounton ovr. As nton n Ston, strut vrton tts yn orton o MSBN. In nr, TstAyty sou pror wnvr t r sunts o MSBN n. Howvr, tr r sp ss wr t xuton o u s TstAyty s unnssry. For xp, nw sunt ony ntrs wt on xstn sunt n t -spst twn t ontns root nos ony, tn t yty o t nw AG unon n onr oy. On usu rton or utur rsr s to nty su sp ss n to vop sp vrton oprtons orny. Anownts Ts wor s support y t Rsr Grnt OGP05545 ro t Ntur Sns n Ennrn Rsr Coun (NSERC) o Cn. Hpu onts on n rr rt ro T. Cu n S. Huntr r now. 4
15 Rrns [] E. Crn. Bysn ntwors wtout trs. AI Mzn, (4):50 6, 99. [] V.R. Lssr n.. Cor. Funtony urt, ooprtv strut systs. IEEE Trns. on Systs, Mn n Cyrnts, SMC-():8 96, 98. [] R.E. Npotn. Prost Rsonn n Exprt Systs. Jon Wy n Sons, 990. [4] J. Pr. Prost Rsonn n Intnt Systs: Ntwors o Pus Inrn. Morn Kunn, 988. [5] S. Srnvs. A prost ppro to rr o-s noss. In Pro. 0t Con. Unrtnty n Art Intn, ps , Stt, Wsnton, 994. [6].F. Stus n N.W. Wr. t Struturs wt Astrt t Typs n Mou-. Broos/Co, 987. [7] Y. Xn. Optzton o ntr-sunt uptn n utpy ston Bysn ntwors. In Pro. t Con. on Unrtnty n Art Intn, ps , Montr, 995. [8] Y. Xn. A prost rwor or ooprtv ut-nt strut ntrprtton n optzton o ounton. Art Intn, 87(-):95 4, 996. [9] Y. Xn. Snts o utpy ston Bysn ntwors or ooprtv ut-nt strut ntrprtton. In G. MC, tor, Avns n Art Intn, ps 6. Sprnr, 996. [0] Y. Xn, B. Pnt, A. Esn, M. P. Bos, n. Poo. Mutpy ston Bysn ntwors or nurousur noss. Art Intn n Mn, 5:9 4, 99. [] Y. Xn,. Poo, n M. P. Bos. Exporn oty n Bysn ntwors or r xprt systs. In Pro. 8t Con. on Unrtnty n Art Intn, ps 44 5, Stnor, 99. [] Y. Xn,. Poo, n M. P. Bos. Mutpy ston Bysn ntwors n unton orsts or r now s systs. Coputton Intn, 9():7 0, 99. 5
Theorem 1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.
Cptr 11: Trs 11.1 - Introuton to Trs Dnton 1 (Tr). A tr s onnt unrt rp wt no sp ruts. Tor 1. An unrt rp s tr n ony tr s unqu sp pt twn ny two o ts vrts. Dnton 2. A root tr s tr n w on vrtx s n snt s t
More informationClosed Monochromatic Bishops Tours
Cos Monoromt Bsops Tours Jo DMo Dprtmnt o Mtmts n Sttsts Knnsw Stt Unvrsty, Knnsw, Gor, 0, USA mo@nnsw.u My, 00 Astrt In ss, t sop s unqu s t s o to sn oor on t n wt or. Ts ms os tour n w t sop vsts vry
More informationGraphs Depth First Search
Grp Dpt Frt Sr SFO 337 LAX 1843 1743 1233 802 DFW ORD - 1 - Grp Sr Aort - 2 - Outo Ø By unrtnn t tur, you ou to: q L rp orn to t orr n w vrt r ovr, xpor ro n n n pt-rt r. q Cy o t pt-rt r tr,, orwr n ro
More informationApplications of trees
Trs Apptons o trs Orgnzton rts Attk trs to syst Anyss o tr ntworks Prsng xprssons Trs (rtrv o norton) Don-n strutur Mutstng Dstnton-s orwrng Trnsprnt swts Forwrng ts o prxs t routrs Struturs or nt pntton
More informationLecture 20: Minimum Spanning Trees (CLRS 23)
Ltur 0: Mnmum Spnnn Trs (CLRS 3) Jun, 00 Grps Lst tm w n (wt) rps (unrt/rt) n ntrou s rp voulry (vrtx,, r, pt, onnt omponnts,... ) W lso suss jny lst n jny mtrx rprsntton W wll us jny lst rprsntton unlss
More informationThe University of Sydney MATH 2009
T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n
More informationCMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk
CMPS 2200 Fll 2017 Grps Crol Wnk Sls ourtsy o Crls Lsrson wt ns n tons y Crol Wnk 10/23/17 CMPS 2200 Intro. to Alortms 1 Grps Dnton. A rt rp (rp) G = (V, E) s n orr pr onsstn o st V o vrts (snulr: vrtx),
More informationPlanar convex hulls (I)
Covx Hu Covxty Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P A oyo P s ovx or y, P, t st s try P. Pr ovx us (I) Coutto Gotry [s 3250] Lur To Bowo Co ovx o-ovx 1 2 3 Covx Hu Covx Hu Covx Hu
More informationL.3922 M.C. L.3922 M.C. L.2996 M.C. L.3909 M.C. L.5632 M.C. L M.C. L.5632 M.C. L M.C. DRIVE STAR NORTH STAR NORTH NORTH DRIVE
N URY T NORTON PROV N RRONOUS NORTON NVRTNTY PROV. SPY S NY TY OR UT T TY RY OS NOT URNT T S TT T NORTON PROV S ORRT, NSR S POSS, VRY ORT S N ON N T S T TY RY. TS NORTON S N OP RO RORS RT SU "" YW No.
