Combinatorial Optimization
|
|
- Dwayne Preston
- 5 years ago
- Views:
Transcription
1 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 u o in ll minimum pnnin r. O our, o lorim only prou inl minimum pnnin r wn i i run, o w w rlly mn i vry minimum pnnin r i poil rul o o o lorim; y mkin ppropri oi urin lorim, ny priulr minimum pnnin r n oun. W in wi lmm wo proo i nilly m proo o Torm. in Ppimiriou n Siliz. Lmm. L G = (V, E) onn rp wi n wi (u, v) or {u, v} E. L (V, T ) minimum pnnin r o G, n l (V, F ) pnnin or u F T. L U V vrx o onn omponn o (V, F ), n l U E o o G vin xly on npoin in U. (T U i ll ounry o U.) Tn mon in U vin minimum wi, r i on i in T. Proo. Suppo or k o onriion o minimum-wi in U onin no o T. L {u, v} U minimum-wi. A {u, v} o T ; i r (uniqu) yl. Bu i yl onin l on vrx in U n l on vrx no in U (nmly, u n v), i mu onin nor {u, v } U irn rom {u, v}, n {u, v } mu in T. By umpion, (u, v) < (u, v ), o i w rmov {u, v } w oin nw pnnin r (V, T ), wi T = T { {u, v} } \ { {u, v } }, wi rily mllr wi n (V, T ). Bu i onri (V, T ) i minimum pnnin r. E irion o Prim lorim (or Krukl lorim) oni o in n o pnnin or, iniilly vin no, unil pnnin or om pnnin r. Tror, pnnin or rul r k irion o ir o lorim xly k. Clim. Fix minimum pnnin r (V, T ) o G. For vry 0 k n, r k irion o Prim lorim (or Krukl lorim), i i poil or rulin pnnin or (V, F ) o iy F T. Proo. By inuion on k. Clrly mn i ru or k = 0, u o Prim lorim n Krukl lorim in wi pnnin or vin no, o F = T. Suppo k. By inuion, i i poil, r k irion, or Prim lorim (or Krukl lorim) o prou pnnin or (V, F ) vin F T ; in k < n, w know F T. In k irion o Prim lorim, minimum-wi i l ou o U o vin xly on npoin in U, wr U i vrx o onn omponn o (V, F ) oninin ix vrx v rirrily on innin o lorim. In k irion o Krukl lorim, minimumwi = {u, v} i l ou o o ll wo npoin r in irn onn omponn o (V, F ); i i in U, wr U i vrx o onn omponn o (V, F ) oninin u. In ir, w y lmm ov mon in U vin minimum wi, r i on i in T, o w my l o n in T, n n F := F {} T. Tror, kin k = n in i lim, w i i poil or Prim lorim (or Krukl lorim) o prou r (V, T ). A (V, T ) w n rirry minimum pnnin r, i ow i i poil or o o lorim o in ny minimum pnnin r.
