13. Binary tree, height 4, eight terminal vertices 14. Full binary tree, seven vertices v 7 v13. v 19

Transcription

1 0. Spnning Trs n Shortst Pths 0. Consir th tr shown blow with root v 0.. Wht is th lvl of v 8? b. Wht is th lvl of v 0? c. Wht is th hight of this root tr?. Wht r th chilrn of v 0?. Wht is th prnt of v 5? f. Wht r th siblings of v? g. Wht r th scnnts of v? 6. Full binry tr, svn vrtics, of which four r intrnl vrtics. Full binry tr, twlv vrtics 8. Full binry tr, nin vrtics 9. Binry tr, hight, svn trminl vrtics 0. Full binry tr, hight, six trminl vrtics v 0. Binry tr, hight, nin trminl vrtics v v. Full binry tr, ight intrnl vrtics, svn trminl vrtics. v v v 5 v8 v 9 v 6 v 0 v v. Binry tr, hight, ight trminl vrtics. Full binry tr, svn vrtics v v v v 5 v 6 v 8 v v 9 5. Full binry tr, nin vrtics, fiv intrnl vrtics 6. Full binry tr, four intrnl vrtics. Drw binry trs to rprsnt th following xprssions:. b (c/( + )) b. /(b c ) In ch of 0 ithr rw grph with th givn spcifictions or xplin why no such grph xists.. Full binry tr, fiv intrnl vrtics 5. Full binry tr, fiv intrnl vrtics, svn trminl vrtics. Binry tr, hight, ightn trminl vrtics 8. Full binry tr, sixtn vrtics 9. Full binry tr, hight, svn trminl vrtics 0. Wht cn you uc bout th hight of binry tr if you know tht it hs th following proprtis?. Twnty-fiv trminl vrtics b. Forty trminl vrtics c. Sixty trminl vrtics Answrs for Tst Yourslf. on vrtx is istinguish from th othrs n is cll th root; th numbr of gs long th uniqu pth btwn it n th root; th mximum lvl of ny vrtx of th tr. vry prnt hs t most two chilrn. vry prnt hs xctly two chilrn. k + ; k + 5.t h, or, quivlntly, log t h 0. Spnning Trs n Shortst Pths I contn tht ch scinc is rl scinc insofr s it is mthmtics. Immnul Knt, 80 An Est Cost irlin compny wnts to xpn srvic to th Miwst n hs rciv prmission from th Frl Avition Authority to fly ny of th routs shown in Figur 0... Minnpolis Milwuk Chicgo Dtroit Cincinnti St. Louis Louisvill Figur 0.. Nshvill

2 0 Chptr 0 Grphs n Trs Th compny wishs to lgitimtly vrtis srvic to ll th citis shown but, for rsons of conomy, wnts to us th lst possibl numbr of iniviul routs to connct thm. On possibl rout systm is givn in Figur 0... Minnpolis Milwuk Chicgo Dtroit Cincinnti St. Louis Louisvill Figur 0.. Nshvill Clrly this systm joins ll th citis. Is th numbr of iniviul routs miniml? Th nswr is ys, n th rson my surpris you. Th fct is tht th grph of ny systm of routs tht stisfis th compny s wishs is tr, bcus if th grph wr to contin circuit, thn on of th routs in th circuit coul b rmov without isconncting th grph (by Lmm 0.5.), n tht woul giv smllr totl numbr of routs. But ny tr with ight vrtics hs svn gs. Thrfor, ny systm of routs tht conncts ll ight vrtics n yt minimizs th totl numbr of routs consists of svn routs. Dfinition A spnning tr for grph G is subgrph of G tht contins vry vrtx of G n is tr. Th prcing iscussion contins th ssnc of th proof of th following proposition: Proposition 0... Evry connct grph hs spnning tr.. Any two spnning trs for grph hv th sm numbr of gs. Proof of (): Suppos G is connct grph. If G is circuit-fr, thn G is its own spnning tr n w r on. If not, thn G hs t lst on circuit C. By Lmm 0.5., th subgrph of G obtin by rmoving n g from C is connct. If this subgrph is circuit-fr, thn it is spnning tr n w r on. If not, thn it hs t lst on circuit C, n, s bov, n g cn b rmov from C to obtin connct subgrph. Continuing in this wy, w cn rmov succssiv gs from circuits, until vntully w obtin connct, circuit-fr subgrph T of G. [This must hppn t som point bcus th numbr of gs of G is finit, n t no stg os rmovl of n g isconnct th subgrph.] Also, T contins vry vrtx of G bcus no vrtics of G wr rmov in constructing it. Thus T is spnning tr for G. Th proof of prt () is lft s n xrcis.

