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Chapter 8: Directed Graphs

Chapter 8: Directed Graphs. digraph:. A digraph is a graph G = (V,E) where V is a finite set of vertices and E is a relation on V . If a, b ∈ V and aEb then there is a directed edge of E from a to b called an arc .

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Chapter 8: Directed Graphs

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  1. Chapter 8: Directed Graphs Discrete Math for CS

  2. Discrete Math for CS digraph: • A digraph is a graph G = (V,E) where V is a finite set of vertices and E is a relation on V. • If a, b ∈ V and aEb then there is a directed edge of E from a to b called an arc. • If u, v ∈ V and (u,v) is an arc we write uv as the arc name. • A simple digraph has no loops and no multiple edges. • If uv is an arc then u is the antecedent of v.

  3. Discrete Math for CS Example: V = { a, b, c, d } E = { ab, cb, dc, bd, db } a b a b c d a 0 1 0 0 b 0 0 0 1 c 0 1 0 0 d 0 1 1 0 c d

  4. Discrete Math for CS More Notation: • A path of length k in a digraph is a sequence of distinct vertices v1, v2, v3, ..., vk where vi-1vi is an arc for i = 1, ... k. • A cycle is a path where v1 = vk and no other vertices are the same. • A graph without cycles is called acyclic. • Directed acyclic graphs (DAGs) are some of the most important graphs. • In task-scheduling problems a DAG is called a PERT chart.

  5. Discrete Math for CS Example: • A student needs to take 8 courses to satisfy a major. The courses and their prerequisites are given below. Draw a PERT chart showing the order in which the courses can be taken.

  6. Discrete Math for CS Answer: A H G B C F D E

  7. Discrete Math for CS Topological Sort Algorithm • A topological sort algorithm produces a consistent labeling of the edges of the above graph. • A labeling 1, 2, 3, ... , n is consisent if uv is an arc, u has label i, v has label j and i < j.

  8. Discrete Math for CS TSA: • G = (V,E) is a digraph. • Let A(v) = { all antecedents of v }. begin for v V do calculate A(v); label := 0; while unlabeled vertices v remain for which A(v) = ∅do begin label := label + 1; u := a vertex with A(u) = ∅ ; assign label to u; for each unlabeled vertex v ∈ V do a(v) := A(v) \ {u} end end end

  9. Discrete Math for CS Example: • Find a consistent labeling of the previous graph. Step 0: A(A) = {B}, A(B) = {C}, A(C) = {H}, A(D) = {C}, A(E) = {D, G}, A(F) = {E}, A(G) = {C}, A(H) = ∅. Step 1: Assign label 1 to H since A(H) = ∅. A(A) = {B}, A(B) = {C}, A(C) = ∅, A(D) = {C}, A(E) = {D, G}, A(F) = {E}, A(G) = {C}, A(H) = ∅. Step 2: Assign label 2 to C since A(C) = ∅. A(A) = {B}, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = {D, G}, A(F) = {E}, A(G) =∅, A(H) = ∅. Step 3: Choose one of the possibilities> Assign label 3 to B. A(A) = ∅, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = {D, G}, A(F) = {E}, A(G) =∅, A(H) = ∅. Step 4: Assign label 4 to A. A(A) = ∅, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = {D, G}, A(F) = {E}, A(G) =∅, A(H) = ∅. Step 5: Assign label 5 to D. A(A) = ∅, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = {G}, A(F) = {E}, A(G) =∅, A(H) = ∅. Step 6: Assign label 6 to G. A(A) = ∅, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = ∅, A(F) = {E}, A(G) =∅, A(H) = ∅. Step 7: Assign label 7 to E. A(A) = ∅, A(B) =∅, A(C) = ∅, A(D) =∅, A(E) = ∅, A(F) = ∅, A(G) =∅, A(H) = ∅. Step 8: Assign label 8 to F. Consistent Labeling is H, C, B, A, D, G, E, F

  10. Discrete Math for CS Paths in Digraphs • Directed paths can represent things line airline routes or networked computers. • We may need to know alternative routes if a link goes down. • So we need to know if there exists a path between two vertices of a digraph. • G = (V,E) with n vertices. Let M be the adjacency matrix. If mij == 1 then there is an arc from vertex vito vertex vj. An arc is a path of length 1. • Consider M2. The boolean product of M by M yields a matrix which shows paths of length 2 in the original graph.

  11. Discrete Math for CS Example: V = { a, b, c, d } E = { ab, cb, dc, bd, db } a b a b c d a 0 1 0 0 b 0 0 0 1 c 0 1 0 0 d 0 1 1 0 c d M a b c d a 0 0 0 1 b 0 1 1 0 c 0 0 0 1 d 0 1 0 1 M2

  12. Discrete Math for CS Reachability Matrix • Calculate M* = M or M2 or M3 or ... or Mn • M* contains all paths of all lengths so shows what vertices are reachable from what vertices. • The reachability matrix of a graph is the graph of the transitive closure of its adjacency matrix.

  13. Discrete Math for CS Example: a b c d a 0 1 0 0 b 0 0 1 1 c 0 0 0 0 d 0 0 1 0 M a b a b c d a 0 0 1 1 b 0 0 1 0 c 0 0 0 0 d 0 0 0 0 c d M2 a b c d a 0 0 1 0 b 0 0 0 0 c 0 0 0 0 d 0 0 0 0 a b c d a 0 0 0 0 b 0 0 0 0 c 0 0 0 0 d 0 0 0 0 M3 M4

  14. Discrete Math for CS Example: a b a b c d a 0 1 1 1 b 0 0 1 1 c 0 0 0 0 d 0 0 1 0 c d M*

  15. Discrete Math for CS Large Matrices • This calculation is labourious for big matrices. • Warshall's Algorithm calculates M* more efficiently. • G = (V,E). with vertices v1, v2, ..., vn. Warshall's Algorithm generates matrices W0 = M, W1, W2, ..., Wn. • For k >= 1, Wk(i,j) = 1 iff there is a path of any length from vi to vjwhere the intermediary vertices in the path lie in the set {v1, ... vk}. • Wn = M*. • Warshall's Algorithm is efficient by a clever use of for-loops. • Successive passes of the outer loop calculate W1, W2, ..., Wn.

  16. Discrete Math for CS Warshall's Algorithm • G = (V,E). M is the adjacency matrix. Calculates W = M*. begin W := M; for k = 1 to n do for i = 1 to n do for j = 1 to n do W(i,j) = W(i,j) or W(i,k) and W(k,j); end Note: On each pass of outer loop the algorithm generates Wk. This is done by updating entries in Wk-1. To find ithrow of Wk we evaluate W(i,j) = W(i,j) or W(i,k) and W(k,j); (*)‏ for various values of j. If W(i,k) = 0 then (W(i,k) and W(k,j)) = 0 and so (*) reduces to W(i,j); ie, row i of the matrix remains unchanged. Otherwise W(i,k) = 1 and (*) reduces to W(i,j) or W(k,j). In this case row i becomes the logical or of the current row i and current row k.

  17. Discrete Math for CS Warshall's Algorithm: • So Warshall's Algorithm reduces to calculating Wkfrom Wk-1 as follows: • Consider the kthcolumn of Wk. • For each row with a 0 entry in this column, copy the row from Wk-1. • For each row with a 1 entry form the logical or of that row with row k and write the resulting row in Wk.

  18. Discrete Math for CS Example (Warshall's Algorithm): 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 0 0 2 3 W0 4 copy rows 1,2,4 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 5 W1 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 W1 W1 row 3 or row 1 row 5 or row 1

  19. Discrete Math for CS Example (Warshall's Algorithm): 0 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 W1 2 3 4 0 0 1 0 0 0 0 0 0 0 W2 1 5 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 W2

  20. Discrete Math for CS Example (Warshall's Algorithm): 0 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 W2 2 3 4 0 0 0 0 0 W3 1 5 Note: W4 = W3 so we are done. 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 W3

  21. Discrete Math for CS Shortest Paths • Find the shortest path between two vertices in a weighted digraph. • Typical situations – transportation networks, communications networks

  22. Discrete Math for CS Shortest Paths: Find the shortest path from A to any other vertex. B C 1 2 5 A F 4 3 1 weight matrix: 2 E D 0 if u = v ∞ if uv is not an arc d if uv is an arc of weight d w(u,v) = A B C D E F A 0 2 ∞ 3 ∞ ∞ B ∞ 0 1 ∞ 4 ∞ C ∞ ∞ 0 ∞ ∞ 5 D ∞ ∞ ∞ 0 2 ∞ E ∞ ∞ ∞ ∞ 0 1 F ∞ ∞ ∞ ∞ ∞ O

  23. Discrete Math for CS Idea: • Initially define d[v] to be the weight of an arc from A to v. • d[v] = ∞ if there is no arc. • We traverse the vertices and improve d[v] as we go. • We mark a value for d[u] once we know for sure the shortest route to u from A. • For the remaining vertices, w, we assign the min of the current value of d[w] and the distance to w via the last marked vertex, u. • The algorithm terminates once all vertices that can be marked are marked.

  24. Discrete Math for CS Step 0: Mark A and let the first row represent the initial values of d[v]. Step 1: Mark B since it is closest to A. Calculate the distances to unmarked vertices via B. If a shorter distance is found, use it.. Vertices not adjacent to the last marked have their d[v] values unchanged. Step 2: Next mark D (we could mark C too). Calculate the remaining distances Step 3: Mark C. F can now be accessed. Step 4 and 5: Mark E and F.

  25. Discrete Math for CS Dijkstra's Algorithm • G = (V,E) is a weighted digraph, A is a vertex. The algorithm finds the shortest path from A to v as well as d[v]. • w(u,v) is the weight of arc uv • PATHTO(v) lists the vertices on the shortest path to v from A. begin for each v in V do begin d[v] = w(A,v); PATHTO(v) := A; end Mark vertex A while unmarked vertices remain do begin u:= unmarked vertex closest to A Mark u; <for-loop> end end for each unmarked vertex v with uv in E do begin d' := d[u] + w(u,v); if d' < d[v] then d[v] := v'; PATHTO(v) := PATHTO(u), v end end

  26. Discrete Math for CS Exercise: • Use Dijkstra's Algorithm with the following graph: B E 4 12 3 2 6 F 11 8 D A 5 10 C

  27. Discrete Math for CS Answer

  28. Discrete Math for CS

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