2 A n xmpl, onir rp own on l low. pnnin r, own prly. I wo minimum I v i on o vrx innin o Prim lorim, n o Prim lorim n Krukl lorim r y lin {, }. In nx irion, o {, } n {, } r ni o l. Slin {, } l o minimum pnnin r own in nr ov, wil lin {, } l o on own on ri.. Explin ow minimum pnnin r lorim n ily u o in mximum pnnin r in rp. Tn xplin wy, i you wn o in lon p wn wo vri in rp, uin i m niqu wi Dijkr lorim o no work. Soluion. A mximum pnnin r n oun y nin ll o wi n n pplyin minimum pnnin r lorim o rulin rp. Alrnivly, i w woul lik o work only wi nonniv wi, w n in lr wi M, rpl vry wi w wi M w, n n pply minimum pnnin r lorim. Ti i quivln o nin ll o wi n n inrin wi y M. Ti ppro work u vry pnnin r on rp wi n vri xly n, o ol wi o vry pnnin r will inr y xly (n )M. T impl ron nin wi o no work o in lon p wi Dijkr lorim i Dijkr lorim rquir ll wi o nonniv. T M w rik on work wi Dijkr lorim ir, u irn p wn ivn pir o vri my oni o irn numr o. Nin wi n n in M o wi will ror irn mulipl o M o ol wi o p, n i will no nrily prrv lon or or p.. Fin mximum low n minimum u in ollowin low nwork. Soluion. T irion o For Fulkron lorim r own low. In iur, pii o r r own irl in ry, no ll r in rn,
3 n nonzro low lon r r in li lu. T no ll r ivn in orm (rom[i], ow-mu[i]). Ar wi zro low r rwn ri lk rrow, ur r r rwn oul r rrow, n unur r wi nonzro low r rwn wvy li lu rrow. Blow iur i LIST o vri uil up For Fulkron lorim pror rou nwork. W r LIST quu r: no r o ri-n i wn y r ll, n y r rmov rom l-n i (ini y roin m ou) wn y r nn. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \, \; ; ;. T no riv ll, o w v oun n umnin p. T vlu o ll ow-mu[] i, o w n umn low y lon i p. To iniy p, w ollow rom ll kwr rom : rom[] =, rom[] =, n rom[] =, o our umnin p i. All o rom ll lon i p wr poiiv (y i no v lin minu in), o low lon vry r in i p will inr y. W ju low n rp llin pro rom innin. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \; \; \; ;. (, ) Followin rom ll kwr rom, w in umnin p. W umn low lon vry r in i p y ow-mu[] = n rp llin pro rom innin.
4 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \; \; \; \;. (, ) T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \, \; \, \; \; \;. (, ) T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \; \; \, \; \;. T umnin p r i ; w umn low lon i p y.
5 (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \; \, \; \;. T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \; \; \;,. T umnin p r i ; w umn low lon i p y. (, ) (, ) (, ) (, ) (, ) (, ) LIST : \; \, \; \; \, \.
6 Now i no ll, o r i no umnin p, o i low i opiml. T vlu o i low i. A minimum u i rmin y priionin no orin o wr y riv ll in i l irion: W = {,,,,, }, W = {,,, }. W W T piy o i u i um o pii ij o r poin rom no i in W o no j in W : = =, wi qul vlu o mximum low, in orn wi mx-low min-u orm. No lo in mximum low ll r o orwr ro u (rom W o W ) r ur wil ll r o kwr ro u (rom W o W ) v zro low.. Crully ri n lorim or ollowin prolm: Givn impl unir rp G = (V, E), rmin wr G i ipri, n i o iv ipriion (i.., priion o vrx V ino wo U n W u vry on npoin in U n on npoin in W ). Illur oprion o your lorim on wo xmpl, on ipri rp n on non-ipri rp. Prov your lorim i orr in nrl. Soluion. Hr r p o n lorim or i prolm.. Bin wi ll vri unolor. S LIST :=.. Coo n unolor vrx v. Color v r. A v o LIST.. Coo vrx w in LIST, n rmov w rom LIST.. I ny nior o w m olor w, op: rp i no ipri.. For unolor nior x o w: olor x lu i w i r, or olor x r i w i lu; n n x o LIST.. I LIST i nonmpy, o k o p.. I r ill xi unolor vri, o k o p.. W r on. T rp i ipri. L U o r vri, n l W o lu vri; n (U, W ) i ipriion. For xmpl, onir ollowin rp, wi i ipri:
7 . W in wi ll vri unolor n n mpy LIST.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST ; i mu, i only lmn o LIST. W rmov i rom LIST, o now LIST i mpy in.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. W olor unolor nior o r n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i mpy, o w kip p.. Tr ill xi unolor vri, o w o k o p.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. Ti i only vrx. Now LIST = {}.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.
8 . No nior o m olor, o w kip p.. W olor unolor nior o r n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. No nior o i unolor, o r i noin o o in p.. LIST i mpy, o w kip p.. All vri v n olor, o w kip p.. T rp i ipri, n U = {,,,, }, W = {,, } i ipriion. For nor xmpl, onir ollowin rp, wi i no ipri:. W in wi ll vri unolor n n mpy LIST.. W oo n unolor vrx, y, n olor i r. W o LIST. Now LIST = {}.. W oo vrx in LIST, wi mu, n rmov i rom LIST. Now LIST i mpy.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. T vri r n. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.
