Eager st-ordering. 1 Introduction. Ulrik Brandes
|
|
- Darleen Barnett
- 5 years ago
- Views:
Transcription
1 Er -Orrin Ulrik Brn Dprmn o Compur & Inormion Sin, Univriy o Konnz ulrik.rn@uni-konnz. Ar. Givn ionn rp G =(V,E) wi{, } E, n -orrin i n orrin v 1,...,v n o V u = v 1, = v n,n vry or vrx o ir-numr n lowr-numr nior. Prviou linr-im -orrin lorim r on prproin p in wi p-ir r i u o ompu lowpoin. T ul orrin i rmin only in on p ovr rp. W prn nw, inrmnl lorim o no rquir lowpoin inormion n, rouou inl p-ir rvrl, minin n -orrin o ionn omponn o {, } in rvr urp. 1 Inrouion T -orrin o vri in n unir rp i unmnl ool or mny rp lorim,.. in plnriy in, rp rwin, or m rouin. I i loly rl o or imporn onp u ionniviy, r ompoiion or ipolr orinion. T ir linr-im lorim or -orrin vri o ionn rp i u o Evn n Trjn [2, 3]. Er [1] prn lily implr lorim, wi i urr implii y Trjn [7]. All lorim, owvr, prpro rp uin p-ir r, nilly o ompu lowpoin wi in urn rmin n (implii) opn r ompoiion. A on rvrl i rquir o ompu ul -orrin. W prn nw lorim voi ompuion o lowpoin n u rquir only inl p ovr rp. I ppr o mor inuiiv, xpliily ompu n opn r ompoiion n ipolr orinion on ly, n i i rou in ppliion o non-ionn rp. Mo noly, i n opp r ny rvrl n will rurn n -orrin o ionn omponn oninin {, } in w n rvr o rp unil n. T lorim n u uiliz in lzy vluion, or inn wn only orrin o w vri i rquir, n on impliily rprn rp r oly o rvr mor n on. T ppr i orniz ollow. In S. 2 w rll i iniion n orrponn wn ionniviy, -orrin n rl onp. Sion 3 i ri rviw o p-ir r n lowpoin ompuion. T nw lorim i vlop in S. 4 n iu in S. 5. R. Mörin n R. Rmn (E.): ESA 2002, LNCS 2461, pp , Sprinr-Vrl Brlin Hilr 2002
2 248 Ulrik Brn 2 Prliminri W onir only unir n impl rp G =(V,E). A (impl) p P =(v 0, 1,v 1,..., k,v k )ing i n lrnin qun o vri V (P )= {v 0,...,v k } V n E(P )={ 1,..., k } E u {v i 1,v i } = i, 1 i k, nv i = v j impli i = j or {i, j} = {0,k}. Tln o P i k. A p i ll lo i v 0 = v k,nopn orwi. ArpG i onn i vry pir o vri i link y p, n i i ionn i i rmin onn r ny vrx i rmov rom G. W r inr in orrin vri o ionn rp in wy urn orwr n kwr onnn. Drminin u n orrin i n nil prproin p in mny ppliion inluin plnriy in, rouin, n rp rwin. Diniion 1 (-orrin [5]). L G =(V,E) ionn rp n V.Anorrin = v 1,v 2,...,v n = o vri o G i ll n -orrin, i or ll vri v j, 1 <j<n,rxi1 i<j<k n u {v i,v j }, {v j,v k } E. Lmm 1 ([5]). A rp G =(V,E) i ionn i n only i, or {, } E, i n -orrin. Svrl linr-im lorim or ompuin -orrin o ionn rp r vill [2, 1, 7]. All o r on priion o rp ino orin p. An orinion in irion o in o. An orinion (lo ll ipolr orinion) o rp G i n orinion u rulin ir rp i yli n n r only our n ink, rpivly. T ollowin lmm i olklor. Lmm 2. A rp G =(V,E) n -orinion i n only i i n -orrin. T n rnorm ino or in linr im. Proo. An -orrin i oin rom n -orinion y opoloil orrin, n n -orinion i oin rom n -orrin y orinin rom lowr-numr o ir-numr vri. A qun D =(P 0,...,P r ) o (opn) p inuin rp G i =(V i,e i ) wi V i = i j=0 V (P j)ne i = i j=0 E(P j), 0 i r, illn(opn) r ompoiion, ie(p 0 ),...,E(P r ) i priion o E n or P i = (v 0, 1,v 1,..., k,v k ), 1 i r, wv{v 0,v k } V i 1 n {v 1,...,v k 1 } V i 1 =. An r ompoiion r wi {, } E, ip 0 =(, {, },). Lmm 3 ([8]). A rp G =(V,E) i ionn i n only i, or {, } E, i n opn r ompoiion rin wi {, }.
