Chapter 22: Elementary Graph Algorithms II

Transcription

1 Chape 22: Elemena Gaph Algoihm II 1

2 Abo hi lece Deph Fi Seach DFS Tee and DFS Foe Popeie of DFS Paenhei heoem (e impoan) Whie-pah heoem (e efl) 2

3 Deph Fi Seach (DFS) An alenaie algoihm o find all eice eachable fom a paicla oce ee Idea: Eploe a banch a fa a poible befoe eploing anohe banch Eail done b ecion o ack 3

4 DFS() The DFS Algoihm { Mak a dicoeed ; } hile ( ha niied neighbo ) DFS(); Mak a finihed ; The hile-loop eploe a banch a fa a poible befoe he ne banch 4

5 Eample ( = oce) finihed dicoeed diecion of edge hen ne node i dicoeed 5

6 Eample ( = oce) finihed dicoeed diecion of edge hen ne node i dicoeed 6

7 Eample ( = oce) finihed dicoeed diecion of edge hen ne node i dicoeed 7

8 Eample ( = oce) finihed dicoeed diecion of edge hen ne node i dicoeed 8

9 Eample ( = oce) finihed dicoeed diecion of edge hen ne node i dicoeed 9

10 Eample ( = oce) Done hen i dicoeed The dieced edge fom a ee ha conain all node eachable fom Called DFS ee of 10

11 Genealizaion J like BFS, DFS ma no ii all he eice of he inp gaph G, becae : G ma be diconneced G ma be dieced, and hee i no dieced pah fom o ome ee In mo applicaion of DFS (a a boine), once DFS ee of i obained, e ill conine o appl DFS algoihm on an niied eice 11

12 Genealizaion (Eample) Sppoe he inp gaph i dieced 12

13 Genealizaion (Eample) 1. Afe appling DFS on 13

14 Genealizaion (Eample) 2. Then, afe appling DFS on 14

15 Genealizaion (Eample) 3. Then, afe appling DFS on 15

16 Genealizaion (Eample) 4. Then, afe appling DFS on 16

17 Genealizaion (Eample) 5. Then, afe appling DFS on 17

18 Genealizaion (Eample) Rel : a collecion of ooed ee called DFS foe 18

19 Pefomance Since no ee i dicoeed ice, and each edge i iied a mo ice (h?) Toal ime: O( V + E ) A menioned, apa fom ecion, e can alo pefom DFS ing a LIFO ack (Do o kno ho?) 19

20 Who ill be in he ame ee? Becae e can onl eploe banche in an niied node DFS() ma no conain all node eachable b in i DFS ee E.g, in he peio n, can each,,, b ee doe no conain an of hem Can e deemine ho ill be in he ame ee? 20

21 Who ill be in he ame ee? Ye, e ill oon ho ha b hie-pah heoem, e can deemine ho ill be in he ame ee a a he ime hen DFS i pefomed on Befoe ha, e ill define he dicoe ime and finihing ime fo each node, and ho ineeing popeie of hem 21

22 Dicoe and Finihing Time When he DFS algoihm i n, le conide a global ime ch ha he ime inceae one ni : hen a node i dicoeed, o hen a node i finihed (i.e., finihed eploing all niied neighbo) Each node ecod : d() = he ime hen i dicoeed, and f() = he ime hen i finihed 22

23 Dicoe and Finihing Time 12/15 1/16 4/9 5/8 13/14 2/11 3/10 6/7 In o fi eample (ndieced gaph) 23

24 Dicoe and Finihing Time 13/14 1/6 7/10 8/9 15/16 2/5 3/4 11/12 In o econd eample (dieced gaph) 24

25 Nice Popeie Lemma: Fo an node, d() f() Lemma: Fo node and, d(), d(), f(), f() ae all diinc Theoem (Paenhei Theoem): Le and be o node ih d() d(). Then, eihe 1. d() d() f() f() [conain], o 2. d() f() d() f() [dijoin] 25

26 Poof of Paenhei Theoem Conide he ime hen i dicoeed Since i dicoeed befoe, hee ae o cae concening he a of : Cae 1: ( i no finihed) Thi implie i a decendan of f() f() (h?) Cae 2: ( i finihed) f() d() 26

27 Coolla Coolla: i a (pope) decendan of if and onl if d() d() f() f() Poof: i a (pope) decendan of d() d() and f() f() d() d() f() f() 27

28 Whie-Pah Theoem Theoem: B he ime hen DFS i pefomed on, fo an a DFS i done, he decendan of ae he ame, and he ae eacl hoe node eachable b ih niied (hie) node onl E.g., If e pefom DFS() no, ill he decendan of ala be he ame e of node? 28

29 Poof (Pa 1) Sppoe ha i a decendan of Le P = (, 1, 2,, k, ) be he dieced pah fom o in DFS ee of Then, apa fom, each node on P m be dicoeed afe The ae all niied b he ime e pefom DFS on Th, a hi ime, hee ei a pah fom o ih niied node onl 29

30 Poof (Pa 2) So, ee decendan of i eachable fom ih niied node onl To complee he poof, i emain o ho he conee : An node eachable fom ih niied node onl become decendan i alo e (We hall poe hi b conadicion) 30

31 Poof (Pa 2) Sppoe on cona he conee i fale Then, hee ei ome, eachable fom ih niied node onl, doe no become decendan If moe han one choice of, le be one ch ee cloe o d() f() d() f() EQ.1 31

32 Poof (Pa 2) Le P = (, 1, 2,, k, ) be an pah fom o ing niied node onl B o choice of (cloe one), all 1, 2,, k become decendan Thi implie: d() d( k ) f( k ) f() Combining ih EQ.1, e hae d( k ) f( k ) d() f() Handle pecial cae: hen = k 32

33 Poof (Pa 2) Hoee, ince hee i an edge (no mae ndieced o dieced) fom k o, if d( k ) d(), hen e m hae d() f( k ) (h??) Coneqenl, i conadic ih : d( k ) f( k ) d() f() Poof complee 33

34 Claificaion of Tee Edge Afe a DFS poce, e can claif he edge of a gaph ino fo pe : 1. Tee : Edge in he DFS foe 2. Back : Fom decendan o anceo hen eploed (inclde elf loop) 3. Foad : Fom anceo o decendan hen eploed (eclde ee edge) 4. Co : Ohe (no anceo-decendan elaion) 34

35 Eample Sppoe he inp gaph i dieced 35

36 Eample Sppoe hi i he DFS foe obained C B F C C Can o claif he pe of each edge? 36

37 Eample Sppoe he DFS foe i diffeen no C B C C C Can o claif he pe of each edge? 37

38 Eample Sppoe he inp gaph i ndieced 38

39 Eample Sppoe hi i he DFS foe obained B B Can o claif he pe of each edge? 39

40 Edge in Undieced Gaph Theoem: Afe DFS of an ndieced gaph, ee edge i eihe a ee edge o a back edge Poof: Le (,) be an edge. Sppoe i dicoeed fi. Then, ill become decenden (hie-pah) o ha f() f() If dicoe (,) i ee edge Ele, (,) i eploed afe dicoeed Then, (,) m be eploed fom becae f() f() (,) i back edge 40

41 Homeok Eecie: , ,