9 . W olor unolor nior o r n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. No nior o m olor, o w kip p.. W olor unolor nior o lu n vri o LIST. Ti i only vrx. Now LIST = {, }.. LIST i nonmpy, o w o k o p.. W oo vrx in LIST, y, n rmov i rom LIST. Now LIST = {}.. W noi on o nior o, nmly, m olor. Tror w rpor rp i no ipri. In orr o prov orrn o i lorim, w ir prov Propoiion rom Sion A. o Ppimiriou n Siliz, n in orr o o w ir prov lmm. Lmm. Evry lo o wlk onin n o yl. Proo. By inuion on ln l o wlk. T or nrl rp i l =. In i, wlk oni o loop (n joinin vrx o il). T loop il i n o yl. No i nno our in impl rp, wi no loop. T or impl rp i l =. In i, wlk mu yl o ln (u rp o no onin loop). For inuiv p, uppo l (or l or nrl rp). I no vrx i rp in wlk (xp ir n l vrx, wi r m), n wlk il i n o yl. Orwi, om vrx v i rp, n w n rk wlk ino wo v v wlk. Sin l i o, ln o on o v v wlk mu o, n y inuion i onin n o yl. Propoiion. A rp i ipri i i no irui o o ln. Proo. (= ) Suppo G i ipri, n l (U, W ) ipriion o vrx. Tn vry on npoin in U n on npoin in W, o vri o wlk lrn wn vri in U n vri in W. Tu wlk in n n m vrx (in priulr, yl) mu v vn ln. So G no yl o o ln. ( =) Suppo G = (V, E) no yl o o ln. W ll onru ipriion (U, W ) o G. Wiou lo o nrliy, w my um G i onn; orwi, w n onru ipriion (U i, W i ) o onn omponn prly n n k U = U i n W = W i. Fix n rirry vrx u V, n or ll v V in (v) o or ln o p rom u o v. [Sin G i onn y umpion, (v) i wll in or ll v V.] L U = { v V : (v) i vn } n W = { v V : (v) i o }. I r i n wn wo vri u, u U, n or p rom u o u, plu rom u o u, plu or p rom u o u yil lo o wlk in U. By prin lmm, u wlk onin n o yl. Bu i onri ypoi G onin no o yl, o r n no u wn vri in U. Likwi, r n no wn vri in W. So (U, W ) i ipriion. Now w li orrn o lorim. Suppo lorim rpor rp i ipri. Ti mu ppn in p, wi nno ppn unil ll vri v n olor n pro y p. I rulin vrx olorin o rp in m olor o o npoin o n, n i woul v n noi in p wn proin on npoin; o i
10 o no ppn. Tror vry on r npoin n on lu npoin, o (U, W ) i ipriion, n rp i in ipri. On or n, uppo lorim rpor rp i no ipri. i our wn i i noi in p wo jn vri w n x v m olor. T vri on v in p r ir vri in ir onn omponn o in olor. Evry or vrx i in r i i i n vn in rom on o v or lu i i i n o in (wr in mn numr o in or p ). I wo jn vri v m olor, n r i lo o wlk in rp [ in ( =) pr o proo o Propoiion ], o rp onin n o yl, o i i in no ipri.. Fin mximum min in ollowin ipri rp. u v w x y z Soluion. On wy o o i i o ru prolm o o inin mximum low in ollowin nwork: u v w x y z 0
11 T orrponin mx-low LP i own low. In i LP, vril v no vlu o low, n or r (i, j) in nwork vril ij no low lon r. T ir onrin nor low ln vry no: n oulow vry no mu 0, xp no wr i i v n no wr i i v. T rminin onrin nor pii o r rom n r o. mximiz uj o v = v u + w = 0 u + v + w + y = 0 x + z = 0 u + v + w + y = 0 x + z = 0 x + z = 0 u u u u = 0 v v v = 0 w w w w = 0 x x x x = 0 y y y = 0 z z z z = 0 u v w x y z = v u v w x y z ll vril nonniv. An opiml oluion o i LP i v =, w = y = x = v = z =, = = = = = v = w = x = y = z =, n ll or vril 0. Tror, mximum min in i rp i own low. u v w x y z (Ti mximum min i no uniqu.)