3 Er -Orrin 249 No, ivn n opn r ompoiion P 0,P 1,...,P r rin wi {, }, i i riorwr o onru n -orinion. Simply orin P 0 rom o, np i =(u,...,w), 1 i r, romu o w (rom w o u) iu li or (r) w in pril orrin inu y P 0,...,P i 1. Sin orinion o n r onorm o orr o i npoin, no yl r inrou, n n r only our n ink. 3 Dp-Fir Sr n Bionniviy Srin rom roo vrx, p-ir r (DFS) o n unir rp G =(V,E) rvr ll o rp, wr nx i on o inin o mo rnly vii vrx n unrvr. An {v, w} rvr rom v o w i ll r, no y v w, iw i nounr or ir im wn rvrin {v, w}, n i i ll k, no y v w, orwi. For onvnin w u v w or v w o no rpiv wll r i u n wn v n w, or p (v, {v, w},w). W no y v w (poily mpy) p o r rvr in orrponin irion. No r orm pnnin DFS r T = T (G) roo, i.. v or ll v V n v w impli w v. Forv V l T (v) ur inu y ll w V wi v w. W will u rp in Fi. 1 our runnin xmpl. DFS i i o mny iin rp lorim [6], wi on mk u o ollowin noion. T lowpoin o vrx u V i vrx w lo o in T (G) wiw = u or u v w. I no u p xi, u i i own lowpoin. Lowpoin r imporn moly or ollowin ron Fi. 1. Runnin xmpl wi numr in DFS rvrl orr, r pi oli, n k (rrwn rom [7])
4 250 Ulrik Brn Lmm 4 ([6]). A rp G =(V,E) i ionn i n only i, in DFS r T (G), only roo i i own lowpoin n r i mo on r u i lowpoin o (in i, i roo). Prviou linr-im -orrin lorim ir onru DFS r n imulnouly ompu lowpoin or ll vri. In on rvrl o rp y u i inormion o rmin n -orrin. A nw ionniviy lorim y Gow [4] rquir only on p ovr rp n in priulr o no rquir lowpoin ri quion wr w n -orr vri o ionn rp in imilr mnnr. 4 An Er Alorim W prn nw linr-im lorim or -orrin vri o ionn rp. I i r in n i minin, urin p-ir r, n orrin o mximum rvr urp or wi u orrin i poil wiou ponil n o moiiy i lr on. I i inrou vi r prliminry p ini ow lorim lo ompu on ly n opn r ompoion n n -orinion. Puo-o or ompl lorim i ivn n o i ion. 4.1 Opn Er Dompoiion L G =(V,E) ionn rp wi {, } E, nlt DFS r o G wi roo n. W in n opn r ompoiion D(T )= (P 0,...,P r ) uin lol inormion only. In priulr, w o no mk u o lowpoin vlu. L P 0 =(, {, },) n um w v in P 0,...,P i, i 0. I r i k l i no in E i, w in P i+1 ollow. L P 0 : P 1 : P 2 : P 3 : P 4 : P 5 : P 6 : (rivil r) Fi. 2. An r ompoiion D(T ) oin rom DFS r T
5 Er -Orrin 251 v, w, x V u w, x V i, v w E i, w x, nx v ( Fi. 2). Uin l vrx u on r p rom x o v wi u V i (ponilly v il), w P i+1 = u v w. Sinw x u, P i+1 i opn. I i ll r o v w, nrivil i u = v. Torm 1. D(T ) i n opn r ompoiion rin wi {, }. Proo. Clrly, D(T )=(P 0,...,P r ) i qun o -ijoin opn p. I rmin o ow y ovr nir rp, i.. V r = V n E r = E. Fir um r i n unovr vrx n oo u V r u u minimum ln. L w lowpoin o u. SinG i ionn n u,, Lmm 4 impli r xi x, v V wi w x u v w n x u. I ollow w, x V r y minimliy o u, o ompoiion i inompl, in v w ii ll oniion or nor r. Sin ll vri r ovr, ll r r ovr y onruion. Finlly, n unovr k ii ll oniion o in nor (rivil) r Orinion W nx rin ov iniion o n opn r ompoiion o oin n -orinion. W y k v w, n likwi i r, pn on uniqurw x or wi x v. ErinD(T ) r orin in ir qunil orr: P 0 i orin rom o, wrp i,0<i r, iorin orin o r i pn on. S Fi. 3 or n xmpl. T ollowin lmm ow orinin k n ir r prlll o r y pn on nily prop ino ur. I {v, w} E i, 0 i r, i orin rom v o w l v i w,0 i r. P 1 = pn on P 2 = pn on P 3 = pn on P 4 = pn on P 5 = pn on P 6 = pn on Fi. 3. Orinion o r in D(T )
6 252 Ulrik Brn Lmm 5. For ll 0 i r, ov orinion o P 0,...,P i yil n -orinion o G i,n i i pril orr iyin: I w x E i n w i x (x i w), n w i v (v i w) or ll v T (x) V i. Proo. T proo i y inuion ovr qun D(T ). T invrin lrly ol or P 0. Aum i ol or om i<rn l P i+1 r o v w. L w x E i r v w pn on n um i i orin rom w o x ( or i ymmri). T l vrx u V i on x v ii w i u. All vri o P i+1 xp w r in T (x), n in P i+1 i orin lik w x, invrin i minin. Corollry 1. T ov orinion o D(T ) yil n -orinion o G Orrin W inlly ow ow o minin inrmnlly n orrin o V i urin onruion o D(T ). Srin wi rivil -orrin o P 0,lP i = u v w, 0<i r, rov w. IP i i orin rom u o w (w o u), inr qun o innr vri V (P i ) \{u, w} o P i in orr ivn y orinion o P i immily r (or) u. Lmm 6. For ll 0 i r, orrin o V i i linr xnion o i. Proo. T proo i in y inuion ovr qun D(T ). T invrin lrly ol or P 0. Aum i ol or om i < r n P i+1 = u v w pn on w x. Iw i x ( or i ymmri), innr vri o P i+1 r inr immily or u. By Lmm 5 n our invrin, u i r w, o orrin i lo linr xnion o i+1. Corollry 2. T ov orrin yil n -orrin o G. No inrin innr vri o n r P = u v w nx o i inion w rr n i oriin u my rul in n vri o nor r in in wron rliv orr urin lr o lorim. An xmpl o i kin i own in Fi Alorim n Implmnion Dil I rmin o ow ow ov p n rri ou in linr im. W our lorim on p-ir r, u unlik prviou lorim, DFS i no u or prproin u rr mpl o lorim. T lorim i ivn in Al. 1, wr o or DFS-r mnmn i implii. E im DFS rvr k v w, w k wr r w x i pn on i lry orin (no x i urrn il o w on DFS p). I w x i orin, r o v w i orin in m irion n inr ino orrin. For r in nwly
