Gomory–Hu tree

In combinatorial optimization, the Gomory–Hu tree^[1] of an undirected graph with capacities is a weighted tree that represents the minimum s-t cuts for all s-t pairs in the graph. The Gomory–Hu tree can be constructed in $| V | - 1$ maximum flow computations. It is named for Ralph E. Gomory and T. C. Hu.

Definition edit

Let $G=(V_{G},E_{G},c)$ be an undirected graph with $c(u,v)$ being the capacity of the edge $(u,v)$ respectively.

Denote the minimum capacity of an

s

-

t

cut by

\lambda _{st}

for each

s,t\in V_{G}

.

Let

T=(V_{G},E_{T})

be a tree, and denote the set of edges in an

s

-

t

path by

P_{st}

for each

s,t\in V_{G}

.

Then $T$ is said to be a Gomory–Hu tree of $G$ , if for each $s,t\in V_{G}$

\lambda _{st}=\min _{e\in P_{st}}c(S_{e},T_{e}),

where

$S_{e},T_{e}\subseteq V_{G}$ are the two connected components of $T\setminus \{e\}$ , and thus $(S_{e},T_{e})$ forms an $s$ - $t$ cut in $G$ .
$c(S_{e},T_{e})$ is the capacity of the $(S_{e},T_{e})$ cut in $G$ .

Algorithm edit

Gomory–Hu Algorithm

Input: A weighted undirected graph

G=((V_{G},E_{G}),c)

Output: A Gomory–Hu Tree

T=(V_{T},E_{T}).

Set $V_{T}=\{V_{G}\},\ E_{T}=\emptyset .$
Choose some $X\in V_{T}$ with $| X | \geq 2$ if such $X$ exists. Otherwise, go to step 6.
For each connected component $C=(V_{C},E_{C})\in T\setminus X,$ let ${\textstyle S_{C}=\bigcup _{v_{T}\in V_{C}}v_{T}.}$
Let $S=\{S_{C}\mid C{\text{ is a connected component in }}T\setminus X\}.$

Contract the components to form $G'=((V_{G'},E_{G'}),c'),$ where: ${\begin{aligned}V_{G'}&=X\cup S\\[2pt]E_{G'}&=E_{G}|_{X\times X}\cup \{(u,S_{C})\in X\times S\mid (u,v)\in E_{G}{\text{ for some }}v\in S_{C}\}\\[2pt]&\qquad \qquad \quad \!\cup \{(S_{C1},S_{C2})\in S\times S\mid (u,v)\in E_{G}{\text{ for some }}u\in S_{C1}{\text{ and }}v\in S_{C2}\}\end{aligned}}$
$c':V_{G'}\times V_{G'}\to R^{+}$ is the capacity function, defined as: ${\begin{aligned}&{\text{if }}\ (u,S_{C})\in E_{G}|_{X\times S}:&&c'(u,S_{C})=\!\!\!\sum _{\begin{smallmatrix}v\in S_{C}:\\(u,v)\in E_{G}\end{smallmatrix}}\!\!\!c(u,v)\\[4pt]&{\text{if }}\ (S_{C1},S_{C2})\in E_{G}|_{S\times S}:&&c'(S_{C1},S_{C2})=\!\!\!\!\!\!\!\sum _{\begin{smallmatrix}(u,v)\in E_{G}:\\u\in S_{C1}\,\land \,v\in S_{C2}\end{smallmatrix}}\!\!\!\!\!c(u,v)\\[4pt]&{\text{otherwise}}:&&c'(u,v)=c(u,v)\end{aligned}}$
Choose two vertices $s, t \in X$ and find a minimum $s - t$ cut $(A', B')$ in $G'$ .
Set $A={\Biggl (}\bigcup _{S_{C}\in A'\cap S}\!\!\!S_{C}\!{\Biggr )}\cup (A'\cap X),\$ $B={\Biggl (}\bigcup _{S_{C}\in B'\cap S}\!\!\!S_{C}\!{\Biggr )}\cup (B'\cap X).$
Set $V_{T}=(V_{T}\setminus X)\cup \{A\cap X,B\cap X\}.$
For each $e=(X,Y)\in E_{T}$ do:
1. Set $e'=(A\cap X,Y)$ if $Y\subset A,$ otherwise set $e'=(B\cap X,Y).$
2. Set $E_{T}=(E_{T}\setminus \{e\})\cup \{e'\}.$
3. Set $w(e')=w(e).$
Set $E_{T}=E_{T}\cup \{(A\cap X,\ B\cap X)\}.$

Set $w((A\cap X,B\cap X))=c'(A',B').$

Go to step 2.
Replace each $\{v\}\in V_{T}$ by $v$ and each $(\{u\},\{v\})\in E_{T}$ by $(u, v)$ . Output $T$ .

