Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Directed Graphs BOS SFO

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1 Prsntation for us with th txtbook, Algorithm Dsign and Applications, by M. T. Goodrich and R. Tamassia, Wily, 2015 Dirctd Graphs BOS ORD JFK SFO LAX DFW MIA 2015 Goodrich and Tamassia Dirctd Graphs 1

2 Digraphs q A digraph is a graph whos dgs ar all dirctd n Short for dirctd graph q Applications n on-way strts n flights C E D B n task schduling A 2015 Goodrich and Tamassia Dirctd Graphs 2

3 E Digraph Proprtis D q A graph G=(V,E) such that n Each dg gos in on dirction: A n Edg (a,b) gos from a to b, but not b to a q If G is simpl, m < n (n - 1) q If w kp in-dgs and out-dgs in sparat adjacncy lists, w can prform listing of incoming dgs and outgoing dgs in tim proportional to thir siz C B 2015 Goodrich and Tamassia Dirctd Graphs 3

4 Digraph Application q Schduling: dg (a,b) mans task a must b compltd bfor b can b startd cs21 cs22 cs46 cs51 cs53 cs52 cs161 cs131 cs141 cs121 cs171 cs151 Th good lif 2015 Goodrich and Tamassia Dirctd Graphs 4

5 Dirctd DFS q q W can spcializ th travrsal algorithms (DFS and BFS) to digraphs by travrsing dgs only along thir dirction In th dirctd DFS algorithm, w hav four typs of dgs E D n n discovry dgs back dgs C n n forward dgs cross dgs B q A dirctd DFS starting at a vrtx s dtrmins th vrtics rachabl from s A 2015 Goodrich and Tamassia Dirctd Graphs 5

6 Th Dirctd DFS Algorithm 2015 Goodrich and Tamassia Dirctd Graphs 6

7 Rachability q DFS tr rootd at v: vrtics rachabl from v via dirctd paths E D E D C A C B F A E C D F 2015 Goodrich and Tamassia Dirctd Graphs 7 A B

8 Strong Connctivity q Each vrtx can rach all othr vrtics a c g d f b 2015 Goodrich and Tamassia Dirctd Graphs 8

9 Strong Connctivity Algorithm q q Pick a vrtx v in G Prform a DFS from v in G n If thr s a w not visitd, print no G: a c g q Lt G b G with dgs rvrsd d q Prform a DFS from v in G f b n If thr s a w not visitd, print no q n Els, print ys Running tim: O(n+m) G : a c g d f b 2015 Goodrich and Tamassia Dirctd Graphs 9

10 Strongly Connctd Componnts q q Maximal subgraphs such that ach vrtx can rach all othr vrtics in th subgraph Can also b don in O(n+m) tim using DFS, but is mor complicatd (similar to biconnctivity). a c g { a, c, g } f d b { f, d,, b } 2015 Goodrich and Tamassia Dirctd Graphs 10

11 Transitiv Closur q q Givn a digraph G, th transitiv closur of G is th digraph G* such that n G* has th sam vrtics as G n if G has a dirctd path from u to v (u v), G* has a dirctd dg from u to v Th transitiv closur provids rachability information about a digraph B A B A D C D C E G E G* 2015 Goodrich and Tamassia Dirctd Graphs 11

12 Computing th Transitiv Closur q W can prform DFS starting at ach vrtx n O(n(n+m)) If thr's a way to gt from A to B and from B to C, thn thr's a way to gt from A to C. Altrnativly... Us dynamic programming: Th Floyd-Warshall Algorithm 2015 Goodrich and Tamassia Dirctd Graphs 12

13 Floyd-Warshall Transitiv Closur q Ida #1: Numbr th vrtics 1, 2,, n. q Ida #2: Considr paths that us only vrtics numbrd 1, 2,, k, as intrmdiat vrtics: i Uss only vrtics numbrd 1,,k (add this dg if it s not alrady in) Uss only vrtics numbrd 1,,k-1 k j Uss only vrtics numbrd 1,,k Goodrich and Tamassia Dirctd Graphs 13

