1. Directed Spanning Trees updated 2 April 2007
EMIS 8374
Directed Spanning Trees
updated 2 April 2007

2. Directed Spanning Tree (DST)
Input
A network (directed graph) G = (N,A)
Arc cost cij for each edge arc (i,j)  A
A designated root node s  N
Optimization Problem
Find a minimum-cost directed-out tree T = (N, A*) routed at node s.
Directed-Out Tree: T = (N, A*) contains a unique, directed path from s to every other node in N (every other node in the tree has in-degree = 1).

3. Is DST a (Pure) Network Flow Problem?
Suggestion:
Let b(s) = |N| - 1
Let b(i) = -1 for i in N\{s}
Let ij = 0
Let uij = |N|

4. DST Example 10 2 3 10 1 1 s 5 5 4 5

5. DST Example: Optimal Solution10 2 3 10 1 1 s 5 5 4
Cost = 26

6. DST Example: MCNF Formulation-1 -1
(0,4,10) 2 3
(0,4,10) 1 +4
(0,4,1)
(0,4,5) 5 4
(0,4,5) -1 -1
(ij,uij,cij) i j
b(i)

7. DST Example: MCNF Solution-1 -1
(1,10) 2 3
(2,10) 1 +4
(0, 1)
(2,5) 5 4
(1,5) -1 -1
(xij, cij) i j
Total Cost = 45
b(i)

8. O(min(m log n, n2))
Chu and Liu [1965], Edmonds [1967], Bock [1971], Tarjan [1977]