More informationDepth First Search. Yufei Tao. Department of Computer Science and Engineering Chinese University of Hong Kong
Dprtmnt o Computr Sn n Ennrn Cns Unvrsty o Hon Kon W v lry lrn rt rst sr (BFS). Toy, w wll suss ts sstr vrson : t pt rst sr (DFS) lortm. Our susson wll on n ous on rt rps, us t xtnson to unrt rps s strtorwr.
More informationOn Hamiltonian Tetrahedralizations Of Convex Polyhedra
O Ht Ttrrzts O Cvx Pyr Frs C 1 Q-Hu D 2 C A W 3 1 Dprtt Cputr S T Uvrsty H K, H K, C. E: @s.u. 2 R & TV Trsss Ctr, Hu, C. E: q@163.t 3 Dprtt Cputr S, Mr Uvrsty Nwu St. J s, Nwu, C A1B 35. E: w@r.s.u. Astrt
More informationHaving a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall
Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s
More informationWeighted Graphs. Weighted graphs may be either directed or undirected.
1 In mny ppltons, o rp s n ssot numrl vlu, ll wt. Usully, t wts r nonntv ntrs. Wt rps my tr rt or unrt. T wt o n s otn rrr to s t "ost" o t. In ppltons, t wt my msur o t lnt o rout, t pty o ln, t nry rqur
More information5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees
/1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our
More informationDetecting Change in Snapshot Sequences
Dttn Cn n Snpsot Squns Mnzn S n Stpn Wntr Dprtnt o Gots, Unvrsty o Mourn, Vtor, 3010, ustr.s@pr.un.u.u, wntr@un.u.u strt. Wrss snsor ntwors r poy to ontor yn orp pnon, or ots, ovr sp n t. Ts ppr prsnts
More informationIn which direction do compass needles always align? Why?
AQA Trloy Unt 6.7 Mntsm n Eltromntsm - Hr 1 Complt t p ll: Mnt or s typ o or n t s stronst t t o t mnt. Tr r two typs o mnt pol: n. Wrt wt woul ppn twn t pols n o t mnt ntrtons low: Drw t mnt l lns on
More informationImproving Union. Implementation. Union-by-size Code. Union-by-Size Find Analysis. Path Compression! Improving Find find(e)
POW CSE 36: Dt Struturs Top #10 T Dynm (Equvln) Duo: Unon-y-Sz & Pt Comprsson Wk!! Luk MDowll Summr Qurtr 003 M! ZING Wt s Goo Mz? Mz Construton lortm Gvn: ollton o rooms V Conntons twn t rooms (ntlly
More informationMinimum Spanning Trees (CLRS 23)
Mnmum Spnnn Trs (CLRS 3) T prolm Rll t nton o spnnn tr: Gvn onnt, unrt rp G = (V, E), sust o s o G su tt ty onnt ll vrts n G n orm no yls s ll spnnn tr (ST) o G. Any unrt, onnt rp s spnnn tr. Atully, rp
More informationSAMPLE CSc 340 EXAM QUESTIONS WITH SOLUTIONS: part 2
AMPLE C EXAM UETION WITH OLUTION: prt. It n sown tt l / wr.7888l. I Φ nots orul or pprotng t vlu o tn t n sown tt t trunton rror o ts pproton s o t or or so onstnts ; tt s Not tt / L Φ L.. Φ.. /. /.. Φ..787.