Analysis edit

Using the submodular property of the capacity function $c$ , one has

c(X)+c(Y)\geq c(X\cap Y)+c(X\cup Y).

Then it can be shown that the minimum

s - t

cut in

G'

is also a minimum

s - t

cut in

G

for any

s, t \in X

.

To show that for all $(P,Q)\in E_{T},$ $w(P,Q)=\lambda _{pq}$ for some $p \in P$ , $q \in Q$ throughout the algorithm, one makes use of the following Lemma,

For any

i, j, k

in

V G

,

\lambda _{ik}\geq \min(\lambda _{ij},\lambda _{jk}).

The Lemma can be used again repeatedly to show that the output $T$ satisfies the properties of a Gomory–Hu Tree.

Example edit

The following is a simulation of the Gomory–Hu's algorithm, where

green circles are vertices of T.
red and blue circles are the vertices in G'.
grey vertices are the chosen s and t.
red and blue coloring represents the s-t cut.
dashed edges are the s-t cut-set.
A is the set of vertices circled in red and B is the set of vertices circled in blue.

	G'	T

	1. Set V_T = {V_G} = { {0, 1, 2, 3, 4, 5} } and E_T = ∅. 2. Since V_T has only one vertex, choose X = V_G = {0, 1, 2, 3, 4, 5}. Note that \| X \| = 6 ≥ 2.
1.
	3. Since T\X = ∅, there is no contraction and therefore G' = G. 4. Choose s = 1 and t = 5. The minimum s-t cut (A', B') is ({0, 1, 2, 4}, {3, 5}) with c'(A', B') = 6. Set A = {0, 1, 2, 4} and B = {3, 5}. 5. Set V_T = (V_T\X) ∪ {A ∩ X, B ∩ X} = { {0, 1, 2, 4}, {3, 5} }. Set E_T = { ({0, 1, 2, 4}, {3, 5}) }. Set w( ({0, 1, 2, 4}, {3, 5}) ) = c'(A', B') = 6. Go to step 2. 2. Choose X = {3, 5}. Note that \| X \| = 2 ≥ 2.
2.
	3. {0, 1, 2, 4} is the connected component in T\X and thus S = { {0, 1, 2, 4} }. Contract {0, 1, 2, 4} to form G', where c'( (3, {0, 1, 2 ,4}) ) = c( (3, 1) ) + c( (3, 4) ) = 4. c'( (5, {0, 1, 2, 4}) ) = c( (5, 4) ) = 2. c'( (3, 5)) = c( (3, 5) ) = 6. 4. Choose s = 3, t = 5. The minimum s-t cut (A', B') in G' is ( {{0, 1, 2, 4}, 3}, {5} ) with c'(A', B') = 8. Set A = {0, 1, 2, 3, 4} and B = {5}. 5. Set V_T = (V_T\X) ∪ {A ∩ X, B ∩ X} = { {0, 1, 2, 4}, {3}, {5} }. Since (X, {0, 1, 2, 4}) ∈ E_T and {0, 1, 2, 4} ⊂ A, replace it with (A ∩ X, Y) = ({3}, {0, 1, 2 ,4}). Set E_T = { ({3}, {0, 1, 2 ,4}), ({3}, {5}) } with w(({3}, {0, 1, 2 ,4})) = w((X, {0, 1, 2, 4})) = 6. w(({3}, {5})) = c'(A', B') = 8. Go to step 2. 2. Choose X = {0, 1, 2, 4}. Note that \| X \| = 4 ≥ 2.
3.
