Grammar-based Compression for Directed and Undirected Generalized

Series-parallel Graphs using Integer Linear Programming

Morihiro Hayashida

, Hitoshi Koyano

and Tatsuya Akutsu

National Institute of Technology, Matsue College, 14-4, Nishiikumacho, Matsue, Shimane 690–8518, Japan

Quantitative Biology Center, Riken, 2-2-3, Minatojima-minamimachi, Chuo-ku, Kobe, Hyogo 650–0047, Japan

Institute for Chemical Research, Kyoto University, Gokasho, Uji, Kyoto 611–0011, Japan

Keywords:

Generalized Series-parallel Graph, Grammar-based Compression, Integer Linear Programming.

Abstract:

We address a problem of ﬁnding generation rules from biological data, especially, represented as directed and

undirected generalized series-parallel graphs (GSPGs), which include trees, outerplanar graphs, and series-

parallel graphs. In the previous study, grammars for edge-labeled rooted ordered and unordered trees, called

SEOTG and SEUTG, respectively, were deﬁned, and it was examined to extract generation rules from glycans

and RNAs that can be represented by rooted tree structures, where integer linear programming-based methods

for ﬁnding the minimum SEOTG and SEUTG that produce only given trees were developed. In nature and or-

ganisms, however, there are various kinds of structures such as gene regulatory networks, metabolic pathways,

and chemical structures that cannot be represented as rooted trees. In this study, we relax the limitation of

structures to be compressed, and propose grammars representing edge-labeled directed and undirected GSPGs

based on context-free grammars by extending SEOTG and SEUTG. In addition, we propose an integer linear

programming-based method for ﬁnding the minimum GSPG grammar in order to analyze more complicated

biological networks and structures.

1 INTRODUCTION

Data compression for a structure is related with the

amount of information that it contains. The amount of

information would be large if the size of compressed

data is still large. Otherwise, the data include redun-

dant data, and the amount of information is small. Our

purpose is to extract useful information and knowl-

edge from data through compression. In particular,

we focus on biological structured data constructed in

nature. Such structures could be often explained by

several simple generation rules.

In previous studies, biological data represented by

rooted trees such as glycans and RNAs were com-

pressed and analyzed (Zhao et al., 2010; Zhao et al.,

2015). It is known that glycans are composed of mul-

tiple monosaccharides bound by glycosidic bonds,

take various structures in accordance with biosyn-

thetic reactions, and the function of a glycan depends

on its structure. Hence, it is important to analyze the

glycan structures, and to extract rules of the biosyn-

theses. They developed integer linear programming-

based methods, called the minimum SEOTG and

SEUTG, for ﬁnding the minimum grammar that pro-

duces only given single ordered and unordered rooted

trees, and applied them to biological data such as gly-

cans with up to 36 nodes and 5 distinct labels, where

these methods are based on a kind of tree grammar,

the simple elementary ordered (unordered) tree gram-

mar (SEO(U)TG) (Akutsu, 2010). Furthermore, they

extended the methods to multiple rooted trees be-

cause generation rules are commonly utilised among

these multiple trees. It, however, is considered that

structures generated in nature cannot be always rep-

resented by rooted trees. In this paper, we extend

their grammar to directed and undirected generalized

series-parallel graphs (GSPGs), which include trees

and outerplanar graphs. In addition, we propose an

integer linear programming-based method for ﬁnding

the minimum GSPG grammar that produces only a

given generalized series-parallel graph.

A series-parallel graph is deﬁned by two proce-

dures, called series and parallel compositions, and

two special nodes in the graph are labeled with source

and sink as terminal nodes (Eppstein, 1992; Eikel

et al., 2015). A generalized series-parallel graph is

deﬁned by the addition of another series-type compo-

sition, called generalized series composition, where

Hayashida, M., Koyano, H. and Akutsu, T.

Grammar-based Compression for Directed and Undirected Generalized Series-parallel Graphs using Integer Linear Programming.

DOI: 10.5220/0006583001050111

In Proceedings of the 11th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2018) - Volume 3: BIOINFORMATICS, pages 105-111

ISBN: 978-989-758-280-6

105

the shared node between two composed graphs is la-

beled with a terminal node (Korneyenko, 1994). Ho

et al. proposed a decomposition method for GSPGs

using many processors in parallel (Ho et al., 1999).

However, it is not guaranteed that their method al-

ways ﬁnds the minimum decomposition tree. It has

been proved that the problem of ﬁnding the mini-

mum SEO(U)TG for a given rooted tree is NP-hard

(Akutsu, 2010). Hence, the problem of ﬁnding the

minimum grammar for a given GSPG is also NP-hard,

and it means that there does not exist any polynomial

time algorithm for ﬁnding the minimum GSPG gram-

mar.