More informationSheet Title: Building Renderings M. AS SHOWN Status: A.R.H.P.B. SUBMITTAL August 9, :07 pm
1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 orthstar expressly reserves its common law copyright and other property rights for all ideas, provisions and plans represented or indicated by these drawings,
More informationCMSC 451: Lecture 4 Bridges and 2-Edge Connectivity Thursday, Sep 7, 2017
Rn: Not ovr n or rns. CMSC 451: Ltr 4 Brs n 2-E Conntvty Trsy, Sp 7, 2017 Hr-Orr Grp Conntvty: (T ollown mtrl ppls only to nrt rps!) Lt G = (V, E) n onnt nrt rp. W otn ssm tt or rps r onnt, t somtms t
More informationDivided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano
RIGHT-ANGLE WEAVE Dv mons Mm t look o ts n rlt tt s ptvly p sn y Py Brnkmn Mttlno Dv your mons nto trnls o two or our olors. FCT-SCON0216_BNB66 2012 Klm Pulsn Co. Ts mtrl my not rprou n ny orm wtout prmsson
More informationAn action with positive kinetic energy term for general relativity. T. Mei
An ton wt post nt ny t fo n tty T (Dptnt of Jon Cnt Cn o Unsty Wn H PRO Pop s Rp of Cn E-: to@nn tow@pwn ) Astt: At fst w stt so sts n X: 7769 n tn sn post nt ny oont onton n y X: 7769 w psnt n ton wt
More informationComputer Graphics. Viewing & Projections
Vw & Ovrvw rr : rss r t -vw trsrt: st st, rr w.r.t. r rqurs r rr (rt syst) rt: 2 trsrt st, rt trsrt t 2D rqurs t r y rt rts ss Rr P usuy st try trsrt t wr rts t rs t surs trsrt t r rts u rt w.r.t. vw vu
More information(Minimum) Spanning Trees
(Mnmum) Spnnn Trs Spnnn trs Kruskl's lortm Novmr 23, 2017 Cn Hrn / Gory Tn 1 Spnnn trs Gvn G = V, E, spnnn tr o G s onnt surp o G wt xtly V 1 s mnml sust o s tt onnts ll t vrts o G G = Spnnn trs Novmr
More informationSpanning Trees. BFS, DFS spanning tree Minimum spanning tree. March 28, 2018 Cinda Heeren / Geoffrey Tien 1
Spnnn Trs BFS, DFS spnnn tr Mnmum spnnn tr Mr 28, 2018 Cn Hrn / Gory Tn 1 Dpt-rst sr Vsts vrts lon snl pt s r s t n o, n tn ktrks to t rst junton n rsums own notr pt Mr 28, 2018 Cn Hrn / Gory Tn 2 Dpt-rst
More informationVOID CABLE REQUIREMENTS
XST & W OTOS 0 SURTY SYOS Orange ounty onvention enter PORTT OTS R OTS W TV R # PX TYP RQURS /" OUT RO R OTO T..., WT US. OR RS OT OT SS, OUT T T, U.O.. W 0 R POR R 0 R R RQURS O () TYP. PX TYP #. OTRTOR
More informationPriority Search Trees - Part I
.S. 252 Pro. Rorto Taassa oputatoal otry S., 1992 1993 Ltur 9 at: ar 8, 1993 Sr: a Q ol aro Prorty Sar Trs - Part 1 trouto t last ltur, w loo at trval trs. or trval pot losur prols, ty us lar spa a optal
More informationSAN JOSE CITY COLLEGE PHYSICAL EDUCATION BUILDING AND RENOVATED LAB BUILDING SYMBOL LIST & GENERAL NOTES - MECHANICAL
S SRUUR OR OORO SU sketch SYO SRPO OO OU O SUPPOR YP SUPPOR O UR SOS OVR SOS (xwx) X W (S) RR RWS U- R R OR ROO OR P S SPR SO S OS OW "Wx00"x0" 000.0, 8/7.0 Z U- R R UPPR ROO S S S SPR SO S OS OW 0"Wx0"x90"
More information4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling.
Cptr 4 4 Intrvl Suln Gry Alortms Sls y Kvn Wyn Copyrt 005 Prson-Ason Wsly All rts rsrv Intrvl Suln Intrvl Suln: Gry Alortms Intrvl suln! Jo strts t s n nss t! Two os omptl ty on't ovrlp! Gol: n mxmum sust
More informationNORTHLAKE APARTMENTS
PRT RW RVTO VTY P PROT RPTO TR # UR P PROPRTY.. /..T..... T. R. V..W.P.. '. OT......... R...U..O. O. O. OT. OOR. PT..T..Y...... P. V.. R... W....... V.. Q. QUP..W.. XT.....O.................. OR OT R OTO
More informationStrongly connected components. Finding strongly-connected components
Stronly onnt omponnts Fnn stronly-onnt omponnts Tylr Moor stronly onnt omponnt s t mxml sust o rp wt rt pt twn ny two vrts SE 3353, SMU, Dlls, TX Ltur 9 Som sls rt y or pt rom Dr. Kvn Wyn. For mor normton
More informationCHELOURANYAN CALENDAR FOR YEAR 3335 YEAR OF SAI RHAVË
CHELOURANYAN CALENDAR FOR YEAR YEAR OF SAI RHAVË I tou woust n unon wt our Motr, now tt tou st nvr t Hr. I tou woust sp t v o mttr, now tt tr s no mttr n no v. ~Cry Mry KEY TO CALENDAR T Dys o t W In t
More informationd e c b a d c b a d e c b a a c a d c c e b
FLAT PEYOTE STITCH Bin y mkin stoppr -- sw trou n pull it lon t tr until it is out 6 rom t n. Sw trou t in witout splittin t tr. You soul l to sli it up n own t tr ut it will sty in pl wn lt lon. Evn-Count
More informationExam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms
CS 542 Avn Dt Stutu n Alotm Exm 2 Soluton Jontn Tun 4/2/202. (5 ont) Con n oton on t tton t tutu n w t n t 2 no. Wt t mllt num o no tt t tton t tutu oul ontn. Exln you nw. Sn n mut n you o u t n t, t n
More informationGraphs Breadth First Search
Grp Brdt Frt Sr SFO ORD LAX DFW - 1 - Outo Ø By undrtndn t tur, you oud to: q L rp ordn to t ordr n w vrt r dovrd n rdt-rt r. q Idnty t urrnt tt o rdt-rt r n tr o vrt tt r prvouy dovrd, ut dovrd or undovrd.
