An Algorithm for enumerating All Sp anning Trees of a Directed Graph

An Algorithm for enumerating All Sp anning Trees of a Directed Graph

- S. Kapoor and H. Ramesh

Speakers: 李孟韓 ¹, 林蔚茵 ², 莊秋芸 ³, 黃稚 穎 ⁴

Reference

• H. N. Gabow and E. W. Myers: Finding all spanning trees of directed and un directed graphs, SIAM J. Comput., Vol. 7, No. 3, 1978.

• S. Kapoor, V. Kumar, and H. Ramesh: An algorithm for generating all spann ing trees of directed graphs, Proceedings of the Workshop on Algorithms an d Data Structures, LNCS, Vol. 955, Springer-Verlag, Berlin, pp. 428–439, 19 95.

• S. Kapoor and H. Ramesh: Algorithms for generating all spanning tree s of undirected and weighted graphs, SIAM J. Comput., Vol. 24, No. 2, 1995.

• W. Mayeda: Graph Theory, Wiley, New York, 1972.

• S. Shinoda: Finding all possible directed trees of a directed graph, Electron Comm. Japan, Vol. 51-A, pp. 45–47, 1968.

Outline

• Introduction and Algorithm outline

– 李孟韓

• Main Algorithm

– 林蔚茵

• Correctness

– 莊秋芸

• Time analysis and conclusion

– 黃稚穎

Definition

• An exchange for a spanning tree T of G rooted at v is a pair of edges (e, f) , where e T∈ , f E – T∈ , and T – {e} {f}∪ is a spanning tree rooted at v.

• A edge p E - T∈ is back edge if its tail is an ancestor of its head in T.

• A edge p’ E - T∈ is forward edge if its tail is a descen dant of its head in T.

v

u T

v

u

z x

G

back edge forward edge

f

x = Tail(f) V = head(f)

cross edge

Property 1

• Every nonback nontree edge, f , relative to a spanning tree T , may r eplace exactly one tree edge, e, in the spanning tree, namely, the edg e having the same tail, to result in a new spanning tree rooted at the same vertex as T.

x

f

e

v

u

z

Property 2

• A back edge cannot be exchanged for any edge in the spanning tree to get a new spanning tree.

x

e

v

u

z

g

Computation Tree

• let CD(G,v) be the computation tree which generates all spanning tr ees of the directed graph G with root v.

a b

1

b

2

CD(G,v) SD_a

x

f

e

v

u

z

g

Computation Tree

• SD_b1 is obtained from SD_a by exchanging f with e, where f is a nontr ee nonback edge and e is the unique tree edge with the same tail as f .

a b

1

b

2

CD(G,v) SD_b1

pick f

x

f

e

v

u

z

g

Computation Tree

• SD_b2 is the same as SD_a. The significance of b₂ is that the subtree ro oted at b₂ will not include f in any spanning tree.

a b

1

b

2

CD(G,v) SD_b2

delete f

x

e

v

u

z

g

Computation Tree

a b

1

b

2

CD(G,v) SD_b1

pick f

x

f

e

v

u

z

Computation Tree

a b

1

b

2

CD(G,v) SD_b2

delete f

x

e

v

u

z

g

Lemma

• CD(G,v) has at its nodes all directed spanning

trees of G rooted at vertex v.

An Algorithm for enumerating All Sp anning Trees of a Directed Graph

--- S. Kapoor and H. Ramesh

Speaker: 林蔚茵 ²

Algorithm Description

• DFS tree of G (rooted at r)

– The root of the computation tree CD(G,r)

• For each node a of the computation tree

– NB: a set of those nontree edges which are nonback w.r.t. the directed spanning tree and which are no t included in

• Maintained as a list of nonempty lists

– Each nonempty list containg edges incident upon a particular v ertex

– Arranged in postorder number

– B: a set of those back edges w.r.t. which are not i ncluded in

SDa

OUTa

SDa

OUTa

Property 3

• Let spanning tree T’ be obtai ned from spanning tree T by applying the exchange (e,f). I f x is a nontree edge which is back w.r.t. T and nonback w.r .t. T’, then head(x) lies in the subtree of T rooted at tail(f), and tail(x) is a vertex which is a proper ancestor of tail(f) an d a proper descendant of lca (head(f), tail(f)) in T

v

f e

a

u

x

head(x) tail(x)

tail(f) lca(head(f),tail(f))

ALGO Main(G,r)

• The first edge in the first list of NB is the one having tail node with th e least postorder number.

