Data Structures

1
Data Structures Algorithms Digraphs and DAGs
Richard Newman based on book by R. Sedgewick and
slides by S. Sahni
2
Digraphs

• Edges are directed
• Number of possible undirected graphs is huge
• 2V(V1)/2
• Number of possible directed graphs is huger(?)
• 2V2

3
Digraphs
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
8 wow Wow wow wow

• Graph enumeration

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.

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.

6
DAGs

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

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.

8
DAGs

• Connected components
• 0
• 0-7-4-5-3
• 2
• 2-6
• 1
• 1

0
2
2
0
7
5
1
1
7
5
6
4
3
6
4
3
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.

10
DAGs

• DAG Components
• 0 0-7-4-5-3
• 1 2-6
• 2 1
• Kernel DAG
• Component 0
• Component 1
• Component 2

0
2
2
0
7
5
1
1
7
5
6
4
3
6
4
3
0
1
2
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.

12
Digraphs

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

13
DFS in Digraphs
Stack 0 572 576 57(2) 541 54 55(6 cross) 53(0
back) 5(4 back) (5 down)
0
down
2
7
0
2
0
2
6
4
1
back
7
5
1
5
7
1
cross
5
6
4
3
4
3
6
3
0
1
2
3
4
5
6
7
pre
0
4
1
7
5
6
2
3
post
7
2
1
3
5
4
0
6
14
DFS Algorithms

• Cycle Detection
• If we find a back edge, it represents a cycle
  including link to parent!
• Cross edges dont make cycles!
• Reachability
• Start from one, DFS until find other (or complete
  DFS)
• Weak connectivity
• If DFS finds all the nodes, then yes!

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

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.

0
2
0
2
0
2
7
5
1
5
7
1
1
5
7
6
4
3
4
3
3
4
6
6
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 0 1 2 3 4 5 6 7 0 0 0 1 0 0 1 0 1 1 0 0 0 0
0 0 0 0 2 0 0 0 0 0 0 1 0 3 0 0 0 0 1 0 0 0 4
0 0 0 0 0 1 1 0 5 1 0 0 1 0 0 0 0 6 0 0 1 0 0 0
0 0 7 0 1 0 0 1 0 0 0
A2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
0
1 1 0 1 1 0 1 0
0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 0 0 1 1 0
1 0 1 1 0 0 0 0
0 0 1 0 1 1 0 1
0 0 0 0 0 0 1 0
0 0 0 0 0 1 1 0
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

0 1 2 3 4 5 6 7 0 1 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 2 0 0 1 0 0 0 1 0 3 0 0 0 0 1 1 1 0 4
1 0 1 1 0 1 1 0 5 1 0 1 1 1 1 0 1 6 0 0 1 0 0 0
1 0 7 0 1 0 0 1 1 1 0
A3 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Alt3
0
0 0 1 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 0 1 1 0 0 0 0
0 0 1 0 1 1 1 1
1 1 0 1 1 1 1 0
0 0 1 0 0 0 0 0
1 1 1 1 1 0 1 0
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

0 1 2 3 4 5 6 7 0 1 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 2 0 0 1 0 0 0 1 0 3 1 0 1 1 1 1 1 0 4
1 0 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 0 0 1 0 0 0
1 0 7 1 1 1 1 1 1 1 0
A4 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Alt4
0
0 0 1 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 1 0 1 1 1 1
1 1 1 1 1 1 1 0
1 0 1 1 1 1 1 1
0 0 0 0 0 0 1 0
0 0 1 0 1 1 1 1
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

0 1 2 3 4 5 6 7 0 1 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 2 0 0 1 0 0 0 1 0 3 1 0 1 1 1 1 1 1 4
1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 0 0 1 0 0 0
1 0 7 1 1 1 1 1 1 1 1
A5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
Alt5
0
0 0 1 0 1 1 1 1
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
1 1 1 1 1 1 1 0
1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1
0 0 1 0 0 0 0 0
1 1 1 1 1 1 1 0
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