More informationThe R-Tree. Yufei Tao. ITEE University of Queensland. INFS4205/7205, Uni of Queensland
Yu To ITEE Unvrsty o Qunsln W wll stuy nw strutur ll t R-tr, w n tout o s mult-mnsonl xtnson o t B-tr. T R-tr supports ntly vrty o qurs (s w wll n out ltr n t ours), n s mplmnt n numrous ts systms. Our
More informationDental PBRN Study: Reasons for replacement or repair of dental restorations
Dntl PBRN Stuy: Rsons or rplmnt or rpr o ntl rstortons Us ts Dt Collton Form wnvr stuy rstorton s rpl or rpr. For nrollmnt n t ollton you my rpl or rpr up to 4 rstortons, on t sm ptnt, urn snl vst. You
More informationPlatform Controls. 1-1 Joystick Controllers. Boom Up/Down Controller Adjustments
Ston 7 - Rpr Prours Srv Mnul - Son Eton Pltorm Controls 1-1 Joystk Controllrs Mntnn oystk ontrollrs t t propr sttns s ssntl to s mn oprton. Evry oystk ontrollr soul oprt smootly n prov proportonl sp ontrol
More informationGraph Search Algorithms
Grp Sr Aortms 1 Grp 2 No ~ ty or omputr E ~ ro or t Unrt or Drt A surprsny r numr o omputton proms n xprss s rp proms. 3 Drt n Unrt Grps () A rt rp G = (V, E), wr V = {1,2,3,4,5,6} n E = {(1,2), (2,2),
More informationLecture II: Minimium Spanning Tree Algorithms
Ltur II: Mnmum Spnnn Tr Alortms Dr Krn T. Hrly Dprtmnt o Computr Sn Unvrsty Coll Cork Aprl 0 KH (/0/) Ltur II: Mnmum Spnnn Tr Alortms Aprl 0 / 5 Mnmum Spnnn Trs Mnmum Spnnn Trs Spnnn Tr tr orm rom rp s
More informationMinimum Spanning Trees (CLRS 23)
Mnmum Spnnn Trs (CLRS 3) T prolm Gvn onnt, unrt rp G = (V, E), sust o s o G su tt ty onnt ll vrts n G n orm no yls s ll spnnn tr (ST) o G. Clm: Any unrt, onnt rp s spnnn tr (n nrl rp my v mny spnnn trs).
More informationBASIC CAGE DETAILS SHOWN 3D MODEL: PSM ASY INNER WALL TABS ARE COINED OVER BASE AND COVER FOR RIGIDITY SPRING FINGERS CLOSED TOP
MO: PSM SY SI TIS SOWN SPRIN INRS OS TOP INNR W TS R OIN OVR S N OVR OR RIIITY. R TURS US WIT OPTION T SINS. R (UNOMPRSS) RR S OPTION (S T ON ST ) IMNSIONS O INNR SIN TO UNTION WIT QU SM ORM-TOR (zqsp+)
More informationAn Optimal and Progressive Algorithm for Skyline Queries
A Opt Prorssv Aort or S Qurs Dtrs Pps Yu To Gr Fu Brr Sr* Dprtt o Coputr S Ho Ko Uvrst o S Too Cr Wtr B, Ho Ko {trs,r}@s.ust. Dprtt o Coputr S Cr Mo Uvrst Pttsur, USA to@s.u.u * Dpt. o Mtts Coputr S Ppps-Uvrst
More informationT H E S C I E N C E B E H I N D T H E A R T
A t t R u r s - L x C t I. xtr turs t Lx Ct Rurs. Rr qurtr s s r t surt strutur. Ts Att Rurs rv ut us, s srt t tr t rtt rt yur t w yu ru. T uqu Lx st ut rv ss ts ss t t y rt t tys t r ts w wr rtts. Atrx
More informationDifferentiation of allergenic fungal spores by image analysis, with application to aerobiological counts
15: 211 223, 1999. 1999 Kuw Puss. Pt t ts. 211 tt u ss y yss, wt t t uts.. By 1, S. s 2,EuR.Tvy 2 St 3 1 tt Ss, R 407 Bu (05), Uvsty Syy, SW, 2006, ust; 2 st ty, v 4 Bu u (6), sttut Rsty, Uvsty Syy, SW,
More informationGraph Search (6A) Young Won Lim 5/18/18
Grp Sr (6A) Youn Won Lm Copyrt () 2015 2018 Youn W. Lm. Prmon rnt to opy, trut n/or moy t oumnt unr t trm o t GNU Fr Doumntton Ln, Vron 1.2 or ny ltr vron pul y t Fr Sotwr Founton; wt no Invrnt Ston, no
More informationIntroduction to Fourier optics. Textbook: Goodman (chapters 2-4) Overview:
Introuton to ourr opts Ttbook: Goon (ptrs -) Ovrv: nr n nvrnt ssts T ourr trnsor Slr rton rsnl n runor pprotons. . nr ssts n ourr trnsor tutorl (rnr) sst onnts n nput to n output su tt: It s s to b lnr
More informationCOMP 250. Lecture 29. graph traversal. Nov. 15/16, 2017
COMP 250 Ltur 29 rp trvrsl Nov. 15/16, 2017 1 Toy Rursv rp trvrsl pt rst Non-rursv rp trvrsl pt rst rt rst 2 Hs up! Tr wr w mstks n t sls or S. 001 or toy s ltur. So you r ollown t ltur rorns n usn ts
More informationA Simple Method for Identifying Compelled Edges in DAGs
A Smpl Mto or Intyn Compll Es n DAGs S.K.M. Won n D. Wu Dprtmnt o Computr Sn Unvrsty o Rn Rn Ssktwn Cn S4S 0A2 Eml: {won, nwu}@s.urn. Astrt Intyn ompll s s mportnt n lrnn t strutur (.., t DAG) o Bysn ntwork.
