About this lecture. Representation of Graph Adjacency List, Adjacency Matrix. Breadth First Search

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1 Abo hi lece Repeenaion of Gaph Adjacenc Li, Adjacenc Mai Beadh Fi Seach 2

2 Gaph ndieced dieced 3

3 Adjacenc Li () Fo each ee, oe i neighbo in a linked li ee neighbo 4

4 Adjacenc Li (2) Fo each ee, oe i neighbo in a linked li ee neighbo 5

5 Adjacenc Li (3) Le G =(V, E) beaninpgaph Uing Adjacenc Li epeenaion : Space : O( V + E ) Ecellen hen E imall Ea o li all neighbo of a ee Take O( V ) ime o check if a ee ianeighboofaee can alo epeen eighed gaph 6

6 7 Adjacenc Mai () Ue a V V maia ch ha A(,) = if (,) ianedge A(,) = oheie

7 8 Adjacenc Mai (2) Ue a V V maia ch ha A(,) = if (,) ianedge A(,) = oheie

8 Adjacenc Mai (3) Le G =(V, E) beaninpgaph Uing Adjacenc Mai epeenaion : Space : O( V 2 ) Bad hen E imall O() ime o check if a ee i a neighbo of a ee ( V ) ime o li all neighbo can alo epeen eighed gaph 9

9 Tanpoe of a Mai Le A be an n m mai Definiion: The anpoe of A, denoed b A T,ian m n mai ch ha A T (,) =A (,) fo ee, If A i an adjacenc mai of an ndieced gaph, hen A = A T

10 Beadh Fi Seach (BFS) A imple algoihm o find all eice eachable fom a paicla ee i called oce ee Idea: Eploe eice in ond A Rond k, ii all eice hoe hoe diance (#edge) fom i k- Alo, dicoe all eice hoe hoe diance fom i k

11 The BFS Algoihm. Mak a dicoeed in Rond 2. Fo Rond k =,2,3,, Fo (each dicoeed in Rond k-) { Mak a iied ; Vii each neighbo of ; If ( no iied and no dicoeed) Mak a dicoeed in Rond k ; } Sop if no eice ee dicoeed in Rond k- 2

12 Eample ( =oce)?? iied (? =dicoeime) dicoeed (? =dicoeime) diecion of edge hen ne node i dicoeed 3

13 Eample ( =oce) ?? iied (? =dicoeime) dicoeed (? =dicoeime) diecion of edge hen ne node i dicoeed 4

14 Eample ( =oce) ?? iied (? =dicoeime) dicoeed (? =dicoeime) diecion of edge hen ne node i dicoeed 5

15 Eample ( =oce) Done hen no ne node i dicoeed The dieced edge fom aeehaconainall node eachable fom 4 Called BFS ee of 6

16 Coecne The coecne of BFS follo fom he folloing heoem : Theoem: A ee i dicoeed in Rond k if and onl if hoe diance of fom oce i k Poof: B indcion 7

17 Pefomance BFS algoihm i eail done if e e an O( V )-ize aa o oe dicoeed/iied infomaion a epaae li fo each ond o oe he eice dicoeed in ha ond Since no ee i dicoeed ice, and each edge i iied a mo ice (h?) Toal ime: O( V + E ) Toal pace: O( V + E ) 8

18 Pefomance (2) Inead of ing a epaae li fo each ond, e can e a common qee When a ee i dicoeed, e p i a he end of he qee To pick a ee o ii in Sep 2, e pick he one a he fon of he qee Done hen no ee i in he qee No impoemen in ime/pace B algoihm i implified Qeion: Can o poe he coecne of ing qee? 9

19 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

20 The DFS Algoihm DFS() { 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

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

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

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

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

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

26 Eample ( =oce) Done hen i dicoeed The dieced edge fom aeehaconainall node eachable fom Called DFS ee of

27 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, andheeino 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

28 Genealizaion (Eample) Sppoe he inp gaph i dieced 2

29 Genealizaion (Eample). Afe appling DFS on 3

30 Genealizaion (Eample) 2. Then, afe appling DFS on 4

31 Genealizaion (Eample) 3. Then, afe appling DFS on 5

32 Genealizaion (Eample) 4. Then, afe appling DFS on 6

33 Genealizaion (Eample) 5. Then, afe appling DFS on 7

34 Genealizaion (Eample) Rel : a collecion of ooed ee called DFS foe 8

35 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?) 9