1. Chu-Liu/Edmonds’ Algorithm an introduction
Qingxia Liu
2019/2/24

2. A Quick Glance Chu-Liu/Edmonds’ Algorithm Contributors input output
a weighted directed graph G(V,E)
root r (reachable to every vertex of G)
output
a minimum spanning tree of G rooted at r
Contributors
Yoeng-jin Chu and Tseng-hong Liu (1965)
Edmonds(1967)
Bock(1971)
Tarjan(1977): O(ElogV), O(V2)
Gabow(1986): O(E+VlogV) ……

3. Basic Concepts Minimum Spanning Tees in directed graphs
a spanning tree rooted at r
a set of n-1 edges containing paths from r to every vertex
of minimum total edge cost
MINT ∑(u,v)∈T c(u,v)

4. Chu-Liu/Edmonds Algorithm
Construct original S
discard entering edges of r
For i∈V\{r}, select the edge (k,i) of mink c(k,i)
Let the selected n-1 edges be the set S.
If no cycle formed, G(N,S) is a MST.
Otherwise, deal with cycle.

5. Chu-Liu/Edmonds Algorithm
Deal with Cycle
For each cycle formed
contract the cycle into a pseudo-node (k)
for each edge(i,j) (i∈V\cycle, j∈cycle) modify the cost: c(i,k)=c(i,j)-c(x(j),j)
for each pseudo-node
select the entering arc which has the smallest modified cost
replace the arc which enters the same real node in S by the new selected arc.
If no cycle formed, G(N,S) is a MST.
else deal with cycle
O(E) time
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any

6. Example Construct original S 1 10 8 5 6 4 7 3 11 92 3 4 5 10 11 9 7 8 1 1 5 2 3 6 4
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any 5 4 3 6

7. Example Cycle Contract k 10 11 9 7 8 10 11 9 7 8 1 5 6 4 32 3 4 5 10 11 9 7 8 1 6 2 3 4 5 10 11 9 7 8 1 1 5 2 k 3 6 4
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any 5 4 3 6

8. Example Edge Replacing modify cost: c(i,k) = c(i,3)-c(4,3)
or c(i,4)-c(5,4)
or c(i,5)-c(3,4))
Edge Replacing 1 6 2 3 4 5 10 11 9 7 8 1 1 1 5 1 6 2 2 4 5 3 3 6 6 4 7
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any 5 5 4 4 3 3 5 2 6 6 6

9. Example Minimum Spanning Tree ∑(u,v)∈T c(u,v)=23 10 11 9 7 8 10 11 9 76 2 3 4 5 10 11 9 7 8 1 6 2 3 4 5 10 11 9 7 8
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any

10. Implementation Tarjan (1977) O(mlogn)
Camerini et.al.(1979) repaired an error
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any

11. Implementation Tarjan (1977)
First step: of any vertex v, choose the edge (u,v) of largest value among its entering edges, add to set H
General step:
for each root component S in G(H)
find the largest unexamined entering edge (u,v) (v∈S)
if u∈S, discard the edge;
else, let W be the weakly connected component of G(H) that v∈W.
if u∈W, add (u,v) to H;
root component: no entering edge

12. Implementation Tarjan (1977) Disjoint Set Priority Queue
else u∉S, u∈W, find the cycle c(between S1……Sk) formed by (u,v), let (i,j) be the minimum edge on the cycle, modify unexamined edge (x,y) y∈Si as
c(x,y)=c(x,y)-c(a,y’)+c(i,j)
add (u,v) to H
Disjoint Set
record weakly/strongly connected component
Priority Queue
keep track of entering edges of each strongly connected compnent
root component: no entering edge

13. Implementation Gabow (1986) O(nlogn+m) improvement on the first phase:
depth-first strategy: growth path
root component: no entering edge

14. References
Tarjan, Robert Endre. "Finding optimum branchings." Networks 7.1 (1977):
H. N. Gabow, Z. Galil, T. Spencer, and R. E. Tarjan, “Efficient algorithms for finding minimum spanning trees in undirected and directed graphs,” Combinatorica 6 (1986),
Camerini, Paolo M., Luigi Fratta, and Francesco Maffioli. "A note on finding optimum branchings." Networks 9.4 (1979):

15. Thank You ~
Any questions？

16. Basic Concepts Branching Optimum Branching
a branching B of G is a set of edges such that
each vertex has at most one in-degree
if (x1,y1),(x2,y2) are distinct edges of B, then y1 != y2
no cycle
Optimum Branching
a branching that ∑(u,v)∈B c(u,v) is maximum/minimum

17. Example Minimum Spanning Tree ∑(u,v)∈T c(u,v)=23 10 11 9 7 8 10 11 9 76 2 3 4 5 10 11 9 7 8 1 6 2 3 4 5 10 11 9 7 8
where c(x(j),j) is the cost of the arc in the cycle which enters j.
find the replacing arc(s) which has the minimum extra cost to eliminate cycle(s) if any