Since production rules of a SEO(U)TG can be re-

garded as two types of series compositions in GSPGs,

we deﬁne a grammar by adding a production rule

corresponding to the parallel composition to their

grammar, and developan integer linear programming-

based method for ﬁnding the minimum GSPG gram-

mar.

2 METHOD

We brieﬂy review the simple elementary ordered (un-

ordered) tree grammar (SEO(U)TG) and the integer

linear programming-based methods for ﬁnding the

minimum SEOTG and SEUTG, and propose gram-

mars for edge-labeled directed and undirected gen-

eralized series-parallel graphs (GSPGs) and an inte-

ger linear programming-based method for ﬁnding the

minimum GSPG grammar.

2.1 SEOTG and SEUTG

SEOTG and SEUTG are context-free grammar

(CFG)-like grammars for edge-labeled ordered and

unordered rooted trees, respectively. In CFG,

a nonterminal symbol is replaced with several

(non)terminal symbols (Hopcroft et al., 2001). In

SEO(U)TG, an edge having a nonterminal symbol is

replaced with one or two edges having (non)terminal

symbols. SEOTG and SEUTG are deﬁned as follows.

Deﬁnition 1 (Simple Elementary Ordered Tree Gram-

mar (SEOTG)). A SEOTG is deﬁned as 4-tuple (Σ, Γ,

S, ∆), where Σ is a set of terminal symbols, Γ is a set

of nonterminal symbols, S is a start nonterminal sym-

bol in Γ, and ∆ is a set of production rules (R1), (R1t),

(R2), (R2t), (R3), (R3r), (R3l) as shown in Fig. 1.

Deﬁnition 2 (Simple Elementary Unordered Tree

Grammar (SEUTG)). A SEUTG is deﬁned as 4-tuple

(Σ, Γ, S, ∆) where Σ is a set of terminal symbols, Γ is

a set of nonterminal symbols, S is a start nonterminal

(R1) (R1t)

(R2)

(R3)

(R3r)

(R3l)

(R2t)

Figure 1: Three main types (R1), (R2), (R3) of production

rules of SEOTG for rooted ordered trees. A black circle

denotes a tag.

(a)

(b)

Figure 2: Example of a SEOTG with

({a,b,c},{S,A, B,C, D, E}, S,∆) and the tree gener-

ated by this grammar. (a) Production rules of ∆. (b) The

derivation of the generated tree.

symbol in Γ, and ∆ is a set of production rules (R1),

(R1t), (R2), (R2t), (R3), (R3r).

It is noted that (R3r) becomes equivalent to (R3l)

because the edge order is ignored.

These production rules do not construct any cycle

but trees. A tree generated from a nonterminal sym-

bol by SEOTG and SEUTG has at most two special

nodes, its root and a tag node, where a tag means a

terminal node to which another tree structure can be

attached.

BIOINFORMATICS 2018 - 9th International Conference on Bioinformatics Models, Methods and Algorithms

106

Fig. 2 shows an example of a SEOTG with

({a,b,c},{S,A, B,C, D,E},S,∆), and the tree gener-

ated by the grammar, where ∆ is shown in Fig. 2(a).

The generation starts from S, production rules are

applied to edges with nonterminal symbols, and the

tree with only terminal symbols is generated (see Fig.

2(b)).

2.2 The Minimum SEOTG and SEUTG

We can obtain a clue of generation mechanisms of bi-

ological structures by ﬁnding the minimum grammar.

For a rooted ordered tree T, the following integer lin-

ear programming problem was formulated for ﬁnding

the minimum SEOTG that produces only the tree T.

Minimize

∑

u∈U

Subject to

i,ε, j, j

= 1 for all i, j ∈ ch(i) (|ch( j)| = 0),

i, j, j, j

= 1 for all i, j ∈ ch(i) (|ch( j)| > 0),

1,ε,lch(1),rch(1)

= 1,

i,ε,h,k

≤

k−1

∑

l=h

i,ε,h,l,k

∑

t∈I(T

i,ε,h,k

)

i,ε,h,k,t

for all i, h ≤ k ∈ ch(i),

i,ε,h,l,k

≤

i,ε,h,l

+ x

i,ε,l+1,k

)

for all i, h ≤ l < k ∈ ch(i),

i,ε,h,k,t

≤

i,t,h,k

+ x

t,ε,lch(t) ,rch(t)