More informationA ' / 1 6 " 5 ' / 4 " A4.2 48' - 0" 3 12' - 7" 13' - 11" 10' - 0" 9' - 0" 2' - 6" 1. 2: 12 INDICATES SHOW MELT TYP ABV ABV
4. 4. 4. K ' - / " ' - / 4 " 0 ' - / " ' - 0 " ' - 0 " ' - / " 4 ' - 0 " 4. M U PPR 48' - 0" ' - ' - " 0' - 0" ' - 0" ' - ". : WOM ' - 0 " OT: PROV URROU TR OUT SVS OR UTUR SP UTTY T OR QUSTR MPUS OTO
More informationSOCKET WELD OR THREADED BODY TYPE (3051SFP FLOWMETER SHOWN THROUGHOUT / 3051CFP, 2051CFP AVAILABLE)
9 10 12 13 14 15 16 RVISION T RVISION O NO. PP' T SI1053953 3/30/17 03/31/17 SRIPTION NOT 10 N RIITION ON ST 10. T 1 - OY INSIONS 2X 1/4" NPT VNT VVS INSIONS IN SIZ 3.4 [86.0] 3.8 [97.0] 4.5 [4.0] 4.7
More informationDistributed Caching of Multi-dimensional Data in Mobile Environments
Dstrut Cn o Mut-mnson Dt n Mo Envronmnts Bn Lu Wn-Cn L D Lun L Dprtmnt o Computr Sn Hon Kon Unvrst o Sn n Tnoo Crwtr B, Hon Kon {un, }@s.ust. Dprtmnt o Computr Sn n Ennrn Pnnsvn Stt Unvrst Unvrst Pr, PA
More informationBASIC CAGE DETAILS D C SHOWN CLOSED TOP SPRING FINGERS INNER WALL TABS ARE COINED OVER BASE AND COVER FOR RIGIDITY
SI TIS SOWN OS TOP SPRIN INRS INNR W TS R OIN OVR S N OVR OR RIIITY. R IMNSIONS O INNR SIN TO UNTION WIT QU SM ORM-TOR (zqsp+) TRNSIVR. R. RR S OPTION (S T ON ST ) TURS US WIT OPTION T SINS. R (INSI TO
More information2 Trees and Their Applications
Trs n Tr Appltons. Proprts o trs.. Crtrzton o trs Dnton. A rp s ll yl (or orst) t ontns no yls. A onnt yl rp s ll tr. Quston. Cn n yl rp v loops or prlll s? Notton. I G = (V, E) s rp n E, tn G wll not
More information16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am
16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)
More informationTelecommunications 1-1/4" RACEWAY WITH DOUBLE-GANG ADAPTER PLATE AND A/V CABLING.
2 3 2 TNOOY SYMO ST NR TNOOY NOTS: NOT: This is a standard symbol list and not all items listed may be used. bbreviations () XSTN OV NS OOR NMW - UNRROUN ONUT T TORY ONTRTOR URNS ONTRTOR NST O ONTRTOR
More informationDetermination of slot leakage inductance for three-phase induction motor winding using an analytical method
ACHIVES OF EECTICA EGIEEIG VO 6 pp 569-59 DOI 78/--6 Dtnton of ot ntn fo t-p nton oto wnn n n nt to JA STASZAK Dptnt of Et Mn n Mton St K Unvt of Tnoo Tą PP 7 K Pon -: j@tp v: v: 5 Att: T t nto o pon fo
More informationFace Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction
F Dtto Roto Lr Alr F Roto C Y I Ursty O solto: tto o l trs s s ys os ot. Dlt to t to ltpl ws. F Roto Aotr ppro: ort y rry s tor o so E.. 56 56 > pot 6556- stol sp A st o s t ps to ollto o pots ts sp. F
More information, each of which is a tree, and whose roots r 1. , respectively, are children of r. Data Structures & File Management
nrl tr T is init st o on or mor nos suh tht thr is on sint no r, ll th root o T, n th rminin nos r prtition into n isjoint susts T, T,, T n, h o whih is tr, n whos roots r, r,, r n, rsptivly, r hilrn o
More informationTelecommunications BUILDING INTERCOM CALL BUTTON WITH 3/4"C AND PULL STRING TO ACCESSIBLE CEILING SPACE. MOUNT 48" AFF.