12 . A mll rukin ompny l o iv ruk, n on rin y vn lo o livr. In ollowin l, pii o ruk n iz o lo r o ivn in uni o 000 poun. Dily Truk Cpiy o $00 $00 $00 $0 $00 Lo Wi A B C D E F G T ily o or ruk mu pi i ruk i o u o mk ny livri. Bu o ir loion, lo A n D nno livr y m ruk, nor n lo B n E. Formul n inr prorm o rmin wi lo oul in o ruk in orr o minimiz ol ily o. Soluion. For i {A, B, C, D, E, F, G} n j {,,,, }, l x ij {0, } ini wr lo i i o livr y ruk j. For j {,,,, }, l y j {0, } ini wr ily o i o pi or ruk j. Our ojiv i o minimiz ol ily o, wi i 00y + 00y + 00y + 0y + 00y. W v vrl o onrin. Fir, vry lo mu livr y xly on ruk, o x ij = or ll i {A, B, C, D, E, F, G}. j= T ruk pii nno x, o x Aj + x Bj + x Cj + x Dj + x Ej + x Fj + x Gj W j or ll j {,,,, }, wr W j i piy o ruk j. T ily o mu pi o u ruk, o x ij y j or ll i {A, B, C, D, E, F, G} n ll j {,,,, }, u i y j = 0 n x ij mu lo 0. Finlly, lo A n D nno livr y m ruk, nor n lo B n E, o x Aj + x Dj or ll j {,,,, }, x Bj + x Ej or ll j {,,,, }.
13 Tror w v ollowin inr prorm. minimiz 00y + 00y + 00y + 0y + 00y uj o x A + x A + x A + x A + x A = x B + x B + x B + x B + x B = x C + x C + x C + x C + x C = x D + x D + x D + x D + x D = x E + x E + x E + x E + x E = x F + x F + x F + x F + x F = x G + x G + x G + x G + x G = x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A + x B + x C + x D + x E + x F + x G x A y, x A y, x A y, x A y, x A y x B y, x B y, x B y, x B y, x B y x C y, x C y, x C y, x C y, x C y x D y, x D y, x D y, x D y, x D y x E y, x E y, x E y, x E y, x E y x F y, x F y, x F y, x F y, x F y x G y, x G y, x G y, x G y, x G y x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x A + x D, x B + x E x ij {0, } or ll i {A, B, C, D, E, F, G} n ll j {,,,, } y j {0, } or ll j {,,,, }. T ormulion i ll i nry or i prolm. I i inr prorm i olv, i i oun r r i opiml il oluion, vin ol o $0: A, B A, B B B C C C C F F A, F A, F A, F A, E B, D F D, E G D, E G D, E G G D, E C, G C, D, E C, G C, D, E B, G B, D, F A, E, F A, B, G For inn, ir oluion ov x A = x B = x C = x D = x E = x F = x G =, y = y = y = y =, n ll or vril 0.
Jonathan 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 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 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 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 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 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 informationEager 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 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 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 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 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 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 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 informationEager st-ordering. 1 Introduction. Ulrik Brandes
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
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationPresent state Next state Q + M N
Qustion 1. An M-N lip-lop works s ollows: I MN=00, th nxt stt o th lip lop is 0. I MN=01, th nxt stt o th lip-lop is th sm s th prsnt stt I MN=10, th nxt stt o th lip-lop is th omplmnt o th prsnt stt I
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 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 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 informationNUCON NRNON CONRNC ON CURRN RN N CHNOOGY, 011 oo uul o w ul x ol volv y y oll. y ov,., - o lo ll vy ul o Mo l u v ul (G) v Gl vlu oll. u 3- [11]. 000
NU O HMB NRM UNVRY, HNOOGY, C 8 0 81, 8 3-1 01 CMBR, 0 1 1 l oll oll ov ll lvly lu ul uu oll ul. w o lo u uol u z. ul l u oll ul. quk, oll, vl l, lk lo, - ul o u v (G) v Gl o oll. ul l u vlu oll ul uj
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 informationMath 266, Practice Midterm Exam 2
Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.
