14 Shortest Paths (November 8)

Transcription

1 CS G Ltur : Shortt Pth Fll 5 Shortt Pth (Novmr ). Introution Givn wight irt grph G = (V, E, w) with two pil vrti, our n trgt t, w wnt to in th hortt irt pth rom to t. In othr wor, w wnt to in th pth p trting t n ning t t minimizing th untion w(p) = p w(). For xmpl, i I wnt to nwr th qution Wht th tt wy to riv rom my ol prtmnt in Chmpign, Illinoi to my wi ol prtmnt in Columu, Ohio?, w might u grph who vrti r iti, g r ro, wight r riving tim, i Chmpign, n t i Columu. Th grph i irt in th riving tim long th m ro might irnt in irnt irtion. Prhp ountr to intuition, w will llow th wight on th g to ngtiv. Ngtiv g mk our liv omplit, in th prn o ngtiv yl might mn tht thr i no hortt pth. In gnrl, hortt pth rom to t xit i n only i thr i t lt on pth rom to t, ut thr i no pth rom to t tht touh ngtiv yl. I thr i ngtiv yl twn n t, thn n lwy in hortr pth y going roun th yl on mor tim. 5 t Thr i no hortt pth rom to t. Evry lgorithm known or olving thi prolm tully olv (lrg portion o) th ollowing mor gnrl ingl our hortt pth or SSSP prolm: in th hortt pth rom th our vrtx to vry othr vrtx in th grph. In t, th prolm i uully olv y ining hortt pth tr root t tht ontin ll th ir hortt pth. It not hr to tht i th hortt pth r uniqu, thn thy orm tr. To prov thi, it nough to orv tht u-pth o hortt pth r lo hortt pth. I thr r multipl hortt pth to th m vrti, w n lwy hoo on pth to h vrtx o tht th union o th pth i tr. I thr r hortt pth to two vrti u n v tht ivrg, thn mt, thn ivrg gin, w n moiy on o th pth o tht th two pth only ivrg on. u x I v n x y u r oth hortt pth, thn u i lo hortt pth. Wt on Churh, north on Propt, t on I-, outh on I-65, t on Airport Exprwy, north on I-65, t on I-, north on Grnviw, t on 5th, north on Olntngy Rivr, t on Dorig, north on High, wt on Klo, outh on Nil. Dpning on tri. W oth liv in Urn now. Thr i p trp on I- jut ini th Ohio orr, ut only or toun tri. y v

2 CS G Ltur : Shortt Pth Fll 5 I houl mphiz hr tht hortt pth tr n minimum pnning tr r uully vry irnt. For on thing, thr i only on minimum pnning tr, ut in gnrl, thr i irnt hortt pth tr or vry our vrtx A minimum pnning tr n hortt pth tr (root t th topmot vrtx) o th m grph. All o th lgorithm I m riing in thi ltur lo work or unirt grph, with om light moiition. Mot importntly, w mut piilly prohiit ltrnting k n orth ro th m unirt ngtiv-wight g. Our unmoii lgorithm woul intrprt ny uh g ngtiv yl o lngth.. Th Only SSSP Algorithm Jut lik grph trvrl n minimum pnning tr, thr r vrl irnt SSSP lgorithm, ut thy r ll pil o th ingl gnri lgorithm. Eh vrtx v in th grph tor two vlu, whih ri tnttiv hortt pth rom to v. it(v) i th lngth o th tnttiv hortt v pth. pr(v) i th pror o v in th tnttiv hortt v pth. Noti tht th pror pointr utomtilly in tnttiv hortt pth tr. W lry know tht it() = n pr() = Null. For vry vrtx v, w initilly t it(v) = n pr(v) = Null to init tht w o not know o ny pth rom to v. W ll n g u v tn i it(u) + w(u v) < it(v). I u v i tn, thn th tnttiv hortt pth v i inorrt, in th pth u v i hortr. Our gnri lgorithm rptly in tn g in th grph n rlx it: Rlx(u v): it(v) it(u) + w(u v) pr(v) u I thr r no tn g, our lgorithm i inih, n w hv our ir hortt pth tr. Th orrtn o th rlxtion lgorithm ollow irtly rom thr impl lim:. I it(v), thn it(v) i th totl wight o th pror hin ning t v: pr(pr(v)) pr(v) v. Thi i y to prov y inution on th numr o rlxtion tp. (Hint, hint.). I th lgorithm hlt, thn it(v) w( v) or ny pth v. Thi i y to prov y inution on th numr o g in th pth v. (Hint, hint.)

