# 5/7/13. Part 10. Graphs. Theorem Theorem Graphs Describing Precedence. Outline. Theorem 10-1: The Handshaking Theorem

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1 Thorm 10-1: Th Hnshkin Thorm Lt G=(V,E) n unirt rph. Thn Prt 10. Grphs CS 200 Alorithms n Dt Struturs v V (v) = 2 E How mny s r thr in rph with 10 vrtis h of r six? 10 * 6 /2= 30 1 Thorm 10-2 An unirt rph hs n vn numr of vrtis of o r. Thorm 10-3 Lt G=(V,E) rph with irt s. Thn Proof: (v) = + (v) = E v V v V inr outr Introution Trminoloy Shortst Pths Outlin Implmntin Grphs Grph Trvrsls Topoloil Sortin Spnnin Trs Minimum Spnnin Trs Ciruits Grphs Dsriin Prn Exmpls: prrquisits for st of ourss pnnis twn prorms E from to inits shoul om for 5 1

2 Grphs Dsriin Prn Grphs Dsriin Prn Wnt n orrin of th vrtis of th rph tht rspts th prn rltion Exmpl: An orrin of CS ourss Th rph os not ontin yls. Btmn ims r from th ook Introu<on to ioinform<s lorithms Topoloil Sortin of DAGs DAG: Dirt Ayli Grph Topoloil sort: listin of nos suh tht if (,) is n, pprs for in th list Is topoloil sort uniqu? Qus%on Is topoloil sort uniqu? A. Ys B. No A irt rph without yls f,,,,,,f,,,,,f, 10 Topoloil Sort - Alorithm 1 topsort1(in G:Grph)!!n= numr of vrtis in G!!for (stp =1 throuh n)!!!slt vrtx v tht hs no sussors!!!list.(first_vill_lo,v)!!!dlt from G vrtx v n its s!!rturn List! Alorithm rlis on th ft tht in DAG thr is lwys vrtx tht hs no sussors Topoloil Sort - Alorithm 1 topsort1(in G:Grph)!!n= numr of vrtis in G!!for (stp =1 throuh n)!!!slt vrtx v tht hs no sussors!!!list. (first_vill_lo,v)!!!dlt from G vrtx v n its s!!rturn List! f f 2

3 Alorithm 2: Exmpl 2 A B C D E F Topoloil Sort - Alorithm 2 Moifition of DFS: Trvrs tr usin DFS strtin from ll nos tht hv no prssor. A no to th list whn ry to ktrk. G H I A, D, E, B, G, C, F, H, I DFS Exmpl: rviw A B C D E F G H I J K L M N O P Itrtiv DFS: rviw fs(in v:vrtx)! s stk for kpin 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!!}! }! Topoloil Sort - Alorithm 2 topsort2( in thgrph:grph):list!!s.rtstk()!!for (ll vrtis v in th rph 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.(1, v)!!!ls!!!!slt n unvisit vrtx u jnt to vrtx on top of th stk!!!!s.push(u)!!!!mrk u s visit!!rturn List! f Alorithm 2: Exmpl 1 f f 18 3

4 Alorithm 2: Exmpl 2 A B C D E F G I H A, D, G, I, E, B, C, F, H Introution Trminoloy Implmntin Grphs Outlin Grph Trvrsls Topoloil Sortin Shortst Pths Spnnin Trs Minimum Spnnin Trs Ciruits 20 Conntnss in Dirt Grphs A irt rph is stronly onnt if thr is pth from to n from to whnvr n r vrtis in th rph. Qus%on: Is Grph A wkly onnt? Exmpl A. Ys B. No A irt rph is wkly onnt if thr is pth twn vry two vrtis in th unrlyin unirt rph. Grph A Grph B Trs s Grphs Tr: n unirt onnt rph tht hs no yls. A B C D Root Trs A root tr is tr in whih on vrtx hs n sint s th root n vry is irt wy from th root E F G H I J K L M N O P 4