0 1 2 3 4 5 6 7 0 1 1 1 1 1 1 1 1 1 0 0 0 0
0 0 0 0 2 0 0 1 0 0 0 1 0 3 1 1 1 1 1 1 1 1 4
1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 6 0 0 1 0 0 0
1 0 7 1 1 1 1 1 1 1 1
Alt6
0

• Keep on multiplying and adding until
• reach fixed point
• Matrix does not change

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

23
Transitive Closure

• Even faster way!
• Warshalls algorithm

for (i 0 i lt V i) for (s 0 s lt V
s) for (t 0 t lt V t) if
(Asi Ait) Ast 1
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 gt
i Inductive step path from s to t w/o i1
(already there) or path via i1 (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
25
Transitive Closure

• Prop. 19.7 Warshalls algorithm computes the
  transitive closure of a digraph in time V3.
• Obvious from structure of Warshalls algorithm
  three nested loops of V each

for (i 0 i lt V i) for (s 0 s lt V
s) for (t 0 t lt V t) if
(Asi Ait) Ast 1
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 Warshalls algorithm

for (i 0 i lt V i) for (s 0 s lt V
s) for (t 0 t lt V t) if
(Asi Ait) Ast 1
27
Transitive Closure

• We can improve Warshalls algorithm by moving the
  test of Asi out of the inner loop, avoiding
  innermost loop when s cannot reach i.

for (i 0 i lt V i) for (s 0 s lt V
s) if (Asi) for (t 0 t lt V
t) if (Ait) Ast 1
28
Shortest Path

• We can modify Warshalls 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 lt V i) for (s 0 s lt V
s) for (t 0 t lt V t) if
(Asi Ait lt Ast) Ast
Asi Ait
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 0 I A AB 0 I B 0 I B 0 0 I 0
0 I
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(EV))

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

32
Topological Sort
Relabel
0
2
0
2
0
2
0
7
7
5
1
7
5
1
7
5
1
1
5
2
6
4
3
6
4
3
6
4
3
6
Rearrange
0
6
7
2
4
5
3
1
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

34
Topological Sort

• Prop. 19.11 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

35
Topological Sort

• Prop. 19.12 Every DAG has at least one source
  and at least one sink.
• Turn this into a topological sort algorithm
• Make indegreeV vector initialized to 0
• Scan through DAG (visiting every edge) and
  increment indegreei each time an edge to node i
  is found.
• Scan through indegree and enqueue all nodes
  with indegree zero (the sources).

36
Topological Sort

• Prop. 19.12 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 indegreej for all edges from node x
  to node j
• If indegreej 0, enqueue node j

37
Topological Sort
0
2
0
2
0
2
0
7
7
5
1
7
5
1
7
5
1
2
5
3
6
4
3
6
4
3
6
4
1
6
indegree
0
1
2
0
2
3
1
1
0
1
2
1
1
0
0
0
0
0
Source Queue
0
6
7
2
4
5
3
1
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, LPv 1 maxLPx (v,x) in E
• Guaranteed that LPx is known by time it is
  needed. Example of
• Dynamic Programming!

39
Topological Sort
0
2
0
2
0
7
7
5
1
7
5
1
2
5
3
6
4
3
6
4
1
6
Longest path
1
5
1
4
3
2
1
4
40
Transitive Closure Redux

• Reachability from each node in DAG
• Reverse topological sort DAG
• Row vector Reachv initially self and
  successors
• In RTS order, for each node v
• Reachv OR Reachx (v,x) in E
• Guaranteed that Reachx is known by time it is
  needed. Another example of
• Dynamic Programming!

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. 19.13 Using DFS and DP, can compute TC of
  a DAG in time O(V2VX) where X is the number of
  cross edges

42
Strongly Connected Components

• Kosarajus 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 VE!

43
Recap

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