3 CS G Ltur : Shortt Pth Fll 5. Th lgorithm hlt i n only i thr i no ngtiv yl rhl rom. Th only i irtion i y i thr i rhl ngtiv yl, thn tr th irt g in th yl i rlx, th yl lwy h t lt on tn g. Th i irtion ollow rom th t tht vry rlxtion tp ru ithr th numr o vrti with it(v) = y or ru th um o th init hortt pth lngth y om poitiv mount. I hvn t i nything out how w tt whih g n rlx, or wht orr w rlx thm in. In orr to mk thi ir, w n rin th rlxtion lgorithm lightly, into omthing loly rmling th gnri grph trvrl lgorithm. W mintin g o vrti, initilly ontining jut th our vrtx. Whnvr w tk vrtx u out o th g, w n ll o it outgoing g, looking or omthing to rlx. Whnvr w uully rlx n g u v, w put v in th g. InitSSSP(): it() pr() Null or ll vrti v it(v) pr(v) Null GnriSSSP(): InitSSSP() put in th g whil th g i not mpty tk u rom th g or ll g u v i u v i tn Rlx(u v) put v in th g Jut with grph trvrl, uing irnt t trutur or th g giv u irnt lgorithm. Thr r thr oviou hoi to try: tk, quu, n hp. Unortuntly, i w u tk, w hv to prorm Θ( V ) rlxtion tp in th wort! (Thi i prolm in th urrnt homwork.) Th othr two poiiliti r muh mor iint.. Dijktr Algorithm I w implmnt th g hp, whr th ky o vrtx v i it(v), w otin n lgorithm irt pulih y Lyzork, Gry, Johnon, Lw, Mkr, Ptry, n Sitz in 5, n thn ltr inpnntly riovr y Egr Dijktr in 5. (A imilr lgorithm w lo ri y Dntzig in 5.) Dijktr lgorithm, it i univrlly known, i prtiulrly wll-hv i th grph h no ngtiv-wight g. In thi, it not hr to how (y inution, o our) tht th vrti r nn in inring orr o thir hortt-pth itn rom. It ollow tht h vrtx i nn t mot on, n thu tht h g i rlx t mot on. Sin th ky o h vrtx in th hp i it tnttiv itn rom, th lgorithm prorm DrKy oprtion vry tim n g i rlx. Thu, th lgorithm prorm t mot E DrKy. Similrly, thr r t mot V Inrt n ExtrtMin oprtion. Thu, i w tor th vrti in Fioni hp, th totl running tim o Dijktr lgorithm i O(E + V log V ). Thi nlyi um tht no g h ngtiv wight. Dijktr lgorithm (in th orm I m prnting hr) i till orrt i thr r ngtiv g, ut th wort- running tim oul xponntil. (I ll lv th proo o thi unortunt t n xtr rit prolm.) Th vrion o Dijktr lgorithm prnt in CLRS giv inorrt rult or grph with ngtiv g.