0 NOOY SYMO S N NOOY NOS: NO: his is a standard symbol list and not all items listed may be used. bbreviations () XSN OV NS OO NMW - UNOUN ONU OY ONO UNS ONO NS O ONO UNS OWN NS OX OX U OP SUON UN OO,
More informationDOCUMENT STATUS: LA-S5302-XXXXS LA, SSS, TRICEPS EXTENSION VERY
RVSON STORY RV T SRPTON O Y //0 RLS OR PROUTON T LN MR ----- L /0/0 UPT SN N OMPONNTS US: S 3-03 (*N TWO PLS ONLY) WS 3-5, PRT 3-00 TO SSMLY. T OLLOWN UPT: 3-30, 3-403, 3-403, 3-40, 3-45, 3-4, 3-5. 30
More informationThe Constrained Longest Common Subsequence Problem. Rotem.R and Rotem.H
T Constrn Lonst Common Susqun Prolm Rotm.R n Rotm.H Prsntton Outln. LCS Alortm Rmnr Uss o LCS Alortm T CLCS Prolm Introuton Motvton For CLCS Alortm T CLCS Prolm Nïv Alortm T CLCS Alortm A Dynm Prormmn
More informationMAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017
MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT
More informationBayesian belief networks: Inference
C 740 Knowd rprntton ctur 0 n f ntwork: nfrnc o ukrcht o@c.ptt.du 539 nnott qur C 750 chn rnn n f ntwork. 1. Drctd ccc rph Nod rndo vr nk n nk ncod ndpndnc. urr rthquk r ohnc rc C 750 chn rnn n f ntwork.
More information24CKT POLARIZATION OPTIONS SHOWN BELOW ARE REPRESENTATIVE FOR 16 AND 20CKT
0 NOTS: VI UNSS OTRWIS SPII IRUIT SMT USR R PORIZTION OPTION IRUIT SMT USR R PORIZTION OPTION IRUIT SMT USR R PORIZTION OPTION. NR: a. PPITION SPIITION S: S--00 b. PROUT SPIITION S: PS--00 c. PIN SPIITION
More informationMATERIAL SEE BOM ANGLES = 2 FINISH N/A
9 NOTS:. SSML N NSPT PR SOP 0-9... NSTLL K STKR N X L STKR TO NS O SROU WT TP. 3. PR-PK LNR RNS WT P (XTRM PRSSUR NL R ) RS OR NNRN PPROV QUVLNT. 4. OLOR TT Y T SLS ORR. RRN T MNS OM OR OMPONNTS ONTNN
More informationCSE 332. Data Structures and Parallelism
Am Blnk Ltur 20 Wntr 2017 CSE 332 Dt Struturs n Prlllsm CSE 332: Dt Struturs n Prlllsm Grps 1: Wt s Grp? DFS n BFS LnkLsts r to Trs s Trs r to... 1 Wr W v Bn Essntl ADTs: Lsts, Stks, Quus, Prorty Quus,
More informationCSE 332. Graphs 1: What is a Graph? DFS and BFS. Data Abstractions. CSE 332: Data Abstractions. A Graph is a Thingy... 2
Am Blnk Ltur 19 Summr 2015 CSE 332: Dt Astrtons CSE 332 Grps 1: Wt s Grp? DFS n BFS Dt Astrtons LnkLsts r to Trs s Trs r to... 1 A Grp s Tny... 2 Wr W v Bn Essntl ADTs: Lsts, Stks, Quus, Prorty Quus, Hps,
More informationDOCUMENT STATUS: MINTP0 E-ST5080, BASE, NO DISPLAY VENDOR: 15.5 INCH MATERIAL SEE BOM FINISH REVISION HISTORY ITEM NO. PART NUMBER DESCRIPTION
RV T RVSON STORY SRPTON O Y 0-0-0 PROUTON RLS K. N NOTS:. SRL LL NORMTON: a) VOLTS: V b) MPS:.0 c) YLS: N/ d) WTTS: W e) PS: N/ f) PX #: PX. RTTON LOOS: S / / LN R WT SOPROPYL LOLOL PROR TO PLN.. PK M:
More informationDOCUMENT STATUS: RELEASE
RVSON STORY RV T SRPTON O Y 0-4-0 RLS OR PROUTON 5 MM -04-0 NS TRU PLOT PROUTON -- S O O OR TLS 30 MM 03-3-0 3-044 N 3-45, TS S T TON O PROTTV RM OVR. 3 05--0 LT 3-004, NOT, 3-050 3 0//00 UPT ST ROM SN,
More informationUNCONTROLLED COPY. ERGON ENERGY SUBSTATION STANDARD CIVIL WORKS - STRUCTURES AND POLES 66kV EARTHING TRANSFORMER CABLE SUPPORT STEELWORK DETAILS
2 3 9 2 RRN RWNS 2 3 UNONTRO OPY RVSON ONTRO RON NRY SUSTTON STNR V WORKS STRUTURS N POS kv RTN TRNSORMR SUPPORT STWORK TS 9 2 x 9 3 O SUR RRSTOR PT 3=;.., PT t/lmm VNT O N WT NTRN O MOUNTN OTS!, q>. O
