10/30/12. Today. CS/ENGRD 2110 Object- Oriented Programming and Data Structures Fall 2012 Doug James. DFS algorithm. Reachability Algorithms

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1 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 Srh (DFS) Explor no y oin pr n pr into th rph. U k trkin to try irnt pth (u tk). Brth Firt Srh (BFS) Explor th no in n orrly mnnr. Look t th no tht r lot to th our. Thn look t thir nihor, t. (u quu) DFS lorithm Lt R th t o vrti rhl rom trtin no x. Lt S tk. DFS(vrtx x){ S.puh(x) whil (S i not mpty){ u =.pop() i (u i not in R) { put u into R or ll (u,v) in E { S.puh(v) // n whil Not: no n n up in th tk mor thn on. Ruriv DFS DFS (vrtx x){ put x into R or ll (x,y) in E i (y i not in R) DFS (y) BFS lorithm Lt R th t o vrti rhl rom trtin no x. Lt Q quu. BFS(vrtx x){ Q.nquu(x) whil (Q i not mpty){ u = Q.quu() i (u i not in R) { put u into R or ll (u,v) in E { Q.nquu(v) // n whil

2 0/0/ Shortt Pth in Grph Finin th hortt (min- ot) pth in rph i prolm tht our oln Bt liht rom Ith, NY to Dulor, Grmny? How loly r two popl onnt on Fook? Drivin irwon rom Ith, NY to Qun, NY? AI pth plnnin in roow Rult pn on our nowon o ot Numr o hop Lt mil Lt Wm Chpt Lt orin All o th ot n rprnt wiht How o w in hortt pth? Sinl Sour, Shortt Pth Prolm: Givn rph G=(V,E) omput th itn o h vrtx x rom our vrtx, whr itn i th lnth o th hortt pth. Unwiht Grph Wiht Grph it[] = 0; it[] = 0;. it[y] = it[x] +, it[y]= it[x]+w(x,y) whr (x,y) in E whr (x,y) in E Brut For Clim: Th hortt pth i impl pth. (i, no vrtx i rpt in th lit) Clim: Thr r only init numr o impl pth in ivn rph. Enumrt ll impl pth trtin t. For h trt vrtx t, ollt ll impl pth with trt t. Comput thir ot, trmin th min. B I Evn in n yli rph, th numr o impl pth my xponntil in n. Exri: trmin th numr o pth to t. Sinl Sour, Shortt Pth Unwiht rph: BFS Moii to kp trk o urrnt itn rom t

3 0/0/ Sinl Sour, Shortt Pth Sinl Sour, Shortt Pth BFS BFS Firt, viit ll no t itn Firt, viit ll no t itn Thn, itn Sinl Sour, Shortt Pth Sinl Sour, Shortt Pth BFS Firt, viit ll no t itn Thn, itn Thn, Not: w hv tully lult hortt pth rom to vry no in rph Not jut rom to In nrl, omputin hortt pth rom to vry othr no i jut xpniv omputin th hortt pth twn ny ivn pir o no BFS or hortt pth or h vrtx x it[x] = ininity; // will rprnt itn rom to x Q.nquu(); it[] = 0; whil ( Q.mpty()) x = Q.quu(); or ll (x,y) in E i (it[y] = ininity) it[y] = it[x] + ; Q.nquu(y); Clim: O(n + m) runtim Will DFS work in thi ontxt? Conir wiht rph whr ll wiht r qul. U th m BFS lorithm. Wht out rph with irnt wiht on?