7 Er -Orrin 253 inrion inion,,,,,,,,,, inrion oriin,,,,,,,,,,,,,,, Fi. 4. Exmpl owin i i imporn o inr n r nx o i oriin in r rr n inion o i inin k Alorim 1: Er -orrin Inpu: rpg =(V,E), {, } E Oupu: li L o vri in ionn omponn o {, } (in -orr) pro r(r w! x) in or v w pnin on w x o rmin u L on x v lo o v; P o u v w; i w x orin rom w o x (rp. x o w) n orin P rom w o u (rp. u o w); inr innr vri o P ino L ri or (rp. r) u; or r w x o P o pro r(w x ); lr pnni on w x; n (vrx v) in or nior w o v o i v w n (w); i v w n l x urrn il o w; mk v w pn on w x; i x L n pro r(w x); n in iniiliz L wi ; (); n
8 254 Ulrik Brn,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Fi. 5. Inrmi p o lorim
9 Er -Orrin 255 orin r, w rurivly pro r o no y orin k pn on i. Ar rvrl o n w r ror l wi n -orrin o ionn omponn o {, } in rvr urp. Conqunly, lorim i rou in ppliion o non-ionn rp, n inpu rp i ionn i n only i orrin rurn onin ll vri. Torm 2. Alorim 1 ompu n -orrin o ionn omponn oninin {, } in linr im. Proo. Givn iuion ov, i i uiin o ow lorim rmin nir opn r ompoiion. E r i r o k ; i k pn on n lry orin r, r i rmin n orin. I r i no y orin k i o o pnn n pro oon r i orin. I ollow rom m rumn in proo o Torm 1 ll r in ionn omponn o {, } will vnully orin. Fiur 5 ow inrmi p o lorim wn ppli o runnin xmpl. No -orrin i irn rom oin in [7], u,.., i rvr or lowpoin- (orwi orr o n woul rvr). An iin implmnion o Al. 1 u ouly-link li L, nor ll pnin on r (inluin il) in yli li n rliz wi n rry (in li r ijoin). Sin only orinion o r i n, i i uiin o or orinion o inomin DFS vrx. In ol w n ouly-link vrx li, our vrx rry (inomin, orinion o inomin, urrn il vrx, poinr o li poiion), wo vrx k (DFS n r rvrl), n on rry (nx pnn ). T rry lo rv o ini wr n lry n rvr. 5 Diuion W v prn impl lorim o ompu n -orrin o vri o ionn rp. I rquir only inl rvrl o rp n minin, r p o DFS, mximum pril oluion or orrin, -orinion, n opn r ompoiion prolm on rvr urp. Wiou moiiion, lorim n lo u o or ionnn (imply k wr ln o rurn li qul numr o vri in rp), n i i rily n rom inuiv proo o Lmm 6 rulin -orrin v ollowin inrin propry. Corollry 3. Alorim 1 yil -orrin in wi vri o vry ur o DFS r orm n inrvl.
10 256 Ulrik Brn I i inrin o no orin (r) orinion vri orrpon o Trjn u o +/- ll in [7]. Sin y n inrpr orin orinion prn rr n il, y n o up, ou. Wil u o lowpoin limin n o kp rk o pnni, lowpoin r known only r rvrin nir rp. Alou Trjn lorim [7] rmin mo primoniou, w u l r lorim i mor lxil n inuiiv. Finlly, lorim n u o rmin nrlizion o ipolr orinion o non-ionn rp, nmly yli orinion wi numr o our n ink i mo on lrr n numr o ionn omponn, n wi i ipolr in omponn. Wnvr DFS krk ovr r no n orin, i ompl ionn omponn. W r n r o orin i r ny wy w wn n u o rmin, y rurivly orinin pnn r, ipolr orinion o i nir omponn. T omin orinion r yli, n or iionl ionn omponn w mo on our or on ink, pnin on ow w oo o orin ir o omponn n wr o i inin vri r u vri. Aknowlmn I woul lik o nk Roro Tmi or iniiin i rr y mkin m wr o Gow work on p- DFS, n Roro Tmi n Lu Vimr or vry lpul iuion on uj. Tnk o r nonymou rviwr or ir il ommn. Rrn [1] J. Er. -orrin vri o ionn rp. Compuin, 30(1):19 33, , 248 [2] S. Evn n R. E. Trjn. Compuin n -numrin. Toril Compur Sin, 2(3): , , 248 [3] S. Evn n R. E. Trjn. Corrinum: Compuin n -numrin. Toril Compur Sin, 4(1):123, [4] H. N. Gow. P- p-ir r or ron n ionn omponn. Inormion Proin Lr, 74: , [5] A. Lmpl, S. Evn, n I. Crum. An lorim or plnriy in o rp. In P. Ronil, ior, Proin o Inrnionl Sympoium on Tory o Grp (Rom, July 1966), p Goron n Br, [6] R. E. Trjn. Dp-ir r n linr rp lorim. SIAM Journl on Compuin, 1: , , 250 [7] R. E. Trjn. Two rmlin p-ir r lorim. Funmn Inormi, 9:85 94, , 248, 249, 255, 256 [8] H. Winy. Non-prl n plnr rp. Trnion o Amrin Mmil Soiy, 34: ,
Eager st-ordering. Universität Konstanz. Ulrik Brandes. Konstanzer Schriften in Mathematik und Informatik Nr. 171, April 2002 ISSN
Univriä Konnz Er -Orrin Ulrik Brn Konnzr Srin in Mmik un Inormik Nr. 171, April 2002 ISSN 1430 3558 Fri Mmik un Siik Fri Inormik un Inormionwin Univriä Konnz F D 188, 78457 Konnz, Grmny Emil: prprin@inormik.uni
More informationCombinatorial Optimization
Cominoril Opimizion Prolm : oluion. Suppo impl unir rp mor n on minimum pnnin r. Cn Prim lorim (or Krukl lorim) u o in ll o m? Explin wy or wy no, n iv n xmpl. Soluion. Y, Prim lorim (or Krukl lorim) n
More informationJonathan Turner Exam 2-10/28/03
CS Algorihm n Progrm Prolm Exm Soluion S Soluion Jonhn Turnr Exm //. ( poin) In h Fioni hp ruur, u wn vrx u n i prn v u ing u v i v h lry lo hil in i l m hil o om ohr vrx. Suppo w hng hi, o h ing u i prorm
More informationCS 541 Algorithms and Programs. Exam 2 Solutions. Jonathan Turner 11/8/01
CS 1 Algorim nd Progrm Exm Soluion Jonn Turnr 11/8/01 B n nd oni, u ompl. 1. (10 poin). Conidr vrion of or p prolm wi mulipliiv o. In i form of prolm, lng of p i produ of dg lng, rr n um. Explin ow or
More informationDesign and Analysis of Algorithms (Autumn 2017)
Din an Analyi o Alorim (Auumn 2017) Exri 3 Soluion 1. Sor pa Ain om poiiv an naiv o o ar o rap own low, o a Bllman-For in a or pa. Simula ir alorim a ru prolm o a layr DAG ( li), or on a an riv rom rurrn.