3 Exmpl 0.. Spnning Trs Fin ll spnning trs for th grph G pictur blow. 0. Spnning Trs n Shortst Pths 0 v 5 v v v 0 v v Solution Th grph G hs on circuit v v v v, n rmovl of ny g of th circuit givs tr. Thus, s shown blow, thr r thr spnning trs for G. v 5 v v v 5 v v v 5 v v v 0 v v v 0 v v v 0 v v Minimum Spnning Trs Th grph of th routs llow by th Frl Avition Authority shown in Figur 0.. cn b nnott by ing th istncs (in mils) btwn ch pir of citis. This is on in Figur 0... Minnpolis 55 Milwuk Dtroit 695 Chicgo Cincinnti 8 St. Louis Louisvill 5 Nshvill Figur 0.. Now suppos th irlin compny wnts to srv ll th citis shown, but with rout systm tht minimizs th totl milg. Not tht such systm is tr, bcus if th systm contin circuit, rmovl of n g from th circuit woul not ffct prson s bility to rch vry city in th systm from vry othr (gin, by Lmm 0.5.), but it woul ruc th totl milg of th systm. Mor gnrlly, grph whos gs r lbl with numbrs (known s wights)is cll wigh grph.a minimum-wight spnning tr, or simply minimum spnning tr, is spnning tr for which th sum of th wights of ll th gs is s smll s possibl.

4 0 Chptr 0 Grphs n Trs Dfinition A wight grph is grph for which ch g hs n ssocit positiv rl numbr wight. Th sum of th wights of ll th gs is th totl wight of th grph. A minimum spnning tr for connct wight grph is spnning tr tht hs th lst possibl totl wight compr to ll othr spnning trs for th grph. If G is wigh grph n is n g of G, thn w() nots th wight of n w(g) nots th totl wight of G. Th problm of fining minimum spnning tr for grph is crtinly solvbl. On solution is to list ll spnning trs for th grph, comput th totl wight of ch, n choos on for which this totl is minimum. (Not tht th wll-orring principl for th intgrs gurnts th xistnc of such minimum totl.) This solution, howvr, is infficint in its us of computing tim bcus th numbr of istinct spnning trs is so lrg. For instnc, complt grph with n vrtics hs n n spnning trs. Evn using th fstst computrs vilbl toy, xmining ll such trs in grph with pproximtly 00 vrtics woul rquir mor tim thn is stimt to rmin in th lif of th univrs. In 956 n 95 Josph B. Kruskl n Robrt C. Prim ch scrib much mor fficint lgorithms to construct minimum spnning trs. Evn for lrg grphs, both lgorithms cn b implmnt so s to tk rltivly short computing tims. Josph Kruskl (born 98) Courtsy of Josph Kruskl Kruskl s Algorithm In Kruskl s lgorithm, th gs of connct wight grph r xmin on by on in orr of incrsing wight. At ch stg th g bing xmin is to wht will bcom th minimum spnning tr, provi tht this ition os not crt circuit. Aftr n gs hv bn (whr n is th numbr of vrtics of th grph), ths gs, togthr with th vrtics of th grph, form minimum spnning tr for th grph. Algorithm 0.. Kruskl Input: G [ connct wight grph with n vrtics, whr n is positiv intgr] Algorithm Boy: [Buil subgrph T of G to consist of ll th vrtics of G with gs in orr of incrsing wight. At ch stg, lt m b th numbr of gs of T.]. Initiliz T to hv ll th vrtics of G n no gs.. Lt E b th st of ll gs of G, n lt m := 0.. whil (m < n ). Fin n g in E of lst wight. b. Dlt from E. c. if ition of to th g st of T os not prouc circuit thn to th g st of T n st m := m + n whil Output: T [T is minimum spnning tr for G.]

5 0. Spnning Trs n Shortst Pths 05 Th following xmpl shows how Kruskl s lgorithm works for th grph of th irlin rout systm. Exmpl 0.. Action of Kruskl s Algorithm Dscrib th ction of Kruskl s lgorithm on th grph shown in Figur 0.., whr n = 8. Minnpolis 55 Milwuk 695 Chicgo St. Louis Figur 0.. Dtroit 0 06 Cincinnti 8 Louisvill 5 Nshvill Solution Itrtion Numbr Eg Consir Wight Action Tkn Chicgo Milwuk Louisvill Cincinnti 8 Louisvill Nshvill 5 Cincinnti Dtroit 0 5 St. Louis Louisvill 6 St. Louis Chicgo 6 Chicgo Louisvill 69 not 8 Louisvill Dtroit 06 not 9 Louisvill Milwuk 8 not 0 Minnpolis Chicgo 55 Th tr prouc by Kruskl s lgorithm is shown in Figur Minnpolis 55 Milwuk Chicgo 6 St. Louis Figur 0..5 Dtroit 0 Cincinnti 8 Louisvill 5 Nshvill Whn Kruskl s lgorithm is us on grph in which som gs hv th sm wight s othrs, mor thn on minimum spnning tr cn occur s output. To mk

