Outline. Graphs Describing Precedence. Graphs Describing Precedence. Topological SorFng of DAGs. Graphs Describing Precedence 4/25/12. Part 10.

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1 4// Outlin Prt. Grphs CS Algorithms n Dt Struturs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits Grphs Dsriing Prn Grphs Dsriing Prn Exmpls: prrquisits for st of ourss pnnis twn progrms Eg from to inits shoul om for Btmn imgs r from th ook IntrouFon to ioinformfs lgorithms Grphs Dsriing Prn Wnt n orring of th vrtis of th grph tht rspts th prn rltion Exmpl: An orring of CS ourss Th grph os not ontin yls. Why? Topologil SorFng of DAGs DAG: Dirt Ayli Grph Topologil sort: listing of nos suh tht if (,) is n g, pprs for in th list Is topologil sort uniqu?

2 4// A irt grph without yls f Topologil Sort - Algorithm topsort(in G:Grph)!!n= numr of vrtis in G!!for (stp = through n)!!!slt vrtx v tht hs no sussors!!!list.(first_vill_lo,v)!!!dlt from G vrtx v n its gs!!rturn List! g,g,,,,,f,,g,,,f, Algorithm rlis on th ft tht in DAG thr is lwys vrtx tht hs no sussors Algorithm : Exmpl Algorithm : Exmpl f A B C D E F G H g g f I A, D, E, B, G, C, F, H, I Topologil Sort - Algorithm DFS Exmpl: rviw Moifition of DFS: Trvrs tr using DFS strting from ll nos tht hv no prssor. A no to th list whn ry to ktrk. A B C D E F G H I J K L M N O P

3 4// ItrFv DFS: rviw fs(in v:vrtx)! s stk for kping trk of tiv vrtis! s.push(v)! mrk v s visit! whil(!s.isempty()) {!!if (no unvisit vrtis jnt to th vrtx on top of th stk) {!!!s.pop() \\ktrk!!ls {!!!slt unvisit vrtx u jnt to vrtx on top of th stk!!!s.push(u)!!!mrk u s visit!!}! }! Topologil Sort - Algorithm topsort( in thgrph:grph):list!!s.rtstk()!!for (ll vrtis v in th grph thgrph)!!!if (v hs no prssors)!!!!s.push(v)!!!!mrk v s visit!!whil (!s.isempty())!!!if (ll vrtis jnt to th vrtx on top of th stk hv n visit)!!!!v = s.pop()!!!!llist.(, v)!!!ls!!!!slt n unvisit vrtx u jnt to vrtx on top of th stk!!!!s.push(u)!!!!mrk u s visit!!rturn List! Algorithm : Exmpl Algorithm : Exmpl f g f g g f A B C D E F G I H A, D, G, I, E, B, C, F, H Outlin Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits Shortst Pth Algorithms (Dijkstr s Algorithm) Grph G(V,E) with non-ngtiv wights ( istns ) Comput shortst istns from vrtx s to vry othr vrtx in th grph 3

4 4// Shortst Pth Algorithms (Dijkstr s Algorithm) Shortst Pth Algorithms (Dijkstr s Algorithm) Algorithm Mintin rry (minimum istn stimts) Init: [s]=, [v]= v V-s Priority quu of vrtis not yt visit slt minimum istn vrtx, visit v, upt nighors Shortst Pth Algorithms (Dijkstr s Algorithm): IniFliz / / Shortst Pth Algorithms (Dijkstr s Algorithm): stp / / / / / 4/ Shortst Pth Algorithms (Dijkstr s Algorithm): stp 4 / / / / / 4/ / / / / Shortst Pth Algorithms (Dijkstr s Algorithm): stp 3 4

5 4// / / / / / 4/ / / / / / / / / Shortst Pth Algorithms (Dijkstr s Algorithm): stp 4 / / / / Shortst Pth Algorithms (Dijkstr s Algorithm): Don Minimum istn from to : Minimum istn from to : Minimum istn from to : Minimum istn from to : Exmpl Dijkstr s Algorithm Dijkstr(G: grph with vrtis v v n- n wights w [u][v])!!// omputs shortst istn of vrtx to vry othr vrtx!!rt st vrtxst tht ontins only vrtx!![] =!!for (v = through n-)!!![v] = infinity!!for (stp = through n)!!!fin th smllst [v] suh tht v is not in vrtxst!!! v to vrtxst!!!for (ll vrtis u not in vrtxst)!!!!if ([u] > [v] + w[v][u])!!!!![u] = [v] + w[v][u]! Shortst Pth Algorithms Using Priority Quu (Dijkstr s Algorithm): stp Shortst Pth Algorithms (Dijkstr s Algorithm): stp 4 [,],[,] [,],[,],[4,]