More informationCSE 421 Algorithms. Warmup. Dijkstra s Algorithm. Single Source Shortest Path Problem. Construct Shortest Path Tree from s
CSE Alorihm Rihr Anron Dijkr lorihm Sinl Sor Shor Ph Prolm Gin rph n r r Drmin in o ry r rom Iniy hor ph o h r Epr onily hor ph r Eh r h poinr o pror on hor ph Conr Shor Ph Tr rom Wrmp - - I P i hor ph
More informationGraphs: Paths, trees and flows
in in grph rph: Ph, r n flow ph-fir rh fin vri rhl from nohr givn vrx. Th ph r no h hor on. rph r = hor in = = Jori orll n Jori Pi prmn of ompur in = in wn wo no: lngh of h hor ph wn hm rh-fir rh rph p.,
More informationShortest Paths. CSE 421 Algorithms. Bottleneck Shortest Path. Negative Cost Edge Preview. Compute the bottleneck shortest paths
Shor Ph CSE Alorihm Rihr Anron Lr 0- Minimm Spnnin Tr Ni Co E Dijkr lorihm m poii o For om ppliion, ni o mk n Shor ph no wll in i rph h ni o yl - - - Ni Co E Priw Topoloil Sor n or olin h hor ph prolm
More informationMathcad Lecture #4 In-class Worksheet Vectors and Matrices 1 (Basics)
Mh Lr # In-l Workh Vor n Mri (Bi) h n o hi lr, o hol l o: r mri n or in Mh i mri prorm i mri mh oprion ol m o linr qion ing mri mh. Cring Mri Thr r rl o r mri. Th "Inr Mri" Wino (M) B K Poin Rr o
More informationJonathan Turner Exam 2-12/4/03
CS 41 Algorim an Program Prolm Exam Soluion S Soluion Jonaan Turnr Exam -1/4/0 10/8/0 1. (10 poin) T igur low ow an implmnaion o ynami r aa ruur wi vral virual r. Sow orrponing o aual r (owing vrx o).
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 informationRUTH. land_of_israel: the *country *which God gave to his people in the *Old_Testament. [*map # 2]
RUTH 1 Elimlk g ln M 1-2 I in im n ln Irl i n *king. Tr r lr rul ln. Ty r ug. Tr n r l in Ju u r g min. Elimlk mn y in n Blm in Ju. H i nm Nmi. S n Elimlk 2 *n. Tir nm r Mln n Kilin. Ty r ll rm Er mily.
More information1 Finite Automata and Regular Expressions
1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o
More informationOUR TEAM SHEET INDEX: Foothill Villas 10, LLC Apple Street, Suite 204, Newhall, Ca Office
Note: rtist s onception; olors, aterials nd pplication ay Vary. OR oothill Villas, LL. pple treet, uite, Newhall, a. Office W. rchitects. lanners. esigners. ontact: ernando Laullon Redhill ve, uite anta
More information1 Introduction to Modulo 7 Arithmetic
1 Introution to Moulo 7 Arithmti Bor w try our hn t solvin som hr Moulr KnKns, lt s tk los look t on moulr rithmti, mo 7 rithmti. You ll s in this sminr tht rithmti moulo prim is quit irnt rom th ons w
More informationECE COMBINATIONAL BUILDING BLOCKS - INVEST 13 DECODERS AND ENCODERS
C 24 - COMBINATIONAL BUILDING BLOCKS - INVST 3 DCODS AND NCODS FALL 23 AP FLZ To o "wll" on this invstition you must not only t th riht nswrs ut must lso o nt, omplt n onis writups tht mk ovious wht h
More informationA1 1 LED - LED 2x2 120 V WHITE BAKED RECESSED 2-3/8" H.E. WILLIAMS PT-22-L38/835-RA-F358L-DIM-BD-UNV 37
O NU +'-" ac I U I K K OUNIN I NIN UN O OUN U IUI INU O INI OO O INI I NIN UNI INUIN IY I UO. N WI Y K ONUI IUI ONO N UN NO () () O O W U I I IIUION N IIN I ONO U N N O IN IO NY. I UIN NY OW O I W OO OION-OO
More informationLabeling Problem & Graph-Based Solution
Lling Prolm & Grph-B Soluion Am M. Ali Lling Prolm In lling Prolm w hv o i P n o ll L : rprn img ur {.g. pixl, g, img gmn,.}. Fur my hv om nurl ruur pixl r rrng in 2D rry. : rprn innii, iprii,. P L Lling
More informationOn the Complexity of Graph Cuboidal Dual Problems for 3-D Floorplanning of Integrated Circuit Design
On Complxiy o Grp Cuoil Dul Prolm or -D Floorplnnin o Inr Circui Din Rnn Wn Uniriy o Cliorni, Sn Dio L Joll, CA 99- rn@c.uc.u Cun-Kun Cn Uniriy o Cliorni, Sn Dio L Joll, CA 99- ckcn@uc.u ABSTRACT Ti ppr
More informationOutline. 1 Introduction. 2 Min-Cost Spanning Trees. 4 Example
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim's Alorithm Introution Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #33 3 Alorithm Gnrl Constrution Mik Joson (Univrsity o Clry)
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 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 informationLibrary Support. Netlist Conditioning. Observe Point Assessment. Vector Generation/Simulation. Vector Compression. Vector Writing
hpr 2 uomi T Prn Gnrion Fundmnl hpr 2 uomi T Prn Gnrion Fundmnl Lirry uppor Nli ondiioning Orv Poin mn Vor Gnrion/imulion Vor omprion Vor Wriing Figur 2- Th Ovrll Prn Gnrion Pro Dign-or-T or Digil I nd
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 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 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 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 informationCycles and Simple Cycles. Paths and Simple Paths. Trees. Problem: There is No Completely Standard Terminology!