6 06 Chptr 0 Grphs n Trs th output uniqu, th gs of th grph cn b plc in n rry n gs hving th sm wight cn b in th orr thy ppr in th rry. It is not obvious from th scription of Kruskl s lgorithm tht it os wht it is suppos to o. To b spcific, wht gurnts tht it is possibl t ch stg to fin n g of lst wight whos ition os not prouc circuit? An if such gs cn b foun, wht gurnts tht thy will ll vntully connct? An if thy o connct, wht gurnts tht th rsulting tr hs minimum wight? Of cours, th mr fct tht Kruskl s lgorithm is print in this book my l you to bliv tht vrything works out. But th qustions bov r rl, n thy srv srious nswrs. Thorm 0.. Corrctnss of Kruskl s Algorithm Whn connct, wight grph is input to Kruskl s lgorithm, th output is minimum spnning tr. Proof: Suppos tht G is connct, wight grph with n vrtics n tht T is subgrph of G prouc whn G is input to Kruskl s lgorithm. Clrly T is circuitfr [sinc no g tht complts circuit is vr to T]. Also T is connct. For s long s T hs mor thn on connct componnt, th st of gs of G tht cn b to T without crting circuit is nonmpty. [Th rson is tht sinc G is connct, givn ny vrtx v in on connct componnt C of T n ny vrtx v in nothr connct componnt C, thr is pth in G from v to v.sincc n C r istinct, thr is n g of this pth tht is not in T. Aing to T os not crt circuit in T, bcus ltion of n g from circuit os not isconnct grph n ltion of woul.] Th prcing rgumnts show tht T is circuit-fr n connct. Sinc by construction T contins vry vrtx of G, T is spnning tr for G. Nxt w show tht T hs minimum wight. Lt T b ny minimum spnning tr for G such tht th numbr of gs T n T hv in common is mximum. Suppos tht T = T. Thn thr is n g in T tht is not n g of T. [Sinc trs T n T both hv th sm vrtx st, if thy iffr t ll, thy must hv iffrnt, but sm-siz, g sts.] Now ing to T proucs grph with uniqu circuit (s xrcis 9 t th n of this sction). Lt b n g of this circuit such tht is not in T. [Such n g must xist bcus T is tr n hnc circuit-fr.] Lt T b th grph obtin from T by rmoving n ing. This sitution is illustrt blow. ' (rmov from T to form T.) ( to T to form T.) Th ntir grph is G. T hs blck gs. is in T but not T. is in T but not T. Not tht T hs n gs n n vrtics n tht T is connct [sinc by Lmm 0.5. th subgrph obtin by rmoving n g from circuit in connct grph is connct]. Consquntly, T is spnning tr for G. In ition, w(t ) = w(t ) w( ) + w(). Now w() w( ) bcus t th stg in Kruskl s lgorithm whn ws to T, ws vilbl to b [sinc it ws not lry in T, n t tht stg its

7 0. Spnning Trs n Shortst Pths 0 ition coul not prouc circuit sinc ws not in T ], n woul hv bn h its wight bn lss thn tht of. Thus w(t ) = w(t ) [w( ) w()] }{{} 0 w(t ). But T is minimum spnning tr. Sinc T is spnning tr with wight lss thn or qul to th wight of T, T is lso minimum spnning tr for G. Finlly, not tht by construction, T hs on mor g in common with T thn T os, which contricts th choic of T s minimum spnning tr for G with mximum numbr of gs in common with T. Thus th supposition tht T = T is fls, n hnc T itslf is minimum spnning tr for G. Robrt Prim (born 9) Courtsy of Alctl-Lucnt Tchnologis Prim s Algorithm Prim s lgorithm works iffrntly from Kruskl s. It buils minimum spnning tr T by xpning outwr in connct links from som vrtx. On g n on vrtx r t ch stg. Th g is th on of lst wight tht conncts th vrtics lry in T with thos not in T, n th vrtx is th npoint of this g tht is not lry in T. Algorithm 0.. Input: G [ connct wight grph with n vrtics whr n is positiv intgr] Algorithm Boy: [Buil subgrph T of G by strting with ny vrtx v of G n ttching gs (with thir npoints) on by on to n s-yt-unconnct vrtx of G, ch tim choosing n g of lst wight tht is jcnt to vrtx of T.]. Pick vrtx v of G n lt T b th grph with on vrtx, v, n no gs.. Lt V b th st of ll vrtics of G xcpt v.. for i := to n. Fin n g of G such tht () conncts T to on of th vrtics in V, n () hs th lst wight of ll gs conncting T to vrtx in V.Ltw b th npoint of tht is in V. b. A n w to th g n vrtx sts of T, n lt w from V. nxt i Output: T [T is minimum spnning tr for G.] Th following xmpl shows how Prim s lgorithm works for th grph of th irlin rout systm. Exmpl 0.. Action of Prim s Algorithm Dscrib th ction of Prim s lgorithm for th grph in Figur 0..6 using th Minnpolis vrtx s strting point.