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 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 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 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 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 informationGraduate Algorithms CS F-18 Flow Networks
Grue Algorihm CS673-2016F-18 Flow Nework Dvi Glle Deprmen of Compuer Siene Univeriy of Sn Frnio 18-0: Flow Nework Diree Grph G Eh ege weigh i piy Amoun of wer/eon h n flow hrough pipe, for inne Single
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 informationRelation between Fourier Series and Transform
EE 37-3 8 Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Rlion bwn ourir Sris n Trnsform Th ourir Trnsform T is riv from h finiion of h ourir Sris S. Consir, for xmpl, h prioic complx sinl To wih prio
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 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 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 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 informationSolutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1
Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):
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 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 informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationb. How many ternary words of length 23 with eight 0 s, nine 1 s and six 2 s?
MATH 3012 Finl Exm, My 4, 2006, WTT Stunt Nm n ID Numr 1. All our prts o this prolm r onrn with trnry strings o lngth n, i.., wors o lngth n with lttrs rom th lpht {0, 1, 2}.. How mny trnry wors o lngth
More informationVisit to meet more individuals who benefit from your time
NOURISHINGN G. Vlz S 2009 BR i y ii li i Cl. N i. J l l. Rl. A y l l i i ky. Vii.li.l. iiil i y i &. 71 y l Cl y, i iil k. 28 y, k My W i ily l i. Uil y, y k i i. T j il y. Ty il iy ly y - li G, y Cl.
More informationTheorem 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 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 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 informationEE1000 Project 4 Digital Volt Meter
Ovrviw EE1000 Projt 4 Diitl Volt Mtr In this projt, w mk vi tht n msur volts in th rn o 0 to 4 Volts with on iit o ury. Th input is n nlo volt n th output is sinl 7-smnt iit tht tlls us wht tht input s
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 information24.1 Sex-Linked Inheritance. Chapter 24 Chromosomal Basis of Inheritance Sex-Linked Inheritance Sex-Linked Inheritance
ptr 24 romosoml sis o Inritn 24. Sx-Link Inritn Normlly, ot mls n mls v 23 pirs o romosoms 22 pirs r ll utosoms On pir is t sx romosoms Mls r XY mls r XX opyrit T Mrw-Hill ompnis, In. Prmission rquir or
More informationWireless & Hybrid Fire Solutions
ic b 8 c b u i N5 b 4o 25 ii p f i b p r p ri u o iv p i o c v p c i b A i r v Hri F N R L L T L RK N R L L rr F F r P o F i c b T F c c A vri r of op oc F r P, u icoc b ric, i fxib r i i ribi c c A K
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 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 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 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 information1K21 LED GR N +33V 604R VR? 1K0 -33V -33V 0R0 MUTE SWTH? JA? T1 T2 RL? +33V 100R A17 CB? 1N N RB? 2K0 QBI? OU T JE182 4K75 RB? 1N914 D?
L P.O. O X 0, N L R. PROROUH, ONRIO N KJ Y PHO N (0) FX (0) 0 WWW.RYSON. ate : Size : 000 File : OVRLL SHMI.Schoc Sheet : 0 of 0 Rev : rawn : 0.0 0K K 0K K 0K0 0K0 0K0 0K0 0K0 00K R K0 R K 0R??? 00N M?
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 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 informationWalk Like a Mathematician Learning Task:
Gori Dprtmnt of Euction Wlk Lik Mthmticin Lrnin Tsk: Mtrics llow us to prform mny usful mthmticl tsks which orinrily rquir lr numbr of computtions. Som typs of problms which cn b on fficintly with mtrics
More informationc 2009 Society for Industrial and Applied Mathematics
SIAM J. DISCRETE MATH. Vol. 0, No. 0, pp. 000 000 2009 Soity or Inustril n Appli Mtmtis THE TWO-COLORING NUMBER AND DEGENERATE COLORINGS OF PLANAR GRAPHS HAL KIERSTEAD, BOJAN MOHAR, SIMON ŠPACAPAN, DAQING
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 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# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.