Outlin Computr Sin 331, Spnnin, n Surphs Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #30 1 Introution 2 3 Dinition 4 Spnnin 5 6 Mik Joson (Univrsity o Clry) Computr Sin 331 Ltur #30 1 / 20 Mik
More informationPlanar Upward Drawings
C.S. 252 Pro. Rorto Tmssi Computtionl Gomtry Sm. II, 1992 1993 Dt: My 3, 1993 Sri: Shmsi Moussvi Plnr Upwr Drwings 1 Thorm: G is yli i n only i it hs upwr rwing. Proo: 1. An upwr rwing is yli. Follow th
More informationFL/VAL ~RA1::1. Professor INTERVI of. Professor It Fr recru. sor Social,, first of all, was. Sys SDC? Yes, as a. was a. assumee.
B Pror NTERV FL/VAL ~RA1::1 1 21,, 1989 i n or Socil,, fir ll, Pror Fr rcru Sy Ar you lir SDC? Y, om um SM: corr n 'd m vry ummr yr. Now, y n y, f pr my ry for ummr my 1 yr Un So vr ummr cour d rr o l
More informationDistributed Algorithms for Secure Multipath Routing in Attack-Resistant Networks
1 Diriu Algorihm or Sur Muliph Rouing in Ak-Rin Nwork Prik P. C. L, Vihl Mir, n Dn Runin Ar To proivly n gin inrur rom rily joprizing ingl-ph ion, w propo iriu ur muliph oluion o rou ro mulipl ph o h inrur
More informationOutline. Computer Science 331. Computation of Min-Cost Spanning Trees. Costs of Spanning Trees in Weighted Graphs
Outlin Computr Sin 33 Computtion o Minimum-Cost Spnnin Trs Prim s Mik Joson Dprtmnt o Computr Sin Univrsity o Clry Ltur #34 Introution Min-Cost Spnnin Trs 3 Gnrl Constrution 4 5 Trmintion n Eiiny 6 Aitionl
More informationComplete Solutions for MATH 3012 Quiz 2, October 25, 2011, WTT
Complt Solutions or MATH 012 Quiz 2, Otor 25, 2011, WTT Not. T nswrs ivn r r mor omplt tn is xpt on n tul xm. It is intn tt t mor omprnsiv solutions prsnt r will vlul to stunts in stuyin or t inl xm. In
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 informationCS 103 BFS Alorithm. Mark Redekopp
CS 3 BFS Aloritm Mrk Rkopp Brt-First Sr (BFS) HIGHLIGHTED ALGORITHM 3 Pt Plnnin W'v sn BFS in t ontxt o inin t sortst pt trou mz? S?? 4 Pt Plnnin W xplor t 4 niors s on irtion 3 3 3 S 3 3 3 3 3 F I you
More informationADDENDUM. The following addendum becomes a part of the above referenced bid. All other terms and conditions remain in effect, unchanged.
RVIS MYRN irector of dministration OUNY O N PRMN O MINISRION PURSIN IVISION Room 4 ity-ounty uilding Martin Luther King Jr. lvd. Madison, WI -4 /-4 X /-44 /-494 RLS IKLIN ontroller NUM : September 4, ROM:
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 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 informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
s s of s Computr Sin & Enginring 423/823 Dsign n Anlysis of Ltur 03 (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) s of s s r strt t typs tht r pplil to numrous prolms Cn ptur ntitis, rltionships twn
More informationPhonics Bingo Ladders
Poi Bio Lddr Poi Bio Lddr Soli Ti Rour Soli I. r r riio o oooy did rroduil or lroo u. No or r o uliio y rrodud i wol or i r, or ord i rrivl y, or rid i y wy or y y, lroi, il, oooyi, rordi, or orwi, wiou
More informationM A. L I O E T O W A R D N O N E A. N I D O H A R I T Y F O R A L L. " An Old Timor's DesorSptlon of HI* Camp Outfit. THE DEATH OF M R L A. R. WEEKS.