8 08 Chptr 0 Grphs n Trs Minnpolis 55 Milwuk Dtroit 695 Chicgo Cincinnti 8 St. Louis Louisvill 5 Nshvill Figur 0..6 Solution Itrtion Numbr Vrtx A Eg A Wight 0 Minnpolis Chicgo Minnpolis Chicgo 55 Milwuk Chicgo Milwuk St. Louis Chicgo St. Louis 6 Louisvill St. Louis Louisvill 5 Cincinnti Louisvill Cincinnti 8 6 Nshvill Louisvill Nshvill 5 Dtroit Cincinnti Dtroit 0 Not tht th tr obtin is th sm s tht obtin by Kruskl s lgorithm, but th gs r in iffrnt orr. As with Kruskl s lgorithm, in orr to nsur uniqu output, th gs of th grph coul b plc in n rry n thos with th sm wight coul b in th orr thy ppr in th rry. It is not hr to s tht whn connct grph is input to Prim s lgorithm, th rsult is spnning tr. Wht is not so clr is tht this spnning tr is minimum. Th proof of th following thorm stblishs tht it is. Thorm 0.. Corrctnss of Prim s Algorithm Whn connct, wight grph G is input to Prim s lgorithm, th output is minimum spnning tr for G. Proof: Lt G b connct, wight grph, n suppos G is input to Prim s lgorithm. At ch stg of xcution of th lgorithm, n g must b foun tht conncts vrtx in subgrph to vrtx outsi th subgrph. As long s thr r vrtics outsi th subgrph, th connctnss of G nsurs tht such n g cn lwys b foun. [For if on vrtx in th subgrph n on vrtx outsi it r chosn, thn by th connctnss of G thr is wlk in G linking th two. As on trvls long this wlk, t som point on movs long n g from vrtx insi th subgrph to vrtx outsi th subgrph.] Now it is clr tht th output T of Prim s lgorithm is tr bcus th g n vrtx to T t ch stg r connct to othr gs n vrtics of T

9 0. Spnning Trs n Shortst Pths 09 n bcus t no stg is circuit crt sinc ch g conncts vrtics in two isconnct sts. [Consquntly, rmovl of nwly g proucs isconnct grph, whrs by Lmm 0.5., rmovl of n g from circuit proucs connct grph.] Also, T inclus vry vrtx of G bcus T, bing trwithn gs, hs n vrtics [n tht is ll G hs]. Thus T is spnning tr for G. Nxt w show tht T hs minimum wight. Lt T b minimum spnning tr for G such tht th numbr of gs T n T hv in common is mximum. Suppos tht T = T. Thn thr is n g in T tht is not n g of T. [Sinc trs T n T both hv th sm vrtx st if thy iffr t ll, thy must hv iffrnt, sm-siz g sts.] Of ll such gs, lt b th lst tht ws whn T ws construct using Prim s lgorithm. Lt S b th st of vrtics of T just bfor th ition of. Thn on npoint, sy v of, isins n th othr, sy w, is not. Sinc T is spnning tr, thr is pth in T joining v to w. An sinc v S n w S, s on trvls long this pth, on must ncountr n g tht joins vrtx in S to on tht is not in S n tht thrfor is not in T bcus ws th lst g to T. Now t th stg whn ws to T, coul lso hv bn n it woul hv bn inst of h its wight bn lss thn tht of. Sinc ws not t tht stg, w conclu tht w( ) w(). Lt T b th grph obtin from T by rmoving n ing. [Thus T hs on mor g in common with T thn T os.] Not tht T is tr. Th rson is tht sinc is prt of pth in T from v to w, n conncts v n w, ing to T crts circuit. Whn is rmov from this circuit, th rsulting subgrph rmins connct. In fct, T is spnning tr for G sinc no vrtics wr rmov in forming T from T. Th rgumnt showing tht w(t ) w(t ) is lft s n xrcis. [It is virtully inticl to prt of th proof of Thorm 0...] It follows tht T is minimum spnning tr for G. By construction, T hs on mor g in common with T thn T, os which contricts th choic of T s minimum spnning tr for G withmximum numbr of gs in common with T. It follows tht T = T, n hnc T itslf is minimum spnning tr for G. Exmpl 0.. Fining Minimum Spnning Trs Fin ll minimum spnning trs for th following grph. Us Kruskl s lgorithm n Prim s lgorithm strting t vrtx. Inict th orr in which gs r to form ch tr. b c f Solution Whn Kruskl s lgorithm is ppli, gs r in on of th following two orrs:. {, f }, {, c}, {, b}, {c, }, {, }. {, f }, {, c}, {b, c}, {c, }, {, }