• CHANGES is used to store the differences from the last spanning tr ee generated

ALGO Gen(T)

b₁

b₂

ALGO Compute-back-to-nonback(f,T)

• The sets NB and B are updated at every exchan ge

– transferring edges from B to NB – Removal of edges from B and NB

• Data structure for B

– For each node v of G

• B[v]: a list of nontree back edges in B having head vertex v

• A[v][p]: each element points to the first edge in its list which i s incident upon a proper ancestor of node p

• BASE[v]: initialized to be v

ALGO Compute-back-to-nonback(f,T)

(cont’d)

An Algorithm for enumerating All Sp anning Trees of a Directed Graph

--- S. Kapoor and H. Ramesh

Speaker: 莊秋芸 ³

PROPERTY 4

u

v f

b lca(u,v)

If f =(u, v) is an edge in NB and b=lca (u,v) in SD_a,

then no edge in NB has its head in the subtree of SD_a rooted at v and its tail on the path from b to u

(b excluded).

• DFS

• The order of the selection of the exchange edge from NB.

• All “exchangeable edges”

must connect a vertex with higher postorder number to a vertex with lower postord er number.

PROPERTY 4 (some observatio ns)

u

v f

b lca(u,v)

• By induction on the level of x, w here x is a node on the comput ation tree.

1. Base case: root node.

2. Induction hypothesis: assume t his is true for any node x of the computation tree.

3. Consider the left and right sons b₁, b₂ of x.

1) It’s trivially true for the right s on b₂.

PROPERTY 4 (proof)

5

3 4

1 2

A DFS tree with postorder number on each node.

4

2 3

2) For the left son b_1.

PROPERTY 4 (proof)

9

5

8

1 4 f

e 7

6

2 3

9

5

8

1 f

7

6 Exchange

e and f

5.1 5.2 5.3

PROPERTY 4 (conclusion)

u

v f

b lca(u,v)

Since no such edges exist, there’s no

nonback edges will

become back edges.

LEMMA 4.1

• During the construction of the computation tree the following changes to NB and B suf fice after an exchange at a node a in the c omputation tree:

a) changes from B to NB by Property 3.

b) deletions from NB according to IN and OUT definitions.

c) removal of RB_a from B.

PROPERTY 5

• If an exchange (e, f), f=(u,v) is made at a nod e x of the computation tree, then the subtree of SD_x rooted at v is preserved as such in each of the trees generated at descendant nodes of x i n the computation tree. So at node x, any edg e in B incident upon a vertex in that subtree, is redundant for future computations at descenda nt nodes of x and may be removed from B.

PROPERTY 5

v f

e

u

A redundant back edge

computation tree

x

All have the same subtree rooted at v

SD_x

An Algorithm for enumerating All Sp anning Trees of a Directed Graph

--- S. Kapoor and H. Ramesh

Speaker: 黃稚穎 ⁴

LEMMA 4.4

• ALGO Main outputs the changes corresponding to the compressed computation tree CD'(G,r) in O(N(r)V+V²) operations.

The time for manipulating the data structures NB, B, and STACK is O(V*(NBC_x+1))

The total time required by ALGO Gen minus the output operations equals O(Σ(V*NBC_x))

At any node x in the compressed computation tree, the above summation gives an O(N(r)V) time bound.

Total output operations are O(N(r)).

Computing B and NB initially require O(V²) time.

DFS requires O(V+E)timeTotal requires O(N(r)V+V. ²+V+E+N(r))

= O(N(r)V+V²) time

.

LEMMA 4.5

• ALGO Main requires O(V

) space.

– Follows from the size of the data structures in volved.

B requires O(V²)-space.

NB requires O(V+E)-space.

Changes stored on STACK is O(V+NBC_x)-space.

The total stack space required is O(V

+ΣNBC

)

= O(V

)

Complexity

• From LEMMA 4.4 and LEMMA 4.5 the following result fol lows if the above procedure is repeated with each vertex in turn as root.

THEOREM 4.6. All rooted directed spanning trees can b e output in O(NV+V³) operations and O(V²) space.

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