5 Exmpl: Buil root trs. A B C D h E F G H I J K L f M N O P Qus%on: Whih no CANNOT root of this tr? A. No E B. No G C. No E D. Non i 5/7/13 Smmi L Pllikr 26 Trs s Grphs Thorm Tr: n unirt onnt rph tht hs no simpl iruits. An unirt rph is tr iff thr is uniqu simpl pth (no rpt vrtis) twn ny two vrtis. Whn is rph Tr? Cn xpliitly hk tht th rph is onnt n hs no yls. (How?) W n n ltrntiv hrtriztion Whn is rph Tr?: Thorm A onnt unirt rph with n vrtis must hv t lst n-1 s (PROOF: y inution on th numr of vrtis) 5

6 Whn is rph Tr?: Thorm Whn is rph Tr? : Thorm A onnt unirt rph tht hs n vrtis n xtly n-1 s nnot ontin yl (PROOF: y ontrition with prvious sttmnt) A onnt unirt rph tht hs n vrtis n mor thn n-1 s must ontin yl. Proof: Lt G onnt unirt rph with n vrtis n n-1 s without ny yl. If w on twn ny pir of vrtis, this will n itionl pth twn tht pir. Tht will form yl. Whn is rph Tr? Conlusion: A onnt rph with n vrtis n n-1 s is tr. In orr to hk if rph is tr w n to hk tht it is onnt n ount th numr of s n vrtis. Spnnin Trs Spnnin tr: A su-rph of onnt unirt rph G tht ontins ll of G s vrtis n nouh of its s to form tr. How to t spnnin tr: Rmov s until you t tr. A s until you hv spnnin tr Spnnin Trs - DFS lorithm Spnnin Tr Dpth First Srh Exmpl fstr(in v:vrtx)!!mrk v s visit!!for (h unvisit vrtx u jnt to v)!!!mrk th from u to v!!!fstr(u)! A B C D E F G H I J K L M N O P 6

7 Exmpl Suppos tht n irlin must ru its fliht shul to sv mony. If its oriinl routs r s illustrt hr, whih flihts n isontinu to rtin srvi twn ll pirs of itis (whr it my nssry to omin flihts to fly from on ity to nothr?) SVl SF LA Sn Dio Dtroit Dnvr Dlls Chio St. Louis Atlnt NYC Boston Bnor Wshinton D.C. Introution Trminoloy Implmntin Grphs Outlin Grph Trvrsls Topoloil Sortin Shortst Pths Spnnin Trs Minimum Spnnin Trs Ciruits Minimum Spnnin Tr Minimum spnnin tr Spnnin tr minimizin th sum of wihts Exmpl: Conntin h hous in th nihorhoo to l Grph whr h hous is vrtx. N th rph to onnt, n minimiz th ost of lyin th ls. Prim s Alorithm I: inrmntlly uil spnnin tr y in th lst-ost to th tr Wiht rph Fin st of s Touhs ll vrtis Miniml wiht Not ll th s my us Prim s Alorithm: Exmpl h 11 i f 1 2 {(,),(,), (,i), (,), (,f), (f,), (,h), (h,) } Strt from A 42 7

8 Prim s Alorithm Implmntin Prim s Alorithm prims(in: G=(V,E):Grph)!!//V T urrnt vrtis in spnnin tr!!//e T s lonin to th spnnin tr!!v T = {w} // w is n ritrrily hosn vrtx!!e T = ϕ //spnnin tr ontins no vrtis initilly!!for i = 1 to V - 1 o!!!fin minimum-wiht =(u,v) mon s tht!onnts vrtx in V T with vrtx in V V T!!! v to V T!!! to E T!!rturn E T! Eh no not in th tr hs n tthin ost th wiht of th smllst tht onnts it to th formin tr (infinity if no suh xists). At h itrtion, w rtriv th no with th smllst tthin ost n upt th tthin ost of its nihors. Cn us priority quu! (n to mtho for uptin prioritis). 8

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