10 0 Chptr 0 Grphs n Trs Whn Prim s lgorithm is ppli strting t, gs r in on of th following two orrs:. {, c}, {, b}, {c, }, {, f }, {, }. {, c}, {b, c}, {c, }, {, f }, {, } Thus, s shown blow, thr r two istinct minimum spnning trs for this grph. b c b 5 f 5 f c ( ) (b) Dijkstr s Shortst Pth Algorithm Although th trs prouc by Kruskl s n Prim s lgorithms hv th lst possibl totl wight compr to ll othr spnning trs for th givn grph, thy o not lwys rvl th shortst istnc btwn ny two points on th grph. For instnc, ccoring to th complt rout systm shown in Figur 0.., on cn fly irctly from Nshvill to Minnpolis for istnc of 695 mils, whrs if you us th minimum spnning tr shown in Figur 0..5 th only wy to fly from Nshvill to Minnpolis is by going through Louisvill, St. Louis, n Chicgo, which givs totl istnc of =, 00 mils n th unplsntnss of thr chngs of pln. In 959 th computing pionr, Esgr Dijkstr (s Sction 5.5), vlop n lgorithm to fin th shortst pth btwn strting vrtx n n ning vrtx in wight grph in which ll th wights r positiv. It is somwht similr to Prim s lgorithm in tht it works outwr from strting vrtx, ing vrtics n gs on by on to construct tr T. Howvr, it iffrs from Prim s lgorithm in th wy it chooss th nxt vrtx to, nsuring tht for ch vrtx v, th lngth of th shortst pth from to v hs bn intifi. At th strt of xcution of th lgorithm, ch vrtx u of G is givn lbl L(u), which inicts th currnt bst stimt of th lngth of th shortst pth from to u. L() is initilly st qul to 0 bcus th shortst pth from to hs lngth zro, but, bcus thr is no prvious informtion bout th lngths of th shortst pths from to ny othr vrtics of G, th lbl L(u) of ch vrtx u othr thn is initilly st qul to numbr, not, tht is grtr thn th sum of th wights of ll th gs of G. As xcution of th lgorithm progrsss, th vlus of L(u) r chng, vntully bcoming th ctul lngths of th shortst pths from to u in G. Bcus T is built up outwr from, t ch stg of xcution of th lgorithm th only vrtics tht r cnits to join T r thos tht r jcnt to t lst on vrtx of T. Thus t ch stg of Dijkstr s lgorithm, th grph G cn b thought of s ivi into thr prts: th tr T tht is bing built up, th st of fring vrtics tht r jcnt to t lst on vrtx of th tr, n th rst of th vrtics of G. Ech fring vrtx is cnit to b th nxt vrtx to T. Th on tht is chosn is th on for which th lngth of th shortst pth to it from through T is minimum mong ll th vrtics in th fring. An ssntil obsrvtion unrlying Dijkstr s lgorithm is tht ftr ch ition of vrtxv to T, th only fring vrtics for which shortr pth from might b foun r thos tht r jcnt to v [bcus th lngth of th pth from to v ws minimum mong ll th pths from to vrtics in wht ws thn th fring]. So ftr ch ition of vrtx v to T, ch fring vrtx u jcnt to v is xmin n two numbrs r