More informationTrader Horn at Strand This Week
- -N { 6 7 8 9 3 { 6 7 8 9 3 O OO O N U R Y Y 28 93 OU XXXX UO ONR ON N N Y OOR U RR NO N O 8 R Y R YR O O U- N O N N OR N RR R- 93 q 925 N 93; ( 928 ; 8 N x 5 z 25 x 2 R x q x 5 $ N x x? 7 x x 334 U 2
More informationMATERIAL SEE BOM ANGLES = 2 > 2000 DATE MEDIUM FINISH
NOTS:. LN MTN SUR WT NTUR/SOPROPYL LOOL PROR TO RN L OR LOO. PPLY LOTT 4 ON TRS. TORQU TO. Nm / 00 lb-in 4. TORQU TO 45-50 Nm / - lb-ft 5. TORQU TO Nm / 4.5 lb-ft. TORQU TO 0 Nm / lb-in. TORQU TO 5.5 Nm
More informationDesigning A Uniformly Loaded Arch Or Cable
Dsinin A Unirmy Ar Or C T pr wit tis ssn, i n t Nxt uttn r r t t tp ny p. Wn yu r n wit tis ssn, i n t Cntnts uttn r r t t tp ny p t rturn t t ist ssns. Tis is t Mx Eyt Bri in Stuttrt, Grmny, sin y Si
More informationGrade 7/8 Math Circles March 4/5, Graph Theory I- Solutions
ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n
More informationlearning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms
rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r
More informationMath 166 Week in Review 2 Sections 1.1b, 1.2, 1.3, & 1.4
Mt 166 WIR, Sprin 2012, Bnjmin urisp Mt 166 Wk in Rviw 2 Stions 1.1, 1.2, 1.3, & 1.4 1. S t pproprit rions in Vnn irm tt orrspon to o t ollowin sts. () (B ) B () ( ) B B () (B ) B 1 Mt 166 WIR, Sprin 2012,
More information/99 $10.00 (c) 1999 IEEE
P t Hw Itt C Syt S 999 P t Hw Itt C Syt S - 999 A Nw Atv C At At Cu M Syt Y ZHANG Ittut Py P S, Uvty Tuu, I 0-87, J Att I t, w tv t t u yt x wt y tty, t wt tv w (LBSB) t. T w t t x t tty t uy ; tt, t x
More informationPhysics 222 Midterm, Form: A
Pysis 222 Mitrm, Form: A Nm: Dt: Hr r som usul onstnts. 1 4πɛ 0 = 9 10 9 Nm 2 /C 2 µ0 4π = 1 10 7 tsl s/c = 1.6 10 19 C Qustions 1 5: A ipol onsistin o two r point-lik prtils wit q = 1 µc, sprt y istn
More informationPRECAST APPROACH SLAB NOTES
ULNS TS ULN RWNS RPRSNT TYPL TLS OR T SN N TLN O PRST PPRO SLS. TS STS R NLU TO PROV N XMPL O T RTN LYOUT O TYPL PRST PPRO SL. TWO RNT PPRO SL SYSTMS R SOWN: SUR PPRO SLS: SLS TT R PL WT T TOP SUR T OR
More informationTRASH ENCLOSURE WITH SOLID GATE 4 STORY BUSINESS / RESIDENTIAL BUILDING CONTAINING 2 BUSINESS SPACES AND 6 DWELLING UNITS 6' - 0"
NSN N. PUN WY R. P 0. SG S 4 SRY USNSS / RSN UNG NNNG USNSS SPS N 6 WNG UNS RS NSUR W S G.. RSRV PRKNG $50 N SGN RV S (7) UR PRKNG SPS ' - PRPRY N M N, YP PU Y SG RNGS S GNR NS 6" G UR rchitecture nteriors
More informationROSEMOUNT 3051S SCALABLE OR 3051SMV MULTIVARIABLE COPLANAR PRESSURE TRANSMITTER COPLANAR FLANGE PROCESS CONNECTION
1 2 3 4 5 6 7 8 ROSOUNT 3051S S OR 3051SV UTIVRI OPNR PRSSUR TRNSITTR OPNR N PROSS ONNTION RVISION T RVISION O NO. PP' T RT1071620 SRIPTION N NTNN NT STNR 2.4..TISON 10/25/18 PNTW OUSIN SOWN WIT OPTION
More informationCS September 2018
Loil los Distriut Systms 06. Loil los Assin squn numrs to msss All ooprtin prosss n r on orr o vnts vs. physil los: rport tim o y Assum no ntrl tim sour Eh systm mintins its own lol lo No totl orrin o
More informationSHELL CANADA PIPING AND INSTRUMENT DIAGRAM QUEST CCS PROJECT LEGENDS AND SYMBOLS QUEST CCS PROJECT UNIT COMMON "!!