How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0
More informationErlkönig. t t.! t t. t t t tj "tt. tj t tj ttt!t t. e t Jt e t t t e t Jt
Gsng Po 1 Agio " " lkö (Compl by Rhol Bckr, s Moifi by Mrk S. Zimmr)!! J "! J # " c c " Luwig vn Bhovn WoO 131 (177) I Wr Who!! " J J! 5 ri ris hro' h spä h, I urch J J Nch rk un W Es n wil A J J is f
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 informationAn undirected graph G = (V, E) V a set of vertices E a set of unordered edges (v,w) where v, w in V
Unirt Grphs An unirt grph G = (V, E) V st o vrtis E st o unorr gs (v,w) whr v, w in V USE: to mol symmtri rltionships twn ntitis vrtis v n w r jnt i thr is n g (v,w) [or (w,v)] th g (v,w) is inint upon
More informationMinimum Spanning Trees
Yufi Tao ITEE Univrsity of Qunslan In tis lctur, w will stuy anotr classic prolm: finin a minimum spannin tr of an unirct wit rap. Intrstinly, vn tou t prolm appars ratr iffrnt from SSSP (sinl sourc sortst
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 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 informationSolutions to Homework 5
Solutions to Homwork 5 Pro. Silvia Frnánz Disrt Mathmatis Math 53A, Fall 2008. [3.4 #] (a) Thr ar x olor hois or vrtx an x or ah o th othr thr vrtis. So th hromati polynomial is P (G, x) =x (x ) 3. ()
More informationMaximum Flow. Flow Graph
Mximum Flow Chper 26 Flow Grph A ommon enrio i o ue grph o repreen flow nework nd ue i o nwer queion ou meril flow Flow i he re h meril move hrough he nework Eh direed edge i ondui for he meril wih ome
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 informationWeighted graphs -- reminder. Data Structures LECTURE 15. Shortest paths algorithms. Example: weighted graph. Two basic properties of shortest paths
Dt Strutur LECTURE Shortt pth lgorithm Proprti of hortt pth Bllmn-For lgorithm Dijktr lgorithm Chptr in th txtook (pp ). Wight grph -- rminr A wight grph i grph in whih g hv wight (ot) w(v i, v j ) >.
More informationAdrian Sfarti University of California, 387 Soda Hall, UC Berkeley, California, USA
Innionl Jonl of Phoonis n Oil Thnolo Vol. 3 Iss. : 36-4 Jn 7 Rliisi Dnis n lonis in Unifol l n in Unifol Roin s-th Gnl ssions fo h loni 4-Vo Ponil in Sfi Unisi of Clifoni 387 So Hll UC Bkl Clifoni US s@ll.n
More informationLecture 21 : Graphene Bandstructure
Fundmnls of Nnolcronics Prof. Suprio D C 45 Purdu Univrsi Lcur : Grpn Bndsrucur Rf. Cpr 6. Nwor for Compuionl Nnocnolog Rviw of Rciprocl Lic :5 In ls clss w lrnd ow o consruc rciprocl lic. For D w v: Rl-Spc:
More informationYour Choice! Your Character. ... it s up to you!
Y 2012 i y! Ti i il y Fl l/ly Iiiiv i R Iiy iizi i Ty Pv Riiliy l Diili Piiv i i y! 2 i l & l ii 3 i 4 D i iv 6 D y y 8 P yi N 9 W i Bllyi? 10 B U 11 I y i 12 P ili D lii Gi O y i i y P li l I y l! iy
More informationBeechwood Music Department Staff
Beechwood Music Department Staff MRS SARAH KERSHAW - HEAD OF MUSIC S a ra h K e rs h a w t r a i n e d a t t h e R oy a l We ls h C o l le g e of M u s i c a n d D ra m a w h e re s h e ob t a i n e d
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 informationDFA Minimization. DFA minimization: the idea. Not in Sipser. Background: Questions: Assignments: Previously: Today: Then:
Assinmnts: DFA Minimiztion CMPU 24 Lnu Tory n Computtion Fll 28 Assinmnt 3 out toy. Prviously: Computtionl mols or t rulr lnus: DFAs, NFAs, rulr xprssions. Toy: How o w in t miniml DFA or lnu? Tis is t
More information