4 0/0/ Brth- Firt Srh or Shortt Pth Unwiht Grph Input: trt no, WnWon no t Put trt no into quu n mrk viit. Whil quu not mpty Poll n o quu. FOR ll (unmrk) uor n o n IF n qul t THEN rturn pth Put n into quu Mrk n viit. Tim omplxity: O(m) Wm Why o BFS in Shortt Pth? Any no in itn i viit or ny no t hop, or ny no t itn hop, Whnvr no i t th top o th quu or th irt Wm, w mut hv ohn thr with th minimum numr o hop. How o w kp trk o th pth tht ot BFS thr? Stor pror no on pth or h no in rph. 8 Wiht E, Shortt Pth BFS lorithm i only rlvnt or unwiht rph Wht out wiht rph? SFO 7 LAX 86 6 DFW 80 ORD MIA JFK 96 8 BWI 87 BOS 090 PVD 8 Brth- Firt Srh or Shortt Pth Wiht Grph Input: trt no, WnWon no t Put trt (,0,null) into min- priority quu. IniWliz mpty iwonry pth. Whil quu not mpty Poll minimum lmnt (n,,prv) o quu. Mrk n on in pth y torin prv. IF n qul t THEN rturn pth IF n i not yt on FOR ll uor n o n tht r not on Put (n,+wiht(n,n ),n) into priority quu Tim omplxity: O(m lo m) Wm uin hp n jny lit Cn improv Gnrl Rul Prototyp Alorithm W mintin n rry it[x]: - initilly it[] = 0, it[x] = or ll othr vrti - t ny tim urin th lorithm, w tor th ot o rl pth rom to x in it[x] (ut not nrily th ot o th hortt pth, w my hv n ovrtimt). - (x,y) rquir ttntion i it[y] > it[x] + ot(x,y) Whn n rom x to y rquir ttntion w rlx it, uptin th timt or it[y]: it[y] = it[x] + ot(x,y) Thu w now hv ttr timt or th hortt pth rom to x. Thi prou prototyp lorithm: initiliz it[]; whil( om (x,y) rquir ttntion ) rlx (x,y);

5 0/0/ Dijktr' Alorithm Dijktr' Alorithm Th prolm i to hoo th riht to rlx. Dijktr' lorithm lwy pik th (x,y) uh tht it[x] i miniml ut work on h x only on. initiliz it[]; inrt ll v in V into PQ; // prioritiz y it whil( PQ not mpty ) x = PQ.ltMin(); orll (x,y) in E o i( (x,y) rquir ttntion ) rlx Dijktr' Alorithm Dijktr lorithm initiliz it[]; inrt ll v in V into PQ; // prioritiz y it whil( PQ not mpty ) x = PQ.ltMin(); orll (x,y) in E o i( it[y] > it[x] + ot[x,y] ) // rlx - upt our urrnt timt // o itn rom to y it[y] = it[x] + ot[x,y]; PQ.promot( y ); // Optionl: ror t pth to y w rom x Initiliztion. St it() = 0. For ll vrti v V, v, t it(v) =. Inrt ll vrti into priority quu P, uin itn th ky 0 P Dijktr lorithm Dijktr lorithm Pro Pro (D = 0) 0

6 0/0/ Dijktr lorithm Dijktr lorithm Pro (D = 0) (D = ) Pro (D = 0) (D = ) (D = ) 6 6 Dijktr lorithm Dijktr lorithm Pro (D = 0) (D = ) (D = ) (D = ) Pro (D = 0) (D = ) (D = ) (D = ) (D = ) Dijktr lorithm Dijktr Alorithm i ry Pro (D = 0) (D = ) (D = ) (D = ) (D = ) (D = 6) (D = 6) (D = ). Optimiztion prolm O th mny il olution, in th optiml (minimum or mximum) olution.. Cn only pro in t no irt olution vill. Gry-hoi proprty: A lolly optiml (ry) hoi will l to lolly optiml olution. Hr, th ltmin tp i th ry hoi Sinl our, hortt itn. Optiml utrutur: An optiml olution ontin within it optiml olution to uprolm 6

7 0/0/ Ftur o Dijktr Alorithm Eh vrtx i pro xtly on (whn it om th top o th priority quu) Eh i pro xtly on Ditn my rvi multipl tim: urrnt vlu rprnt t u on our orvtion o r On vrtx i pro w r urnt to hv oun th hortt pth to tht vrtx. why? Prormn (uin hp) Initiliztion: O(n) Viittion loop: n ll ltmin(): O(lo n) Eh i onir only on urin ntir xution, or totl o m upt o th priority quu, h O(lo n) Ovrll ot: O( (n+m) lo n ) Ai Hp i u unvnly: n lt-min ut m promot oprtion. Cn xploit y uin ttr t trutur (Fioni hp) to t runnin tim O(n lo n + m). Rprntin hortt pth W now hv n lorithm to omput th lnth o th hortt pth twn n x. But wht i w tully wnt to in th vrti on th hortt pth? Ft: i = 0,,..., n =x i th hortt pth rom to x, thn = 0,,..., n- i th hortt pth rom to n-. I: With h it(x), rmmr th prviou no prv(x) = n- in th hortt pth. Thouht Prolm: Ntiv Wiht Wht i th minimum ot itn twn n? Thouht Prolm: Ntiv Wiht Wht o w o whn thr r ntiv wiht? Othr i n lorithm my n