J : UO XOW YOU ONY «00 V DVZ WOW R KO L O O W R D N O N N D O R Y O R L L VOL LOWLL KN OUNY NOVR 25 893 NO 22 W L L L K Y O WNR K 0? 0 LOR W Y K YUU U O LO L ND YL LOW R N D R O ND N L O O LL 0R8 D KOR
More information(2) If we multiplied a row of B by λ, then the value is also multiplied by λ(here lambda could be 0). namely
. DETERMINANT.. Dtrminnt. Introution:I you think row vtor o mtrix s oorint o vtors in sp, thn th gomtri mning o th rnk o th mtrix is th imnsion o th prlllppi spnn y thm. But w r not only r out th imnsion,
More informationEE Control Systems LECTURE 11
Up: Moy, Ocor 5, 7 EE 434 - Corol Sy LECTUE Copyrigh FL Lwi 999 All righ rrv POLE PLACEMET A STEA-STATE EO Uig fc, o c ov h clo-loop pol o h h y prforc iprov O c lo lc uil copor o oi goo y- rcig y uyig
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 informationCSE 373: More on graphs; DFS and BFS. Michael Lee Wednesday, Feb 14, 2018
CSE 373: Mor on grphs; DFS n BFS Mihl L Wnsy, F 14, 2018 1 Wrmup Wrmup: Disuss with your nighor: Rmin your nighor: wht is simpl grph? Suppos w hv simpl, irt grph with x nos. Wht is th mximum numr of gs
More informationSeven-Segment Display Driver
7-Smnt Disply Drivr, Ron s in 7-Smnt Disply Drivr, Ron s in Prolm 62. 00 0 0 00 0000 000 00 000 0 000 00 0 00 00 0 0 0 000 00 0 00 BCD Diits in inry Dsin Drivr Loi 4 inputs, 7 outputs 7 mps, h with 6 on
More informationV={A,B,C,D,E} E={ (A,D),(A,E),(B,D), (B,E),(C,D),(C,E)}
Introution Computr Sin & Enginring 423/823 Dsign n Anlysis of Algorithms Ltur 03 Elmntry Grph Algorithms (Chptr 22) Stphn Sott (Apt from Vinohnrn N. Vriym) I Grphs r strt t typs tht r pplil to numrous
More informationLaplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011
plc Trnorm Nionl Chio Tung Univriy Chun-Jn Ti /9/ Trnorm o Funcion Som opror rnorm uncion ino nohr uncion: d Dirniion: x x, or Dx x dx x Indini Ingrion: x dx c Dini Ingrion: x dx 9 A uncion my hv nicr
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 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 informationDominator Tree Certification and Independent Spanning Trees
Domintor Tr Crtiition n Inpnnt Spnnin Tr Louk Gorii 1 Rort E. Trjn 2 rxiv:1210.8303v3 [.DS] 7 Mr 2013 Otor 29, 2018 Atrt How o on vriy tht th output o omplit prorm i orrt? On n ormlly prov tht th prorm
More informationThe Procedure Abstraction Part II: Symbol Tables and Activation Records
Th Produr Absrion Pr II: Symbol Tbls nd Aivion Rords Th Produr s Nm Sp Why inrodu lxil soping? Provids ompil-im mhnism for binding vribls Ls h progrmmr inrodu lol nms How n h ompilr kp rk of ll hos nms?
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 informationApplications of these ideas. CS514: Intermediate Course in Operating Systems. Problem: Pictorial version. 2PC is a good match! Issues?
CS514: Inmi Co in Oing Sm Poo Kn imn K Ooki: T liion o h i O h h k h o Goi oool Dii monioing, h, n noiiion gmn oool, h 2PC n 3PC Unling hm: om hing n ong om o onin, om n mng ih k oi To, l look n liion
More informationP a g e 3 6 of R e p o r t P B 4 / 0 9
P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J
More informationEX. WOODS 7.37± ACRES (320,826± SQ. FT.) BM# EX. WOODS UNKNOWN RISER 685
Y RUUR - - Ø R. ( P=. ( " P=. ( " P=. ( " P=. RY -B - Ø R. ( P=. ( P=. ( " P=.. O OUR RY OPY R.. #-. YR PR. R.= PROPRY RO = PROPRY RO OU R L POL R P O BOLL L PPRO LOO O Y R R. L., P. (OU UL O R L POL LOO
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 informationDerivation of the differential equation of motion
Divion of h iffnil quion of oion Fis h noions fin h will us fo h ivion of h iffnil quion of oion. Rollo is hough o -insionl isk. xnl ius of h ll isnc cn of ll (O) - IDU s cn of gviy (M) θ ngl of inclinion
More informationOpenMx Matrices and Operators
OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(
More informationCSC 373: Algorithm Design and Analysis Lecture 9
CSC 373: Algorihm Deign n Anlyi Leure 9 Alln Boroin Jnury 28, 2013 1 / 16 Leure 9: Announemen n Ouline Announemen Prolem e 1 ue hi Friy. Term Te 1 will e hel nex Mony, Fe in he uoril. Two nnounemen o follow
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 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 informationPIPE ELBOW DOWN PIPE ELBOW DOWN PIPE ELBOW UP PIPE ELBOW UP PIPE TEE BELOW WITH BRANCH ELBOW AT DOWN PIPE TEE BELOW WITH BRANCH ELBOW AT DOWN
6 V I I N # NI VIION X 4 4/4 W OUN OINION FI/OK F O O IFIION FO ONUION N INION I. W OUN OK F O O IFIION FO ONUION N INION I. FI F INI FI ; IION INION Y O U, U NU O INI IN FI, FO O 3 OU, N/O INIIN YO Y
More informationx, x, e are not periodic. Properties of periodic function: 1. For any integer n,
Chpr Fourir Sri, Igrl, d Tror. Fourir Sri A uio i lld priodi i hr i o poiiv ur p uh h p, p i lld priod o R i,, r priodi uio.,, r o priodi. Propri o priodi uio:. For y igr, p. I d g hv priod p, h h g lo
More information11 6'-0" 9'-8 1 2" SLOPE DN. FLOOR DRAIN W/ OIL SEPARATOR TO SEWER (TYP.) C.J. 101A SLOPE PER GRADING PLAN
'- " '- ". '- " '-" '- " N X OO TU @ " O.. IU.. PRIMINRY RIN.. I T.. UIIN PRMIT T '-" '-" '- " '- " X OO TU @ " O.. / () YR ". I (-R: T.) X OO TU @ " O.. ONRIP OF OUMNT: This document, and the ideas and
More information(4, 2)-choosability of planar graphs with forbidden structures
1 (4, )-oosility o plnr rps wit orin struturs 4 5 Znr Brikkyzy 1 Cristopr Cox Mil Diryko 1 Kirstn Honson 1 Moit Kumt 1 Brnr Liiký 1, Ky Mssrsmit 1 Kvin Moss 1 Ktln Nowk 1 Kvin F. Plmowski 1 Drrik Stol
More informationA Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique
Inrnionl hmil orum no. 667-67 Sud of h Soluions of h o Volrr r rdor Ssm Using rurion Thniqu D.Vnu ol Ro * D. of lid hmis IT Collg of Sin IT Univrsi Vishnm.. Indi Y... Thorni D. of lid hmis IT Collg of
More information1. Introduction and notations.
