Lecture 6: Topological Ordering and Greedy Algorithms in Graph Theory, Study notes of Computer Science

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9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

Adam Smith

Algorithm Design and Analysis LECTURE 6

• Topological ordering
• Greedy Algorithms

Notes

• Read on your own
  • Graph bipartiteness
  • DFS implementation
• Homework notation:
  • log a

(n) = (log n)

a

• Useful property: 9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne

!x" = smallest integer ≥ x

= “ceiling”

x ≤ "x# < x + 1

Strong Connectivity: Algorithm

Lemma : G is strongly connected if and only if for any node s , every other node t has paths to and from s. Theorem. Can determine if G is strongly connected in O(m + n) time. Pf.  Pick any node s.  Run BFS from s in G.  Run BFS from s in Grev.  Return true iff all nodes reached in both BFS executions.  Correctness follows immediately from the previous lemma. reverse orientation of every edge in G strongly connected not strongly connected

Directed Acyclic Graphs

Def. An DAG is a directed graph that contains no directed cycles. Ex. Precedence constraints: edge (vi, vj) means vi must precede vj. Def. A topological order of a directed graph G = (V, E) is an ordering of its nodes as v 1 , v 2 , …, vn so that for every edge (vi, vj) we have i < j. a DAG a topological ordering v 2 v 3 v 6 v 5 v 4 v 7 v 1 v 1 v 2 v 3 v 4 v 5 v 6 v 7

Directed Acyclic Graphs

Lemma. If G has a topological order, then G is a DAG. Pf. (by contradiction)  Suppose that G has a topological order v 1 , …, vn and that G also has a directed cycle C. Let's see what happens.  Let vi be the lowest-indexed node in C, and let vj be the node just before vi; thus (vj, vi) is an edge.  By our choice of i, we have i < j.  On the other hand, since (vj, vi) is an edge and v 1 , …, vn is a topological order, we must have j < i, a contradiction. ▪ v 1 vi vj vn the supposed topological order: v 1 , …, vn the directed cycle C

Directed Acyclic Graphs

Lemma. If G has a topological order, then G is a DAG. Q. Does every DAG have a topological ordering? Q. If so, how do we compute one?

Directed Acyclic Graphs

Lemma. If G is a DAG, then G has a topological ordering. Pf. (by induction on n)  Base case: true if n = 1.  Given DAG on n > 1 nodes, find a node v with no incoming edges.  G - { v } is a DAG, since deleting v cannot create cycles.  By inductive hypothesis, G - { v } has a topological ordering.  Place v first in topological ordering; then append nodes of G - { v }  in topological order. This is valid since v has no incoming edges. ▪ DAG v

Topological Sorting Algorithm: Running Time

Theorem. Algorithm finds a topological order in O(m + n) time. Pf.  Maintain the following information:

• count[w] = remaining number of incoming edges
• S = set of remaining nodes with no incoming edges  Initialization: O(m + n) via single scan through graph.  Update: to delete v
• remove v from S
• decrement count[w] for all edges from v to w, and add w to S if c count[w] hits 0
• this is O(1) per edge ▪ Alternative algorithm: Modification of DFS (exercise?)

9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne Greedy Algorithms

• Build up a solution to an optimization problem at

each step shortsightedly choosing the option that

currently seems the best.

• Sometimes good
• Often does not work

9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne Interval Scheduling Problem

• Job j starts at s j

and finishes at f

j

• Two jobs are compatible if they do not overlap.
• Find: maximum subset of mutually compatible jobs. 0 1 2 3 4 5 6 7 8 9 10 11 f g h e a b c d Time

9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne Greedy: Counterexamples

for earliest start time

for shortest interval

for fewest conflicts

Next lecture

• We will see that adding jobs greedily in order of

earliest finishing time gives an optimal solution

9/5/ A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne