1. Data Structures & Algorithms Digraphs and DAGs
Richard Newman
based on book by R. Sedgewick
and slides by S. Sahni 1 1 1 1

2. Digraphs Edges are directed
Number of possible undirected graphs is huge
2V(V+1)/2
Number of possible directed graphs is … huger(?)
2V^2 2 2 2 2

3. Digraphs Graph enumeration V Undir Graphs Digraphs 2 8 16 3 64 512 4
1,024
65,536 5
32,768
33,554,432 6
2,097,152
68,719,476,736 7
268,435,456
562,949,953,421,312
wow
Wow wow wow 3 3 3 3

4. Digraphs
Defn. 19.1: A digraph is a set of nodes V and a set of distinct directed edges E, each from one node to another node in V. (self-loop allowed)
Defn. 19.2: A directed path in a digraph is a list of nodes for which there is an edge from each node to its successor. A node t is reachable from node s iff there is a d.p. from s to t. 4 4 4 4

5. Digraphs
Defn. 19.3: A directed acyclic graph (DAG) is a digraph with no directed cycles (tours). A node with only out- edges is a source; a node with only in-edges is a sink.
Defn. 19.4: A digraph is strongly connected iff every node is reachable from every node. 5 5 5 5

6. DAGs DAGs can be used to model many real-life problems Scheduling
Precedence
Pre-requisite structures
Causality
Etc. 6 6 6 6

7. Digraphs
Prop. 19.1: A digraph that is not strongly connected comprises a set of strongly connected components, which are maximal strongly connected subgraphs, and a set of directed edges that go from one component to another. 7 7 7 7

8. DAGs Connected components 0-7-4-5-3 2 2-6 1 2 2 1 1 7 7 5 5 6 6 4 4 3 2
2-6 1 2 2 1 1 7 7 5 5 6 6 4 4 3 3 8 8 8 8

9. Digraphs
Prop. 19.2: Given a digraph D, define another digraph K(D) with one node corresponding to each strongly connected component of D, and an edge from u to v iff there is one or more edge from the component corresponding to u to the component corresponding to v. K(D) is a DAG called the kernel DAG of D. 9 9 9 9

10. DAGs DAG Components 0: 0-7-4-5-3 1: 2-6 2: 1 Kernel DAG Component 01 1 7 7 5 5 6 6 4 4 3 3 1 2 10 10 10 10

11. Digraphs
In undirected graph, we just say two nodes are connected if there is a path between them
In a digraph, node t is reachable from node s if there is a directed path from s to t.
In a digraph, s and t are strongly connected if they are mutually reachable. 11 11 11 11

12. Digraphs Classify edges in DFS Tree – recursive calls
Back – to ancestor (including parent!)
Down – to visited descendent
Cross – neither ancestor nor descendent (cousins) 12 12 12 12

13. DFS in Digraphs
Stack:
572
576
57(2)
541 54
55(6 cross)
53(0 back)
5(4 back)
(5 down)
down 2 7 2 2 6 4
back 1 1 1 7 7 5 5
cross 5 6 6 4 4 3 3 3 1 2 3 4 5 6 7
pre 4 1 7 5 6 2 3
post 7 2 1 3 5 4 6 13 13 13 13

14. DFS Algorithms Cycle Detection
If we find a back edge, it represents a cycle – including link to parent!
Cross edges don’t make cycles!
Reachability
Start from one, DFS until find other (or complete DFS)
Weak connectivity
If DFS finds all the nodes, then yes! 14 14 14 14

15. DFS Algorithms Convert digraph to DAG Remove back edges!
Use to generate large DAGs from large digraphs
Note that DFS in a digraph only gives reachability from the start node, not from all nodes 15 15 15 15

16. Transitive Closure
Defn. 19.5: The transitive closure of a digraph D is a digraph T with the same vertices but with an edge from s to t in T iff t is reachable from s in D. 2 2 2 1 7 5 1 1 7 7 5 5 6 4 3 6 6 4 4 3 3 16 16 16 16

17. Transitive Closure
Can also view (and compute) transitive closure by Boolean matrix multiplication
Use logical AND as x
Use logical OR as +
Ai represents (any) path of length i A A 1 2 3 4 5 6 7 17 17 17 17

18. Transitive Closure
Can also view (and compute) transitive closure by Boolean matrix multiplication
Use logical AND as x
Use logical OR as +
Ai represents (any) path of length i
A<3 A 1 2 3 4 5 6 7 18 18 18 18

19. Transitive Closure
Can also view (and compute) transitive closure by Boolean matrix multiplication
Use logical AND as x
Use logical OR as +
Ai represents (any) path of length i
A<4 A 1 2 3 4 5 6 7 19 19 19 19

20. Transitive Closure
Can also view (and compute) transitive closure by Boolean matrix multiplication
Use logical AND as x
Use logical OR as +
Ai represents (any) path of length i
A<5 A 1 2 3 4 5 6 7 20 20 20 20

21. Transitive Closure
Can also view (and compute) transitive closure by Boolean matrix multiplication
Use logical AND as x
Use logical OR as +
Ai represents (any) path of length i
A<6
Keep on multiplying and adding until…
… reach fixed point
Matrix does not change 21 21 21 21

22. Transitive Closure
Prop. 19.5: We can compute the transitive closure of a digraph by adding self-loops, then computing AV, taking time V4.
Self-loops allow path to be of any length up to exponent
Must reach fixed point by V – why?
Efficient approach:
A, A2, A4, A8, … successive squaring
Takes lg V matrix multiplies, each V3
Total time is V3 lg V 22 22 22 22

23. Transitive Closure Even faster way! Warshall’s algorithm:
for (i = 0; i < V; ++i)
for (s = 0; s < V; ++s)
for (t = 0; t < V; ++t)
if (A[s][i] && A[i][t])
A[s][t] = 1; 23 23 23 23

24. Transitive Closure Correctness by induction on i:
Base: After first iteration, s-t or s-0-t
After second iteration, s-t, s-0-t, s-1-t, s-0-1-t, s-1-0-t.
IH: After ith iteration – all paths w/o inner nodes > i
Inductive step: path from s to t w/o i+1 (already there)
or path via i+1 (tested by if statement)
for (every intermediate node i)
for (every source s)
for (every destination t)
if (s reaches i & i reaches t)
s reaches t; 24 24 24 24

25. Transitive Closure
Prop. 19.7: Warshall’s algorithm computes the transitive closure of a digraph in time V3.
Obvious from structure of Warshall’s algorithm – three nested loops of V each:
for (i = 0; i < V; ++i)
for (s = 0; s < V; ++s)
for (t = 0; t < V; ++t)
if (A[s][i] && A[i][t])
A[s][t] = 1; 25 25 25 25

26. Transitive Closure
Prop. 19.8: We can support constant-time reachability testing for a digraph with V nodes using space O(V2) and preprocessing time O(V3).
Can improve Warshall’s algorithm:
for (i = 0; i < V; ++i)
for (s = 0; s < V; ++s)
for (t = 0; t < V; ++t)
if (A[s][i] && A[i][t])
A[s][t] = 1; 26 26 26 26

27. Transitive Closure
We can improve Warshall’s algorithm by moving the test of A[s][i] out of the inner loop, avoiding innermost loop when s cannot reach i.
for (i = 0; i < V; ++i)
for (s = 0; s < V; ++s)
if (A[s][i])
for (t = 0; t < V; ++t)
if (A[i][t])
A[s][t] = 1; 27 27 27 27

28. Shortest Path
We can modify Warshall’s algorithm to compute shortest path, if A[][] contains the length of the minimum path from s to t (initialized with 1 for an edge and sentinel value V for no edge).
for (i = 0; i < V; ++i)
for (s = 0; s < V; ++s)
for (t = 0; t < V; ++t)
if (A[s][i] + A[i][t] < A[s][t])
A[s][t] = A[s][i] + A[i][t]; 28 28 28 28

29. Reduction
Prop. 19.9: We can use any transitive- closure algorithm to compute the product of two Boolean matrices with at most a constant factor difference in running time.
Prf: Construct a 3v x 3v matrix using A, B, and VxV identity matrix I. TC is square. 2
I A I A AB
0 I B = 0 I B
0 0 I I 29 29 29 29