	3. { {3}, {5} } is the connected component in T\X and thus S = { {3, 5} }. Contract {3, 5} to form G', where c'( (1, {3, 5}) ) = c( (1, 3) ) = 3. c'( (4, {3, 5}) ) = c( (4, 3) ) + c( (4, 5) ) = 3. c'(u,v) = c(u,v) for all u,v ∈ X. 4. Choose s = 1, t = 2. The minimum s-t cut (A', B') in G' is ( {1, {3, 5}, 4}, {0, 2} ) with c'(A', B') = 6. Set A = {1, 3, 4, 5} and B = {0, 2}. 5. Set V_T = (V_T\X) ∪ {A ∩ X, B ∩ X} = { {3}, {5}, {1, 4}, {0, 2} }. Since (X, {3}) ∈ E_T and {3} ⊂ A, replace it with (A ∩ X, Y) = ({1, 4}, {3}). Set E_T = { ({1, 4}, {3}), ({3}, {5}), ({0, 2}, {1, 4}) } with w(({1, 4}, {3})) = w((X, {3})) = 6. w(({0, 2}, {1, 4})) = c'(A', B') = 6. Go to step 2. 2. Choose X = {1, 4}. Note that \| X \| = 2 ≥ 2.
4.
	3. { {3}, {5} }, { {0, 2} } are the connected components in T\X and thus S = { {0, 2}, {3, 5} } Contract {0, 2} and {3, 5} to form G', where c'( (1, {3, 5}) ) = c( (1, 3) ) = 3. c'( (4, {3, 5}) ) = c( (4, 3) ) + c( (4, 5) ) = 3. c'( (1, {0, 2}) ) = c( (1, 0) ) + c( (1, 2) ) = 2. c'( (4, {0, 2}) ) = c( (4, 2) ) = 4. c'(u,v) = c(u,v) for all u,v ∈ X. 4. Choose s = 1, t = 4. The minimum s-t cut (A', B') in G' is ( {1, {3, 5}}, {{0, 2}, 4} ) with c'(A', B') = 7. Set A = {1, 3, 5} and B = {0, 2, 4}. 5. Set V_T = (V_T\X) ∪ {A ∩ X, B ∩ X} = { {3}, {5}, {0, 2}, {1}, {4} }. Since (X, {3}) ∈ E_T and {3} ⊂ A, replace it with (A ∩ X, Y) = ({1}, {3}). Since (X, {0, 2}) ∈ E_T and {0, 2} ⊂ B, replace it with (B ∩ X, Y) = ({4}, {0, 2}). Set E_T = { ({1}, {3}), ({3}, {5}), ({4}, {0, 2}), ({1}, {4}) } with w(({1}, {3})) = w((X, {3})) = 6. w(({4}, {0, 2})) = w((X, {0, 2})) = 6. w(({1}, {4})) = c'(A', B') = 7. Go to step 2. 2. Choose X = {0, 2}. Note that \| X \| = 2 ≥ 2.
5.
	3. { {1}, {3}, {4}, {5} } is the connected component in T\X and thus S = { {1, 3, 4, 5} }. Contract {1, 3, 4, 5} to form G', where c'( (0, {1, 3, 4, 5}) ) = c( (0, 1) ) = 1. c'( (2, {1, 3, 4, 5}) ) = c( (2, 1) ) + c( (2, 4) ) = 5. c'( (0, 2) ) = c( (0, 2) ) = 7. 4. Choose s = 0, t = 2. The minimum s-t cut (A', B') in G' is ( {0}, {2, {1, 3, 4, 5}} ) with c'(A', B') = 8. Set A = {0} and B = {1, 2, 3 ,4 ,5}. 5. Set V_T = (V_T\X) ∪ {A ∩ X, B ∩ X} = { {3}, {5}, {1}, {4}, {0}, {2} }. Since (X, {4}) ∈ E_T and {4} ⊂ B, replace it with (B ∩ X, Y) = ({2}, {4}). Set E_T = { ({1}, {3}), ({3}, {5}), ({2}, {4}), ({1}, {4}), ({0}, {2}) } with w(({2}, {4})) = w((X, {4})) = 6. w(({0}, {2})) = c'(A', B') = 8. Go to step 2. 2. There does not exist X∈V_T with \| X \| ≥ 2. Hence, go to step 6.
6.
	6. Replace V_T = { {3}, {5}, {1}, {4}, {0}, {2} } by {3, 5, 1, 4, 0, 2}. Replace E_T = { ({1}, {3}), ({3}, {5}), ({2}, {4}), ({1}, {4}), ({0}, {2}) } by { (1, 3), (3, 5), (2, 4), (1, 4), (0, 2) }. Output T. Note that exactly \| V \| − 1 = 6 − 1 = 5 times min-cut computation is performed.