)

for all i, h ≤ k ∈ ch(i),t ∈ I(T

i,ε,h,k

i, j,h,k

≤

k−1

∑

l=h

i, j,h,l,k

∑

t∈anc( j)

i, j,h,k,t

for all i, h ≤ k ∈ ch(i), j ∈ I(T

i,ε,h,k

i, j,h,l,k

≤

i,ε,h,l

+ x

i, j,l+1,k

)

for all i, h ≤ l < k ∈ ch(i), j ∈ I(T

i,ε,l+1,k

i, j,h,l,k

≤

i, j,h,l

+ x

i,ε,l+1,k

)

for all i, h ≤ l < k ∈ ch(i), j ∈ I(T

i,ε,h,l

i, j,h,k,t

≤

i,t,h,k

+ x

t, j,lch(t),rch(t)

)

for all i, h ≤ k ∈ ch(i), j ∈ I(T

i,ε,h,k

), t ∈ anc( j),

≥

|S(u)|

∑

i, j,h,k

∈S(u)

i, j,h,k

for all u ∈ U,

< 1 +

|S(u)|

∑

i, j,h,k

∈S(u)

i, j,h,k

for all u ∈ U,

i, j,h,k

i, j,h,l,k

i, j,h,k,t

, p

∈ {0,1},

where lch(i), rch(i), and ch(i) denote the leftmost

child of the node v

in T, the rightmost child of v

and the set of child nodes of v

, respectively. T

i,t,h,k

denotes the subtree rooted at v

, with the child nodes

(h ≤ j ≤ k) and v

labeled with a tag in T, which

does not have a tag when t = ε. I(T) denotes the set

of internal nodes, except for the root and leaves of

tree T. anc( j) denotes the set of ancestor nodes of v

where j /∈ anc( j) and anc(ε) =

Each variable of x

i, j,h,k

i, j,h,l,k

i, j,h,k,t

takes ei-

ther 0 or 1. x

i, j,h,k

= 1 if T

i, j,h,k

is generated by the

grammar, x

i, j,h,k

= 0 otherwise. y

i, j,h,l,k

= 1 if both of

i, j,h,l

and T

i, j,l+1,k

are generated, y

i, j,h,l,k

= 0 other-

wise. z

i, j,h,k,t

= 1 if both of T

i,t,h,k

and T

i, j,lch(t),rch(t)

are generated, z

i, j,h,k,t

= 0 otherwise.

In this formulation, the Euler string es(T) is used

to determine if two edge-labeled rooted trees T

and

are isomorphic to each other, where es(T) for a tree

T is deﬁned by the sequence of edge labels l and its

opposite

l, along the depth-ﬁrst search traversal of T

(Akutsu, 2010). It is noted that for two edge-labeled

rooted trees T

and T

, T

is isomorphic to T

if (and

only if) es(T

) = es(T

). U denotes the set of all Euler

strings for all connected subtrees of T. S(u) denotes

the set of all subtrees T

i, j,h,k

of T such that es(T

i, j,h,k

)

is equivalent to u. Then, p

= 1 means that the mini-

mum grammar generates the subtree corresponding to

u, and

∑

u∈U

represents the number of nonterminal

symbols.

Similarly to the minimum SEOTG, the minimum

SEUTG was formulated.

2.3 Directed and Undirected

Generalized Series-parallel Graph

Grammars (GSPGGs)

Let G(V,E) be an undirected GSPG with a set V of

nodes and a set E of edges labeled with l(e) for e ∈ E.

For example, Fig. 4(a) shows the benzene ring, which

is regarded as an undirected GSPG with six nodes and

six edges, and is constructed by several series compo-

sitions after one parallel composition.

We deﬁne an undirected generalized series-

parallel graph grammar (GSPGG) as follows.

Deﬁnition 3 (Undirected generalized series-parallel

graph grammar). An undirected GSPGG is deﬁned as

4-tuple (Σ, Γ, S, ∆), where Σ and Γ are sets of nonter-

minal and terminal symbols, every terminal symbol is

an undirected labeled edge, S is a start nonterminal

symbol, and ∆ is a set of production rules as shown in

Fig. 3.

In Fig. 3, a head and a tail of each arrow denote

two terminal nodes of its edge. If the graph with only

terminal symbols generated from a nonterminal sym-

bol is symmetric, then the source and sink nodes can

be changed to each other. White and black squares

mean that in a production rule, the node with a white

(black) square in the left-hand side corresponds to the

node with a white (black) square in the right-hand

Grammar-based Compression for Directed and Undirected Generalized Series-parallel Graphs using Integer Linear Programming

107

(R1)

(R2a)

(R3a)

(R4a)

(R2b)

(R2c)

(R2d)

(R3b)

(R3c)

(R3d)

(R4b)

(R4c)

(R4d)

Figure 3: Four main types of production rules of undirected GSPGGs. A head and a tail of each arrow denote two terminal

nodes of its edge. White and black squares mean that the node with a white (black) square in the left-hand side corresponds

to the node with a white (black) square in the right-hand side.