11 0. Spnning Trs n Shortst Pths compr: th currnt vlu of L(u) n th vlu of L(v) + w(v, u), whr L(v) is th lngth of th shortst pth to v (in T ) n w(v, u) is th wight of th g joining v n u. IfL(v) + w(v, u) <L(u), thn th vlu of L(u) is chng to L(v) + w(v, u). At th bginning of xcution of th lgorithm, th tr consists only of th vrtx, n L() = 0. Whn xcution trmints, L(z) is th lngth of shortst pth from to z. As with Kruskl s n Prim s lgorithms for fining minimum spnning trs, thr is simpl but rmticlly infficint wy to fin th shortst pth from to z: comput th lngths of ll th pths n choos on tht is shortst. Th problm is tht vn for rltivly smll grphs using this mtho to fin shortst pth coul rquir billions of yrs, whrs Dijkstr s lgorithm coul o th job in fw scons. Algorithm 0.. Dijkstr Input: G[ connct simpl grph with positiv wight for vry g], [ numbr grtr thn th sum of th wights of ll th gs in th grph], w(u,v) [th wight of g {u,v}],[th strting vrtx], z[th ning vrtx] Algorithm Boy:. Initiliz T to b th grph with vrtx n no gs. Lt V (T ) b th st of vrtics of T, n lt E(T ) b th st of gs of T.. Lt L() = 0, n for ll vrtics in G xcpt, lt L(u) =. [Th numbr L(x) is cll th lbl of x.]. Initiliz v to qul n F to b {}. [Th symbol v is us to not th vrtx most rcntly to T.]. whil (z V (T )). F := (F {v}) {vrtics tht r jcnt to v n r not in V (T )} [Th st F is cll th fring. Ech tim vrtx is to T, it is rmov from th fring n th vrtics jcnt to it r to th fring if thy r not lry in th fring or th tr T.] b. For ch vrtx u tht is jcnt to v n is not in V (T ), if L(v) + w(v, u) <L(u) thn L(u) := L(v) + w(v, u) D(u) := v Not Th uniqu pth in th tr T from to z is th shortst pth in G from to z. [Not tht ing v to T os not ffct th lbls of ny vrtics in th fring F xcpt thos jcnt to v. Also, whn L(u) is chng to smllr vlu, th nottion D(u) is introuc to kp trck of which vrtx in T gv ris to th smllr vlu.] c. Fin vrtx x in F with th smllst lbl A vrtx x to V (T ), n g {D(x), x} to E(T ) v := x [This sttmnt sts up th nottion for th nxt itrtion of th loop.] n whil Output: L(z) [L(z), nonngtiv intgr, is th lngth of th shortst pth from toz.] Th ction of Dijkstr s lgorithm is illustrt by th flow of th rwings in Exmpl 0..5.

12 Chptr 0 Grphs n Trs Exmpl 0..5 Action of Dijkstr s Algorithm Show th stps in th xcution of Dijkstr s shortst pth lgorithm for th grph shown blow with strting vrtx n ning vrtx z. b 6 5 z c 6 Solution Stp : Going into th whil loop: V (T ) ={}, E(T ) =, n F ={} b c 5 6 z During itrtion: F ={b, c}, L(b) =, L(c) =. Sinc L(b) <L(c), b is to V (T ) n {, b} is to E(T ). Stp : Going into th whil loop: V (T ) ={, b}, E(T ) ={{, b}} b c 5 6 z During itrtion: F = {c,, }, L(c) =, L() = 9, L() = 8. Sinc L(c) <L() n L(c) <L(), c is to V (T ) n {, c} is to E(T ). Stp : Going into th whil loop: V (T ) ={, b, c}, E(T ) ={{, b}, {, c}} b 6 During itrtion: F ={, }, L() = 9, L() = 5 L() bcoms 5 bcus c, which hs 5 z lngth 5, is shortr pth to thn b, which hs lngth 8. c Sinc L() <L(), is to V (T ) n {c, } is to E(T ). Stp : Going into th whil loop: V (T ) ={, b, c, }, E(T ) ={{, b}, {, c}, {c, }} During itrtion: b 6 F ={, z}, L() =, L(z) = L() bcoms bcus c, which hs 5 z lngth, is shortr pth to thn b, which hs lngth 9. c Sinc L() <L(z), is to V (T ) n {, } is to E(T ). Stp 5: Going into th whil loop: V (T ) ={, b, c,, }, E(T ) ={{, b},{, c}, {c, }, {, }} During itrtion: F ={z}, L(z) = b 6 L(z) bcoms bcus cz, which hs lngth, is shortr pth to thn 5 z bz, which hs lngth. Sinc z is th only vrtx in F, its lbl c is minimum, n so z is to V (T ) n {, z} is to E(T ).