14 Floyd-Warshall s Algorithm: High-Lvl Viw q Numbr vrtics v 1,, v n q Comput digraphs G 0,, G n n G 0 =G n G k has dirctd dg (v i, v j ) if G has a dirctd path from v i to v j with intrmdiat vrtics in {v 1,, v k } q W hav that G n = G* q In phas k, digraph G k is computd from G k - 1 q Running tim: O(n 3 ), assuming aradjacnt is O(1) (.g., adjacncy matrix) 2015 Goodrich and Tamassia Dirctd Graphs 14

15 Th Floyd-Warshall Algorithm q Th running tim is clarly O(n 3 ) Goodrich and Tamassia Dirctd Graphs 15

16 Floyd-Warshall Exampl BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 16

17 Floyd-Warshall, Itration 1 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 17

18 Floyd-Warshall, Itration 2 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 18

19 Floyd-Warshall, Itration 3 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 19

20 Floyd-Warshall, Itration 4 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 20

21 Floyd-Warshall, Itration 5 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 21

22 Floyd-Warshall, Itration 6 BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 22

23 Floyd-Warshall, Conclusion BOS v 7 ORD v 4 ot Th v Th 2 ot b SFO JFK Th v6 ot v 1 ot b LAX Th DFW Th v3 ot b MIA v5 Th ot b 2015 Goodrich and Tamassia Dirctd Graphs 23

24 DAGs and Topological Ordring q q A dirctd acyclic graph (DAG) is a digraph that has no dirctd cycls A topological ordring of a digraph is a numbring v 1,, v n of th vrtics such that for vry dg (v i, v j ), w hav i < j q Exampl: in a task schduling digraph, a topological ordring a task squnc that satisfis th prcdnc constraints Thorm A digraph admits a topological ordring if and only if it is a DAG 2015 Goodrich and Tamassia Dirctd Graphs 24 v 2 v 1 B A B A D C E DAG G v 4 v 5 D E C v3 Topological ordring of G

25 Topological Sorting q Numbr vrtics, so that (u,v) in E implis u < v 1 wak up A typical studnt day 2 3 at study computr sci nap mor c.s. play 8 writ c.s. program 6 9 work out bak cookis 10 slp 11 dram about graphs 2015 Goodrich and Tamassia Dirctd Graphs 25

26 Algorithm for Topological Sorting q Not: This algorithm is diffrnt than th on in th book Algorithm TopologicalSort(G) H G // Tmporary copy of G n G.numVrtics() whil H is not mpty do Lt v b a vrtx with no outgoing dgs Labl v n n n - 1 Rmov v from H q Running tim: O(n + m) 2015 Goodrich and Tamassia Dirctd Graphs 26

27 Implmntation with DFS q q Simulat th algorithm by using dpth-first sarch O(n+m) tim. Algorithm topologicaldfs(g) Input dag G Output topological ordring of G n G.numVrtics() for all u G.vrtics() stlabl(u, UNEXPLORED) for all v G.vrtics() if gtlabl(v) = UNEXPLORED topologicaldfs(g, v) Algorithm topologicaldfs(g, v) Input graph G and a start vrtx v of G Output labling of th vrtics of G in th connctd componnt of v stlabl(v, VISITED) for all G.outEdgs(v) { outgoing dgs } w opposit(v,) if gtlabl(w) = UNEXPLORED { is a discovry dg } topologicaldfs(g, w) ls { is a forward or cross dg } Labl v with topological numbr n n n Goodrich and Tamassia Dirctd Graphs 27

28 Topological Sorting Exampl 2015 Goodrich and Tamassia Dirctd Graphs 28

29 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 29

30 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 30

31 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 31

32 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 32

33 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 33

34 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 34

35 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 35

36 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 36

37 Topological Sorting Exampl Goodrich and Tamassia Dirctd Graphs 37