30. Reduction
What this means is that if we can perform transitive closure faster, then we can compute Boolean matrix products faster.
Likewise, a faster Boolean matrix multiply algorithm will speed up our TC algorithm.
Note that we can compute TC faster for sparse graphs – time O(V(E+V)) 30 30 30 30

31. Topological Sort
Relabel: Given a DAG, relabel its nodes such that every directed edge points from a lower-numbered node to a higher- numbered node.
Rearrange: Given a DAG, rearrange its nodes on a horizontal line such that all the directed edges point from left to right.
Key = turn partial order into total order that is consistent with the partial order 31 31 31 31

32. Topological Sort Relabel: 2 2 7 2 1 7 5 1 1 2 7 7 1 5 5 5 6 4 3 6 6 62 7 2 1 7 5 1 1 2 7 7 1 5 5 5 6 4 3 6 6 6 4 4 3 3
Rearrange: 3 7 1 4 5 6 2 32 32 32 32

33. Reverse Topological Sort
Relabel: Given a DAG, relabel its nodes such that every directed edge points from a higher-numbered node to a lower- numbered node.
Rearrange: Given a DAG, rearrange its nodes on a horizontal line such that all the directed edges point from right to left.
Just reverse the regular topological sort 33 33 33 33

34. Topological Sort
Prop : Postorder numbering in DFS yields a reverse topological sort for any DAG.
It is easy to turn reverse topological sort into a regular topological sort 34 34 34 34

35. Topological Sort
Prop : Every DAG has at least one source and at least one sink.
Turn this into a topological sort algorithm:
Make indegree[V] vector initialized to 0
Scan through DAG (visiting every edge) and increment indegree[i] each time an edge to node i is found.
Scan through indegree[] and enqueue all nodes with indegree zero (the sources). 35 35 35 35

36. Topological Sort
Prop : Every DAG has at least one source and at least one sink.
Set currentID = 0.
While the queue is non-empty,
remove a node x and label it currentID
currentID++
Decrement indegree[j] for all edges from node x to node j
If indegree[j] == 0, enqueue node j 36 36 36 36

37. Topological Sort 2 2 7 2 1 7 5 1 3 1 7 2 7 5 5 5 6 4 3 6 6 6 4 4 3 1
indegree: 1 2 1 1 2 3 2 1 1 1
Source Queue: 3 7 1 4 5 6 2 37 37 37 37

38. Topological Sort Application
Finding longest path from each node, longest path in DAG
“Critical path” for scheduling
What is the most urgent thing to do?
Reverse topological sort DAG
In RTS order, for each node v, longest path from v, LP[v] = 1 + max{LP[x] | (v,x) in E}
Guaranteed that LP[x] is known by time it is needed. Example of…
Dynamic Programming! 38 38 38 38

39. Topological Sort 2 2 7 1 3 1 7 2 7 5 5 5 6 6 6 4 4 1 3 Longest path: 51 3 1 7 2 7 5 5 5 6 6 6 4 4 1 3
Longest path: 5 4 4 1 3 1 2 1 39 39 39 39

40. Transitive Closure Redux
Reachability from each node in DAG
Reverse topological sort DAG
Row vector Reach[v] – initially self and successors
In RTS order, for each node v
Reach[v] = OR {Reach[x] | (v,x) in E}
Guaranteed that Reach[x] is known by time it is needed. Another example of…
Dynamic Programming! 40 40 40 40

41. Transitive Closure Redux
Matrix approach takes time O(VE)
Direct recursive DFS approach:
No back edges (no cycles)
Tree edges – recursive call
Cross edges – do OR, no call
Down edges – skip, no call
Prop : Using DFS and DP, can compute TC of a DAG in time O(V2+VX) where X is the number of cross edges 41 41 41 41

42. Strongly Connected Components
Kosaraju’s method
Do DFS on reversed digraph
Compute post-order numbering
Do DFS of original graph using reverse of postorder computed on reverse graph
Time and space are both linear in V+E! 42 42 42 42

43. Recap Digraphs Strong connectivity Connected components Reachability
Digraph kernel
Transitive closure
Shortest paths (special case)
Reduction (from Boolean matrix multiply)
Topological Sort 43 43 43 43