Implementations: Sequential and Parallel edit

Gusfield's algorithm can be used to find a Gomory–Hu tree without any vertex contraction in the same running time-complexity, which simplifies the implementation of constructing a Gomory–Hu Tree.^[2]

Andrew V. Goldberg and K. Tsioutsiouliklis implemented the Gomory-Hu algorithm and Gusfield algorithm, and performed an experimental evaluation and comparison.^[3]

Cohen et al. report results on two parallel implementations of Gusfield's algorithm using OpenMP and MPI, respectively.^[4]

Related concepts edit

In planar graphs, the Gomory–Hu tree is dual to the minimum weight cycle basis, in the sense that the cuts of the Gomory–Hu tree are dual to a collection of cycles in the dual graph that form a minimum-weight cycle basis.^[5]

References edit

^ Gomory, R. E.; Hu, T. C. (1961). "Multi-terminal network flows". Journal of the Society for Industrial and Applied Mathematics. 9 (4): 551–570. doi:10.1137/0109047.
^ Gusfield, Dan (1990). "Very Simple Methods for All Pairs Network Flow Analysis". SIAM J. Comput. 19 (1): 143–155. doi:10.1137/0219009.
^ Goldberg, A. V.; Tsioutsiouliklis, K. (2001). "Cut Tree Algorithms: An Experimental Study". Journal of Algorithms. 38 (1): 51–83. doi:10.1006/jagm.2000.1136.
^ Cohen, Jaime; Rodrigues, Luiz A.; Silva, Fabiano; Carmo, Renato; Guedes, André Luiz Pires; Duarte, Jr., Elias P. (2011). "Parallel implementations of Gusfield's cut tree algorithm". In Xiang, Yang; Cuzzocrea, Alfredo; Hobbs, Michael; Zhou, Wanlei (eds.). Algorithms and Architectures for Parallel Processing – 11th International Conference, ICA3PP, Melbourne, Australia, October 24–26, 2011, Proceedings, Part I. Lecture Notes in Computer Science. Vol. 7016. Springer. pp. 258–269. doi:10.1007/978-3-642-24650-0_22.{{cite conference}}: CS1 maint: multiple names: authors list (link)
^ Hartvigsen, D.; Mardon, R. (1994). "The all-pairs min cut problem and the minimum cycle basis problem on planar graphs". SIAM J. Discrete Math. 7 (3): 403–418. doi:10.1137/S0895480190177042..

B. H. Korte, Jens Vygen (2008). "8.6 Gomory–Hu Trees". Combinatorial Optimization: Theory and Algorithms (Algorithms and Combinatorics, 21). Springer Berlin Heidelberg. pp. 180–186. ISBN 978-3-540-71844-4.

[1] Gomory, R. E.; Hu, T. C. (1961). "Multi-terminal network flows". Journal of the Society for Industrial and Applied Mathematics. 9 (4): 551–570. doi:10.1137/0109047.

[2] Gusfield, Dan (1990). "Very Simple Methods for All Pairs Network Flow Analysis". SIAM J. Comput. 19 (1): 143–155. doi:10.1137/0219009.

[3] Goldberg, A. V.; Tsioutsiouliklis, K. (2001). "Cut Tree Algorithms: An Experimental Study". Journal of Algorithms. 38 (1): 51–83. doi:10.1006/jagm.2000.1136.

[4] Cohen, Jaime; Rodrigues, Luiz A.; Silva, Fabiano; Carmo, Renato; Guedes, André Luiz Pires; Duarte, Jr., Elias P. (2011). "Parallel implementations of Gusfield's cut tree algorithm". In Xiang, Yang; Cuzzocrea, Alfredo; Hobbs, Michael; Zhou, Wanlei (eds.). Algorithms and Architectures for Parallel Processing – 11th International Conference, ICA3PP, Melbourne, Australia, October 24–26, 2011, Proceedings, Part I. Lecture Notes in Computer Science. Vol. 7016. Springer. pp. 258–269. doi:10.1007/978-3-642-24650-0_22.{{cite conference}}: CS1 maint: multiple names: authors list (link)

[5] Hartvigsen, D.; Mardon, R. (1994). "The all-pairs min cut problem and the minimum cycle basis problem on planar graphs". SIAM J. Discrete Math. 7 (3): 403–418. doi:10.1137/S0895480190177042..

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