Alyi Ar om plii orml or q o ory mr Rol Gro Lyé olyl Roièr, r i lir ill, B 5 837 Tolo Fr Emil : rolgro@orgr W y hr q o ory mr, o ll h o ory polyomil o gi rm om orhogol or h mr Th mi rl i orml mig plii h
More informationIndices. Indices. Curriculum Ready ACMNA: 209, 210, 212,
Inis Inis Curriulum Ry ACMNA: 09, 0,, 6 www.mtltis.om Inis INDICES Inis is t plurl or inx. An inx is us to writ prouts o numrs or pronumrls sily. For xmpl is tully sortr wy o writin #. T is t inx. Anotr
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 informationLAUREL HILL VERMONT APRIL 2014
MY Y / OVR O OP MP R OP Y R Y K R U M R PK R R OM R P U ROU P O R RMP OO R MP MOU R RR UR R OU V OR V M O OR R OP R R R OO JOV Y V R V OO Y OUR PKY U V O VY MP O R R UR R R O O V R R R R RO Y P Y QU RU
More informationSAMPLE PAGES. Primary. Primary Maths Basics Series THE SUBTRACTION BOOK. A progression of subtraction skills. written by Jillian Cockings
PAGES Primry Primry Mths Bsis Sris THE SUBTRACTION BOOK A prorssion o sutrtion skills writtn y Jillin Cokins INTRODUCTION This ook is intn to hlp sur th mthmtil onpt o sutrtion in hilrn o ll s. Th mstry
More informationImage Modeling & Segmentation
Img Moling & Sgmnion Aly Frg n Am Ali Lur #7 MGRF- Img Anlyi Fiing MRF mol o n img rquir h h prmr o h mol im rom mpl o h img Th lirur i rih wih work h propo irn MGRF mol whih r uil or pii ym hvior. Uully,
More informationStable Matching for Spectrum Market with Guaranteed Minimum Requirement
Sl g Spum Gun mum Rqumn Yno n T S Ky Sw ngg ompu Sool Wun Uny nyno@wuun Yuxun Xong T S Ky Sw ngg ompu Sool Wun Uny xongyx@mlluun Qn Wng ompu Sool Wun Uny qnwng@wuun STRT Xoyn Y Sool mon Tlogy ow Uny X
More informationA-1 WILDCARD BEER TASTING ROOM 3 DRAWING INDEX & STATE CODES 4 PROJECT DATA SOLANO AVE KAINS AVE. 5 AERIAL VIEW 6 PROJECT TEAM TENANT IMPROVEMENTS
NN : 11 ONO VNU, BNY OWN: WI BWING O. XIING NN U: I B ING NN NB : FI FOO 1,090 F NO HNG MZZNIN 80 F 6 F O 1,70 F 1,5 F MZZNIN : MZZNIN I U O % 1 FOO NN OUPN O: 1 FOO FO O ING OOM 59 F 15 0.5 B 117 F 100
More informationSolutions to assignment 3
D Sruure n Algorihm FR 6. Informik Sner, Telikeplli WS 03/04 hp://www.mpi-.mpg.e/~ner/oure/lg03/inex.hml Soluion o ignmen 3 Exerie Arirge i he ue of irepnie in urreny exhnge re o rnform one uni of urreny
More informationTCI SERIES, 3 CELL THE COOLING TOWER CO., L.C. TCI 3 CELL SERIES GA
of T SIN IS T PROPRTY O T OOLIN TOWR OMPNY, L, N IS LON OR MUTUL SSISTN. IT IS NOT TO ORWR NOR RPRINT IN NY ORM WITOUT WRITTN PRMISSION. L W OPTIONL SS PLTORM " x " STNR PLTORM PR LL OPTIONL VIRTION SWIT
More information10/30/12. Today. CS/ENGRD 2110 Object- Oriented Programming and Data Structures Fall 2012 Doug James. DFS algorithm. Reachability Algorithms
0/0/ CS/ENGRD 0 Ojt- Orint Prormmin n Dt Strutur Fll 0 Dou Jm Ltur 9: DFS, BFS & Shortt Pth Toy Rhility Dpth-Firt Srh Brth-Firt Srh Shortt Pth Unwiht rph Wiht rph Dijktr lorithm Rhility Alorithm Dpth Firt
More informationCSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp http://www.s.wshington.u/373/ Univrsity o Wshington, ll rights rsrv. 1 Wht is grph? 56 Tokyo Sttl Soul 128 16 30 181 140
More information-Z ONGRE::IONAL ACTION ON FY 1987 SUPPLEMENTAL 1/1
-Z-433 6 --OGRE::OA ATO O FY 987 SUPPEMETA / APPR)PRATO RfQUEST PAY AD PROGRAM(U) DE ARTMET OF DEES AS O' D 9J8,:A:SF ED DEFS! WA-H ODM U 7 / A 25 MRGOPf RESOUTO TEST HART / / AD-A 83 96 (~Go w - %A uj
More information1- I. M. ALGHROUZ: A New Approach To Fractional Derivatives, J. AOU, V. 10, (2007), pp
Jourl o Al-Qus Op Uvrsy or Rsrch Sus - No.4 - Ocobr 8 Rrcs: - I. M. ALGHROUZ: A Nw Approch To Frcol Drvvs, J. AOU, V., 7, pp. 4-47 - K.S. Mllr: Drvvs o or orr: Mh M., V 68, 995 pp. 83-9. 3- I. PODLUBNY:
More information176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s
A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps
More informationQUESTIONS BEGIN HERE!
