CSE 373. Graphs 1: Concepts, Depth/Breadth-First Search reading: Weiss Ch. 9. slides created by Marty Stepp
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1 CSE 373 Grphs 1: Conpts, Dpth/Brth-First Srh ring: Wiss Ch. 9 slis rt y Mrty Stpp Univrsity o Wshington, ll rights rsrv. 1
2 Wht is grph? 56 Tokyo Sttl Soul Nw York Syny L.A. 2
3 Grphs grph: A t strutur ontining: st o vrtisv, (somtims ll nos) st o gse, whr n g rprsnts onntion twn 2 vrtis. Grph G= (V, E) n g is pir (v, w) whr v, wr in V th grph t right: V= {,,, } E= {(, ), (, ), (, ), (, )} gr: numr o gs touhing givn vrtx. t right: =1, =2, =3, =2 3
4 Grph xmpls For h, wht r th vrtis n wht r th gs? W pgs with links Mthos in progrm tht ll h othr Ro mps (.g., Googl mps) Airlin routs Fook rins Cours pr-rquisits Fmily trs Pths through mz 4
5 Pths pth: A pth rom vrtx to is squn o gs tht n ollow strting rom to rh. n rprsnt s vrtis visit, or gs tkn xmpl, on pth rom Vto Z: {, h} or {V, X, Z} Wht r two pths rom U to Y? pth lngth: Numr o vrtis or gs ontin in th pth. nighor or jnt: Two vrtis onnt irtly y n g. xmpl: V n X U V W X Y g h Z 5
6 Rhility, onntnss rhl: Vrtx is rhl rom i pth xists rom to. V onnt: A grph is onnti vry vrtx is rhl rom ny othr. Is th grph t top right onnt? U W X g Y h Z strongly onnt: Whn vry vrtx hs n g to vry othr vrtx. 6
7 Loops n yls yl: A pth tht gins n ns t th sm no. xmpl: {, g,,, } or {V, X, Y, W, U, V}. xmpl: {,, } or {U, W, V, U}. yli grph: On tht os not ontin ny yls. V loop: An g irtly rom no to itsl. Mny grphs on't llow loops. U W X g h Z Y 7
8 Wight grphs wight: Cost ssoit with givn g. Som grphs hv wight gs, n som r unwight. Egs in n unwight grph n thought o s hving qul wight (.g. ll 0, or ll 1, t.) Most grphs o not llow ngtiv wights. xmpl: grph o irlin lights, wight y mils twn itis: SFO 1843 ORD PVD HNL LAX DFW LGA 1099 MIA 8
9 Dirt grphs irt grph ("igrph"): On whr gs r on-wy onntions twn vrtis. I grph is irt, vrtx hs sprt in/out gr. A igrph n wight or unwight. Is th grph low onnt? Why or why not? g 9
10 Digrph xmpl Vrtis = UW CSE ourss (inomplt list) Eg (, ) = is prrquisit or
11 Link Lists, Trs, Grphs A inry tris grph with som rstritions: Th tr is n unwight, irt, yli grph (DAG). Eh no's in-gr is t most 1, n out-gr is t most 2. Thr is xtly on pth rom th root to vry no. A link listis lso grph: F Unwight DAG. In/out gr o t most 1 or ll nos. B K A B C D A E H G J 11
12 Srhing or pths Srhing or pth rom on vrtx to nothr: Somtims, w just wnt ny pth (or wnt to know thr is pth). Somtims, w wnt to minimiz pth lngth(# o gs). Somtims, w wnt to minimiz pth ost(sum o g wights). Wht is th shortst pth rom MIA to SFO? Whih pth hs th minimum ost? HNL $130 $250 SFO $60 LAX $70 $170 $120 $500 $80 DFW ORD $50 $140 $110 MIA PVD $200 LGA $100 12
13 Dpth-irst srh pth-irst srh(dfs): Fins pth twn two vrtis y xploring h possil pth s r s possil or ktrking. Otn implmnt rursivly. Mny grph lgorithms involv visiting or mrking vrtis. Dpth-irst pths rom to ll vrtis (ssuming ABC g orr): to :{, } to : {,,,, } to :{, } to :{,, } to : {,,, } to g:{,, g} to h: {,, g, h} g h 13
14 DFS psuoo untion s(v 1, v 2 ): s(v 1, v 2, { }). untion s(v 1, v 2, pth): pth+= v 1. mrk v 1 s visit. i v 1 is v 2 : pth is oun! g h or h unvisit nighor no v 1 : i s(n, v 2, pth) ins pth: pth is oun! pth-= v 1. // pth is not oun. Th pthprm ov is us i you wnt to hv th pth vill s list on you r on. Tr s(, ) in th ov grph. 14
15 DFS osrvtions isovry: DFS is gurnt to in pth i on xists. rtrivl: It is sy to rtriv xtly wht th pth is (th squn o gs tkn) i w in it g h optimlity: not optiml. DFS is gurnt to in pth, not nssrily th st/shortst pth Exmpl: s(, ) rturns {,,, } rthr thn {,, }. 15
16 Brth-irst srh rth-irst srh(bfs): Fins pth twn two nos y tking on stp own ll pths n thn immitly ktrking. Otn implmnt y mintining quu o vrtis to visit. BFS lwys rturns th shortst pth (th on with th wst gs) twn th strt n th n vrtis. to :{, } to : {,,, } to :{, } to :{, } to : {,, } to g:{,, g} to h: {,, h} g h 16
17 BFS psuoo untion s(v 1, v 2 ): quu:= {v 1 }. mrk v 1 s visit. whil quu is not mpty: v:= quu.rmovfirst(). i vis v 2 : pth is oun! g h or h unvisit nighor no v: mrk ns visit. quu.lst(n). // pth is not oun. Tr s(, ) in th ov grph. 17
18 BFS osrvtions optimlity: lwys ins th shortst pth (wst gs). in unwight grphs, ins optiml ost pth. In wight grphs, not lwys optiml ost. g h rtrivl: hrr to ronstrut th tul squn o vrtis or gs in th pth on you in it onptully, BFS is xploring mny possil pths in prlll, so it's not sy to stor pth rry/list in progrss solution: W n kp trk o th pth y storing prssors or h vrtx (h vrtx n stor rrn to prvious vrtx). DFS uss lss mmory thn BFS, sir to ronstrut th pth on oun; ut DFS os not lwys in shortst pth. BFS os. 18
19 DFS, BFS runtim Wht is th xpt runtim o DFS n BFS, in trms o th numr o vrtis V n th numr o gs E? Answr: O( V + E ) whr V = numr o vrtis, E = numr o gs Must potntilly visit vry no n/or xmin vry g on. why not O( V * E )? Wht is th sp omplxity o h lgorithm? (How muh mmory os h lgorithm rquir?) 19
20 BFS tht ins pth untion s(v 1, v 2 ): quu:= {v 1 }. mrk v 1 s visit. whil quu is not mpty: v:= quu.rmovfirst(). g h i vis v 2 : pth is oun! (ronstrut it y ollowing.prvk to v 1.) or h unvisit nighor no v: mrk ns visit. (st n.prv= v.) quu.lst(n). // pth is not oun. prv By storing som kin o "prvious" rrn ssoit with h vrtx, you n ronstrut your pth k on you in v 2. 20
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