.. 2 S 222... 2. SSU TON Y K PS S M P PM T S N QUST S POJT. S NON N N NSTUMNT M QUST S POJT S W NO.. 2... NT M T \\\2\WNS\UTTS\2\2..pid MO T22 PM Yahm 2. UNT 2 OMMON NS N SYMOS .. SSU T 2 2 2. TON Y K
More informationCHICAGO, O'HARE INTERNATIONAL AIRPORT
O' o dern izat ion r re ogr am a O, O'R NTRNTON RORT TY O O R NU YOR O'R NTRNTON RORT TY O O O'R ORNZTON ROR R NU YOR O RTNT O VTON O'R ORNZTON ROR ROSR S. NONO OSSONR ROSSON SS ROSR S. NONO OSSONR t:
More informationCLKOUT CLKOUT VCC CLKOUT RESOUT OSCOUT ALE TEST AD0 66 AD2 INT0 INT0 AD INT1 AD INT2/INTA0 AD5 AD7 AD7 INT AD8 AD8 AD10
I U N R 00K RSIN* RST S N.0u Y LK TP RP K L TP USY INT0 INT RISMINT P.0 P. P. P. P. P. P. RY OL RX0 TX0 T P.0 P. P. P. S* S* S* S* RROR* SLK U LKIN LKOUT LKOUT LKIN LKOUT OSOUT 0 OSOUT L L RSIN* L 0 0
More informationAn Introduction to Clique Minimal Separator Decomposition
Alortms 2010, 3, 197-215; o:10.3390/3020197 Rvw OPEN ACCESS lortms ISSN 1999-4893 www.mp.om/ournl/lortms An Introuton to Clqu Mnml Sprtor Domposton Ann Brry 1,, Romn Poorln 1 n Gnvèv Smont 2 1 LIMOS UMR
More information(4, 2)-choosability of planar graphs with forbidden structures
(4, )-ooslty o plnr rps wt orn struturs Znr Brkkyzy 1 Crstopr Cox Ml Dryko 1 Krstn Honson 1 Mot Kumt 1 Brnr Lký 1, Ky Mssrsmt 1 Kvn Moss 1 Ktln Nowk 1 Kvn F. Plmowsk 1 Drrk Stol 1,4 Dmr 11, 015 Astrt All
More informationTHE in-loop deblocking filter in the MPEG-4 AVC/H.264. Parallel Deblocking Filtering in MPEG-4 AVC/H.264 on Massively-Parallel Architectures
4205 1 Pr D Ftr MPEG-4 AVC/H.264 Mssvy-Pr Artturs Brt Ptrs, Crs-Frr J. Hrs, J D C, Mr, IEEE, Ptr Lrt, Mr, IEEE, Wsy D Nv, R V W, Mr, IEEE. Astrt T tr t MPEG-4 AVC/H.264 str s utty x us ts tt tvty, rsut
More informationDesigning A Concrete Arch Bridge
This is th mous Shwnh ri in Switzrln, sin y Rort Millrt in 1933. It spns 37.4 mtrs (122 t) n ws sin usin th sm rphil mths tht will monstrt in this lsson. To pro with this lsson, lik on th Nxt utton hr
More informationMINI POST SERIES BALUSTRADE SYSTEM INSTALLATION GUIDE PRODUCT CODE: MPS-RP
MN POST SRS LUSTR SYSTM NSTLLTON U PROUT O: MPS-RP 0 R0 WLL LN 0 RONT LVTON VW R0 N P 0 T RUR LOK LOT ON LSS. SLON SL TYP. OT SS 000 LSS T 0 00 SRS LSS WT 00/00 (0mm NRMNTS VLL) MX. 000 00-0 (ROMMN) 00
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationFun sheet matching: towards automatic block decomposition for hexahedral meshes
DOI 10.1007/s00366-010-0207-5 ORIGINAL ARTICLE Fun st mtn: towrs utomt lok omposton or xrl mss Nols Kowlsk Frnk Loux Mttw L. Sttn Stv J. Own Rv: 19 Frury 2010 / Apt: 22 Dmr 2010 Ó Sprnr-Vrl Lonon Lmt 2011
More information( ) ( ) ( ) 0. Conservation of Energy & Poynting Theorem. From Maxwell s equations we have. M t. From above it can be shown (HW)
8 Conson o n & Ponn To Fo wll s quons w D B σ σ Fo bo n b sown (W) o s W w bo on o s l us n su su ul ow ns [W/ ] [W] su P su B W W 4 444 s W A A s V A A : W W R o n o so n n: [/s W] W W 4 44 9 W : W F
More informationCENTER POINT MEDICAL CENTER
T TRI WTR / IR RISR S STR SRST I TT, SUIT SRST, RI () X () VUU T I Y R VU, SUIT 00 T, RI 0 () 00 X () RISTRTI UR 000 "/0 STY RR I URT VU RT STY RR, RI () 0 X () 00 "/0 STIR # '" TRV IST TRI UIIS UII S,
More informationPART NUMBERS ACCESSORIES: Genuine & Optomize
PRT NUMRS SSORS: enuine & Optomize V(S) SSORY PRT NUMR V(S) SSORY PRT NUMR R-S TR R TR PTR43-00085 i OOR ST NNMNT PT948-1M160 R-S TR R PS PTR09-18130 im RO NT PT347-12160 t TR O TR PTR43-00082 R-S RPT
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationSingle Source Shortest Paths (with Positive Weights)
Snl Sour Sortst Pts (wt Postv Wts) Yuf To ITEE Unvrsty of Qunslnd In ts ltur, w wll rvst t snl sour sortst pt (SSSP) problm. Rll tt w v lrdy lrnd tt t BFS lortm solvs t problm ffntly wn ll t ds v t sm
More information3WN6 Circuit-Breakers
i WN Ciruit-Brrs WN iruit-rrs fix-out, usw- u -po Bstt Horizot otio 0 70 S- S- 7 7 0 0 0 S-7 S-, 70 f Fixi os for outi rt S- 0 $ Sp for rov of r r % Cr for uxiiry pu & r vti sp ( uxiiry pu ) Cui oor *
More information