Points miss: Stunt's Nm: Totl sor: /100 points Est Tnnss Stt Univrsity Dprtmnt o Computr n Inormtion Sins CSCI 2710 (Trno) Disrt Struturs TEST or Sprin Smstr, 2005 R this or strtin! This tst is los ook
More informationENJOY ALL OF YOUR SWEET MOMENTS NATURALLY
ENJOY ALL OF YOUR SWEET MOMENTS NATURALLY I T R Fily S U Wi Av I T R Mkr f Sr I T R L L All-Nrl Sr N Yrk, NY (Mr 202) Crl Pki Cr., kr f Sr I T R Svi I T R v x ll-rl I T R fily f r il Av I T R, 00% ri v
More informationGraph Algorithms and Combinatorial Optimization Presenters: Benjamin Ferrell and K. Alex Mills May 7th, 2014
Grp Aloritms n Comintoril Optimiztion Dr. R. Cnrskrn Prsntrs: Bnjmin Frrll n K. Alx Mills My 7t, 0 Mtroi Intrstion In ts ltur nots, w mk us o som unonvntionl nottion or st union n irn to kp tins lnr. In
More informationJHC series electrical connector
i lil oo i iouio oli wi I-- Ⅲ i i- ui ouli wi i-looi i ll iz, li i wi, i o iy I/I ili ovl i o, oo-oo i ii i viio u i u, li i vio li wi,, oi,. liio: i il ii [il] oui: luiu lloy, il l li: - y iu li lol il
More informationMIL-DTL SERIES 2
o ll oo I--26482 I 2 I--26482 I 2 OI O 34 70 14 4 09 70 14 4 71 l, l o 74 l, u 75 lu, I ou 76 lu, luu, l oz luu, lol l luu, olv u ov lol l l l, v ll z 8, 10, 12, 14,, 18,, 22, o 24 I o lyou I--69 o y o
More informationSingle Correct Type. cos z + k, then the value of k equals. dx = 2 dz. (a) 1 (b) 0 (c)1 (d) 2 (code-v2t3paq10) l (c) ( l ) x.
IIT JEE/AIEEE MATHS y SUHAAG SIR Bhopl, Ph. (755)3 www.kolsss.om Qusion. & Soluion. In. Cl. Pg: of 6 TOPIC = INTEGRAL CALCULUS Singl Corr Typ 3 3 3 Qu.. L f () = sin + sin + + sin + hn h primiiv of f()
More information- ASSEMBLY AND INSTALLATION -
- SSEMLY ND INSTLLTION - Sliin Door Stm Mot Importnt! Ti rmwork n ml to uit 100 mm ini wll tikn (75 mm tuwork) or 125 mm ini wll tikn (100 mm tuwork) HOWEVER t uppli jm kit i pii to itr 100 mm or 125 mm
More information16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Ideal Nozzle Fluid Mechanics
6.5, Rok ropulsion rof. nul rinz-snhz Lur 3: Idl Nozzl luid hnis Idl Nozzl low wih No Sprion (-D) - Qusi -D (slndr) pproximion - Idl gs ssumd ( ) mu + Opimum xpnsion: - or lss, >, ould driv mor forwrd
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 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 information2. The Laplace Transform
Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin
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 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 informationFourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t
Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih
More informationSTRUCTURAL GENERAL NOTES
UILIN OS: SIN LOS: RUTURL NRL NOTS NRL NOTS: US ROUP: - SSMLY USS INTN OR PRTIIPTION IN OR VIWIN OUTOOR TIVITIS PR MIIN UILIN O STION. SSONL. T UNTION O TIS ILITY IS NOT OR QUIPP OR OUPNY URIN WINTR/ TIN
More informationAmphenol RNJ LOW PROFILE. Harsh Environment Rack & Panel Cylindrical Connectors
ol O O vo & l yll oo O O
More informationINVENTORY MODEL FOR DETERIORATING ITEMS WITH QUADRATIC DEMAND, PARTIAL BACKLOGGING AND PARTIAL TRADE CREDIT
Oprions Rsr n ppliions : n nrnionl Journl ORJ Vol. No.4 Novmr 5 NVENORY ODEL FOR DEERORNG ES WH QUDR DEND RL BKLOGGNG ND RL RDE RED D. Srmil n R.Uykumr Dprmn of mis Gnigrm Rurl nsiu Dm Univrsiy Gnigrm
More informationMidterm. Answer Key. 1. Give a short explanation of the following terms.
ECO 33-00: on nd Bnking Souhrn hodis Univrsi Spring 008 Tol Poins 00 0 poins for h pr idrm Answr K. Giv shor xplnion of h following rms. Fi mon Fi mon is nrl oslssl produd ommodi h n oslssl sord, oslssl
More information