6 4// Shortst Pth Algorithms (Dijkstr s Algorithm): stp 3 Shortst Pth Algorithms (Dijkstr s Algorithm): stp 4 [,],[,] [,] Shortst Pth Algorithms (Dijkstr s Algorithm): Don Dijkstr s Algorithm How to otin th shortst pths? At h vrtx mintin pointr tht tlls you th vrtx from whih you rriv. Outlin Conntnss in Dirt Grphs Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits A irt grph is strongly onnt if thr is pth from to n from to whnvr n r vrtis in th grph. A irt grph is wkly onnt if thr is pth twn vry two vrtis in th unrlying unirt grph

7 4// Exmpl Trs s Grphs Tr: n unirt onnt grph tht hs no yls. A B C D E F G H Grph A Grph B I J K L 3 M N O P Root Trs Exmpl: Buil root trs. A root tr is tr in whih on vrtx hs n signt s th root n vry g is irt wy from th root A B C D E F G H I J K L M N O P Trs s Grphs Tr: n unirt onnt grph tht hs no simpl iruits. Thorm - An unirt grph is tr iff thr is uniqu simpl pth (no rpt vrtis) twn ny two vrtis.

8 4// Whn is grph Tr? Cn xpliitly hk tht th grph is onnt n hs no yls. (How?) W n n ltrntiv hrtriztion Whn is grph Tr?: Thorm - A onnt unirt grph with n vrtis must hv t lst n- gs (PROOF: y inution on th numr of vrtis) Whn is grph Tr?: Thorm - 3 A onnt unirt grph tht hs n vrtis n xtly n- gs nnot ontin yl (PROOF: y ontrition with prvious sttmnt) Whn is grph Tr? : Thorm - 4 A onnt unirt grph tht hs n vrtis n mor thn n- gs must ontin yl. Whn is grph Tr? Conlusion: A onnt grph with n vrtis n n- gs is tr. In orr to hk if grph is tr w n to hk tht it is onnt n ount th numr of gs n vrtis. Spnning Trs Spnning tr: A sugrph of onnt unirt grph G tht ontins ll of G s vrtis n nough of its gs to form tr. How to gt spnning tr: Rmov gs until you gt tr. A gs until you hv spnning tr

9 4// Spnning Trs - DFS lgorithm Spnning Tr Dpth First Srh Exmpl fstr(in v:vrtx)!!mrk v s visit!!for (h unvisit vrtx u jnt to v)!!!mrk th g from u to v!!!fstr(u)! A B C D E F G H I J K L M N O P Exmpl Outlin Suppos tht n irlin must ru its flight shul to sv mony. If its originl routs r s illustrt hr, whih flights n isontinu to rtin srvi twn ll pirs of itis (whr it my nssry to omin flights to fly from on ity to nothr?) Chigo Bngor Shl Boston Dtroit SF NYC LA Sn Digo Dnvr Dlls St. Louis Atlnt Wshington D.C. Introution Trminology Implmnting Grphs Grph Trvrsls Topologil Sorting Shortst Pths Spnning Trs Minimum Spnning Trs Ciruits Minimum Spnning Tr Prim s Algorithm Minimum spnning tr Spnning tr minimizing th sum of g wights Exmpl: Connting h hous in th nighorhoo to l Grph whr h hous is vrtx. N th grph to onnt, n minimiz th ost of lying th ls. I: inrmntlly uil spnning tr y ing th lst-ost g to th tr Wight grph Fin st of gs Touhs ll vrtis Miniml wight Not ll th gs my us

10 4// Prim s Algorithm: Exmpl Prim s Algorithm 4 h g i 4 6 f prims(in: G=(V,E):Grph)!!//V T urrnt vrtis in spnning tr!!//e T gs longing to th spnning tr!!v T = {w} // w is n ritrrily hosn vrtx!!e T = ϕ //spnning tr ontins no vrtis initilly!!for i = to V - o!!!fin minimum-wight g =(u,v) mong gs tht!onnts vrtx in V T with vrtx in V V T!!! v to V T!!! to E T!!rturn E T! {(,),(,), (,i), (,), (,f), (f,g), (g,h), (h,) } ImplmnFng Prim s Algorithm Eh no not in th tr hs n tthing ost th wight of th smllst g tht onnts it to th forming tr (infinity if no suh g xists). At h itrtion, w rtriv th no with th smllst tthing ost n upt th tthing ost of its nighors. Cn us priority quu! (n to mtho for upting prioritis).