13 0. Spnning Trs n Shortst Pths Excution of th lgorithm trmints t this point bcus z V (T ). Th shortst pth from to z hs lngth L(z) =. Kping trck of th stps in tbl is convnint wy to show th ction of Dijkstr s lgorithm. Tbl 0.. os this for th grph in Exmpl Tbl 0.. Stp V (T ) E(T ) F L() L(b) L(c) L() L() L(z) 0 {} {} 0 {} {b, c} 0 {, b} {{, b}} {c,, } {, b, c} {{, b}, {, c}} {, } {, b, c, } {{, b}, {, c}, {c, }} {, z} {, b, c,, } {{, b}, {, c}, {c, }, {, }} {z} {, b, c,,, z} {{, b}, {, c}, {c, }, {, }, {, z}} It is clr tht Dijkstr s lgorithm kps ing vrtics to I until it hs z.th proof of th following thorm shows tht whn th lgorithm trmints, th lbl z gos th lngth of th shortst pth to it from. Thorm 0.. Corrctnss of Dijkstr s Algorithm Whn connct, simpl grph with positiv wight for vry g is input to Dijkstr s lgorithm with strting vrtx n ning vrtx z, th output is th lngth of shortst pth from to z. Proof: Lt G b connct, wight grph with no loops or prlll gs n with positiv wight for vry g. Lt T b th grph built up by Dijkstr s lgorithm, n for ch vrtx u in G, ltl(u) b th lbl givn by th lgorithm to vrtx u. For ch intgr n 0, lt th proprty P(n) b th sntnc Aftr th nth itrtion of th whil loop in Dijkstr s lgorithm, () T is tr, n () for vry vrtx v in T, L(v) is th lngth of P(n) shortst pth in G from to v. W will show by mthmticl inuction tht P(n) is tru for ll intgrs n from 0 through th trmintion of th lgorithm. Show tht P(0) istru:whn n = 0, th grph T is tr bcus it is fin to consist only of th vrtx n no gs. In ition, L() is th lngth of th shortst pth from to bcus th initil vlu of L() is 0. Show tht for ll intgrs k 0, if P(k) is tru thn P(k + ) is lso tru: Lt k b ny intgr with k 0 n suppos tht Aftr th kth itrtion of th whil loop in Dijkstr s lgorithm, () T P(k) is tr, n () for vry vrtx v in T, L(v) is th lngth of inuctiv hypothsis shortst pth in G from to v. W must show tht Aftr th (k + )st itrtion of th whil loop in Dijkstr s lgorithm, () T is tr, n () for vry vrtx v in T, P(k + ) L(v) is th lngth of shortst pth in G from to v. continu on pg

14 Chptr 0 Grphs n Trs So suppos tht ftr th (k + )st itrtion of th whil loop in Dikjstr s lgorithm, th vrtx v n g {x,v} hv bn to T, whr x is in V (T ). Clrly th nw vlu of T is tr bcus ing nw vrtx n g to tr os not crt circuit n os not isconnct th tr. By inuctiv hypothsis for ch vrtx y in th tr bfor th ition of v, L(y) is th lngth of shortst pth from to y. So it rmins only to show tht L(v) is th lngth of shortst pth from to v. Now, ccoring to th lgorithm, th finl vlu of L(v) = L(x) + w(x,v). Consir ny shortst pth from to v, n lt {s, t} b th first g in this pth to lv T, whr s V (T ) n t V (T ). This sitution is illustrt blow. s t x Lt LSP(, v)b th lngth of shortst pth from to v, n lt LSP(, s) b th lngth of shortst pth from to s. Obsrv tht LSP(, v) LSP(, s) + w(s, t) bcus th pth from t to v hs lngth 0 L(s) + w(s, t) by inuctiv hypothsis bcus s is vrtx in T t is in th fring of th tr, n so if L(s)+w(s, t) L(x) + w(x,v) wr lss thn L(x)+w(x,v)thnt woul hv bn to T inst of v. On th othr hn bcus L(x)+w(x, v) is th lngth of pth from L(x) + w(x, v) LSP(,v) to v n so it is grtr thn or qul to th lngth of th shortst pth from to v. It follows tht LSP(,v) = L(x) + w(x,v), n, sinc L(v) = L(x) + w(x,v), L(v) is th lngth of shortst pth from to v. This complts th proof by mthmticl inuction. Th lgorithm trmints s soon s z is in T, n, sinc w hv prov tht th lbl of vry vrtx in th tr givs th lngth of th shortst pth to it from, thn, in prticulr, L(z) is th lngth of shortst pth from to z. v Tst Yourslf. A spnning tr for grph G is.. A wight grph is grph for which, n th totl wight of th grph is.. A minimum spnning tr for connct, wight grph is.. In Kruskl s lgorithm, th gs of connct, wight grph r xmin on by on in orr of strting with. 5. In Prim s lgorithm, minimum spnning tr is built by xpning outwr from n in squnc of. 6. In Dijkstr s lgorithm, vrtx is in th fring if it is vrtx in th tr tht is bing built up.. At ch stg of Dijkstr s lgorithm, th vrtx tht is to th tr is vrtx in th fring whos lbl is.