4 CS G Ltur : Shortt Pth Fll 5 Four ph o Dijktr lgorithm run on grph with no ngtiv g. At h ph, th h vrti r in th hp, n th ol vrtx h jut n nn. Th ol g ri th volving hortt pth tr.. Th A Huriti A light gnrliztion o Dijktr lgorithm, ommonly known th A lgorithm, i rquntly u to in hortt pth rom ingl our no to ingl trgt no t. A u lkox untion GuDitn(v,t) tht rturn n timt o th itn rom v to t. Th only irn twn Dijktr n A i tht th ky o vrtx v i it(v) + GuDitn(v,t). Th untion GuDitn i ll miil i GuDitn(v, t) nvr ovrtimt th tul hortt pth itn rom v to t. I GuDitn i miil n th tul g wight r ll non-ngtiv, th A lgorithm omput th tul hortt pth rom to t t lt quikly Dijktr lgorithm. Th lor GuDitn(v,t) i to th rl itn rom v to t, th tr th lgorithm. Howvr, in th wort, th running tim i till O(E + V log V ). Th huriti i pilly uul in itution whr th tul grph i not known. For xmpl, A n u to olv puzzl (5-puzzl, Frll, Shnghi, Minwpr, Sokon, Atomix, Ruh Hour, Ruik Cu,...) n othr pth plnning prolm whr th trting n gol onigurtion r givn, ut th grph o ll poil onigurtion n thir onntion i not givn xpliitly..5 Shiml Algorithm ( Bllmn-For ) I w rpl th hp in Dijktr lgorithm with quu, w gt n lgorithm tht w irt pulih y Shiml in 55, thn inpnntly riovr y Moor in 5, y Wooury n Dntzig in 5, n y Bllmn in 5. Sin Bllmn u th i o rlxing g, whih w irt propo y For in 56, thi lgorithm i uully ll Bllmn-For. Shiml lgorithm i iint vn i thr r ngtiv g, n it n u to quikly tt th

5 CS G Ltur : Shortt Pth Fll 5 prn o ngtiv yl. I thr r no ngtiv g, howvr, Dijktr lgorithm i tr. (In t, in prti, Dijktr lgorithm i otn tr vn or grph with ngtiv g.) Th it wy to nlyz th lgorithm i to rk th xution into ph. Bor w vn gin, w inrt tokn into th quu. Whnvr w tk th tokn out o th quu, w gin nw ph y jut rinrting th tokn into th quu. Th th ph onit ntirly o nning th our vrtx. Th lgorithm n whn th quu ontin only th tokn. A impl inutiv rgumnt (hint, hint) impli th ollowing invrint: At th n o th ith ph, or vry vrtx v, it(v) i l thn or qul to th lngth o th hortt pth v oniting o i or wr g. Sin hortt pth n only p through h vrtx on, ithr th lgorithm hlt or th V th ph, or th grph ontin ngtiv yl. In h ph, w n h vrtx t mot on, o w rlx h g t mot on, o th running tim o ingl ph i O(E). Thu, th ovrll running tim o Shiml lgorithm i O(V E). 6 6 Four ph o Shiml lgorithm run on irt grph with ngtiv g. No r tkn rom th quu in th orr, whr i th tokn. Sh vrti r in th quu t th n o h ph. Th ol g ri th volving hortt pth tr. Now tht w unrtn how th ph o Shiml lgorithm hv, w n impliy th lgorithm onirly. Int o uing quu to prorm prtil rth-irt rh o th grph in h ph, lt jut n through th jny lit irtly n try to rlx vry g in th grph. 5

6 CS G Ltur : Shortt Pth Fll 5 ShimlSSSP() InitSSSP() rpt V tim: or vry g u v i u v i tn Rlx(u v) or vry g u v i u v i tn rturn Ngtiv yl! Thi i lor to how CLRS prnt th Bllmn-For lgorithm. Th O(V E) running tim o thi vrion o th lgorithm houl oviou, ut it my not lr tht th lgorithm i till orrt. To prov orrtn, w jut hv to how tht our rlir invrint hol; or, thi n prov y inution on i..6 Gry Shortt Pth? Hr nothr lgorithm tht it our gnri rmwork, ut whih I v nvr n nlyz. Rptly rlx th tnt g. Spiilly, lt in th tnn o n g u v ollow: tnn(u v) = mx {, it(v) it(u) w(u v)} (Thi i in vn whn it(v) = or it(u) =, long w trt jut lik om inrily lrg ut init numr.) I n g h zro tnn, it not tn. I w rlx n g u v, thn it(v) r y th g tnn. Intuitivly, w n kp th g o th grph in om ort o hp, whr th ky o h g i it tnn. Thn w rptly pull out th tnt g u v n rlx it. Thn w n to romput th tnn o othr g jnt to v. Eg lving v poily om mor tn, n g oming into v poily om l tn. So w n hp tht iintly upport th oprtion Inrt, ExtrtMx, InrKy, n DrKy. I thr r no ngtiv yl, thi lgorithm vntully hlt with hortt pth tr, ut how quikly? Cn th m g rlx mor thn on, n i o, how mny tim? I it tr i ll th g wight r poitiv? Hmm... 6