15 0. Spnning Trs n Shortst Pths 5 Exrcis St 0. Fin ll possibl spnning trs for ch of th grphs in n.. b. v 0 v c v v Fin spnning tr for ch of th grphs in n.. b c. z g r s t y x w f v Us Kruskl s lgorithm to fin minimum spnning tr for ch of th grphs in 5 n 6. Inict th orr in which gs r to form ch tr. 5. b c g 6 f 6. v 0 v 0 5 v 0 5 v v v 8 5 v 6 v 8 9 Us Prim s lgorithm strting with vrtx or v 0 to fin minimum spnning tr for ch of th grphs in n 8. Inict th orr in which gs r to form ch tr.. Th grph of xrcis Th grph of xrcis 6. For ch of th grphs in 9 n 0, fin ll minimum spnning trs tht cn b obtin using () Kruskl s lgorithm n (b) Prim s lgorithm strting with vrtx or t. Inict th orr in which gs r to form ch tr. 9. g 5 f b u c 0 0. t u 5 8 w 5 x y 5 z v 0. A piplin is to b built tht will link six citis. Th cost (in hunrs of millions of ollrs) of constructing ch potntil link pns on istnc n trrin n is shown in th wight grph blow. Fin systm of piplins to connct ll th citis n yt minimiz th totl cost. Slt Lk City.5 Chynn Dnvr Amrillo Phonix Albuqurqu. Us Dijkstr s lgorithm for th irlin rout systm of Figur 0.. to fin th shortst istnc from Nshvill to Minnpolis. Mk tbl similr to Tbl 0.. to show th ction of th lgorithm. Us Dijkstr s lgorithm to fin th shortst pth from to z for ch of th grphs in 6. In ch cs mk tbls similr to Tbl 0.. to show th ction of th lgorithm.. b c 8 5 z 0. b c 8 f 5. Th grph of xrcis 9 with = n z = f 6. Th grph of xrcis 0 with = u n z = w. Prov prt () of Proposition 0..: Any two spnning trs for grph hv th sm numbr of gs. 8. Givn ny two istinct vrtics of tr, thr xists uniqu pth from on to th othr.. Giv n informl justifiction for th bov sttmnt. b. Writ forml proof of th bov sttmnt. g 0 z

16 6 Chptr 0 Grphs n Trs 9. Prov tht if G is grph with spnning tr T n is n g of G tht is not in T, thn th grph obtin by ing to T contins on n only on st of gs tht forms circuit. 0. Suppos G is connct grph n T is circuit-fr subgrph of G. Suppos lso tht if ny g of G not in T is to T, th rsulting grph contins circuit. Prov tht T is spnning tr for G... Suppos T n T r two iffrnt spnning trs for grphg. MustT n T hv n g in common? Prov or giv countrxmpl. b. Suppos tht th grph G in prt () is simpl. Must T n T hv n g in common? Prov or giv countrxmpl. H. Prov tht n g is contin in vry spnning tr for connct grph G if, n only if, rmovl of isconncts G.. Consir th spnning trs T n T in th proof of Thorm 0... Prov tht w(t ) w(t ).. Suppos tht T is minimum spnning tr for connct, wight grph G n tht G contins n g (not loop) tht is not in T.Ltv n w b th npoints of.byxrcis 8 thr is uniqu pth in T from v to w. Lt b ny g of this pth. Prov tht w( ) w(). H Prov tht if G is connct, wight grph n is n g of G (not loop) tht hs smllr wight thn ny othr g of G, thn is in vry minimum spnning tr for G. If G is connct, wight grph n no two gs of G hv th sm wight, os thr xist uniqu minimum spnning tr for G? Us th rsult of xrcis 9 to hlp justify your nswr.. Prov tht if G is connct, wight grph n is n g of G tht () hs grtr wight thn ny othr g of G n () is in circuit of G, thn thr is no minimum spnning tr T for G such tht is in T. 8. Suppos isconnct grph is input to Kruskl s lgorithm. Wht will b th output? 9. Suppos isconnct grph is input to Prim s lgorithm. Wht will b th output? 0. Prov tht if connct, wight grph G is input to Algorithm 0.. (shown blow), th output is minimum spnning tr for G. Algorithm 0.. Input: G [ connct grph] Algorithm Boy:. T := G.. E := th st of ll gs of G, m := th numbr of gs of G.. whil (m > 0). Fin n g in E tht hs mximl wight. b. Rmov from E n st m := m. c. if th subgrph obtin whn is rmov from th g st of T is connct thn rmov from th g st of T n whil Output: T [ minimum spnning tr for G]. Moify Algorithm 0.. so tht th output consists of th squnc of gs in th shortst pth from to z. Answrs for Tst Yourslf. subgrph of G tht contins vry vrtx of G n is tr.. ch g hs n ssocit positiv rl numbr wight; th sum of th wights of ll th gs of th grph. spnning tr tht hs th lst possibl totl wight compr to ll othr spnning trs for th grph. wight; n g of lst wight 5. initil vrtx; jcnt vrtics n gs 6. jcnt to. minimum mong ll thos in th fring