C C

a a

(a)

(c)

(b)

Figure 4: Example of an undirected generalized series-

parallel graph and its grammar. (a) The benzene ring. (b)

An undirected GSPGG of the benzene ring. (c) The deriva-

tion of the benzene ring using the grammar, where terminal

symbol ’a’ denotes a bond with order 1.5 of the benzene

ring.

side. In the production rule (R4a-d), an edge between

the source and sink nodes is replaced with two edges,

and a cycle is generated.

Fig. 4(b) shows an example of an undirected

GSPGG that produces the benzene ring (Fig. 4(a)),

where each bond in the benzene ring is represented as

an edge with label ’a’ because six bonds are equiv-

alent to each other. Fig. 4(c) shows the derivation

of the benzene ring using the undirected GSPGG.

The start symbol ’S’ is replaced with two nontermi-

nal symbols ’A’ making a cycle. ’A’ is replaced with

’B’ and ’C’. ’B’ is replaced with two ’C’s. ’C’ is

replaced with ’a’. Then, the number of production

rules is equal to the number of nonterminal symbols,

|Σ| = 4 (Σ = {S, A,B,C}). For ﬁnding the minimum

GSPGG, it is enough to ﬁnd GSPGGs with the mini-

mum number of nonterminal symbols.

(R1a)

(R1b)

Figure 5: Production rules of replacement of a nontermi-

nal symbol with a terminal symbol in directed GSPGGs. In

each production rule, the arrow in the right-hand side de-

notes a directed edge.

Similarly to the deﬁnition of undirected GSPGGs,

we deﬁne a directed GSPGG as follows.

Deﬁnition 4 (Directed generalized series-parallel

graph grammar). A directed GSPGG is deﬁned as 4-

tuple (Σ, Γ, S, ∆), where Σ and Γ are sets of nonter-

minal and terminal symbols, every terminal symbol is

a directed labeled edge, S is a start nonterminal sym-

bol, ∆ is a set of the same types of production rules of

undirected GSPGGs except (R1), and (R1) is replaced

with (R1a-b) as shown in Fig. 5.

Fig. 6 shows an example of a directed GSPG and

its grammar that produces only the graph, where the

chemical structure of the purine (Fig. 6(a)) is trans-

formed to a directed graph as shown in Fig. 6(b).

If it is transformed to an undirected graph, two end-

points of a terminal symbol cannot be distinguished,

and atom types are not determined in the graph pro-

duced by an undirected GSPGG.

2.4 The Minimum Directed and

Undirected GSPGGs

Let G(V,E) be a directed (undirected) GSPG with a

set V of nodes and a set E of labeled edges. To con-

sider all combinations of compositions of subgraphs

BIOINFORMATICS 2018 - 9th International Conference on Bioinformatics Models, Methods and Algorithms

108

(c)

(a) (b)

Figure 6: Example of a directed generalized series-parallel

graph and its directed GSPGG. (a) The purine. (b) A trans-

formed directed graph. (c) A directed GSPGG that gener-

ates the graph (b).

of G, we repeatedly partition subgraphs into two con-

nected components until only edges remain. A GSPG

has two terminal nodes, whereas G does not have any

terminal node. Hence, we require that a partitioned

subgraph has at most two terminal nodes. Suppose

that G

i,S, j,T

represents a connected subgraph with ter-

minal nodes i and j, where S and T are subsets of ad-

jacent nodes of i and j, respectively. If a partitioned

subgraph has one terminal node, we represent the sub-

graph as G

i,S

, G

i,S,ε,

, or G

ε,

0,i,S

. G is also represented

as G

ε,

0,ε,

. If G

i,S, j,T

is isomorphic to G

′

except

node j, and is generated by a production rule, then

′

is also generated by the same production rule.

A subgraph with at most two terminal nodes can

be partitioned into two subgraphs with one or two ter-

minal nodes.

Fig. 7 shows an example of an undirected GSPG

3 3

(a)

(b)

(c)

(d)

(e)

4,{3}

3,{1,2}

3,{4}

1,{2},3,{2}

1,{3},3,{1,4}

4,{5}

3,{4},4,{3}

1,{3},3,{1}

Figure 7: Example of an undirected GSPG and its partition-

ing. (a) An example graph G with ﬁve nodes. (b) The parti-

tioned graphs G

4,{3}

and G

4,{5}

at node 4 of G. (c) The par-

titioned graphs G

3,{1,2}

and G

3,{4},4,{3}

at node 3 of G

4,{3}

(d) The partitioned graphs G

1,{2},3,{2}

and G

1,{3},3,{1,4}

nodes 1 and 3 of G. (e) The partitioned graphs G

1,{3},3,{1}

and G

3,{4}

at node 3 of G

1,{3},3,{1,4}

. A black circle denotes

a terminal node.

G with ﬁve nodes and its partitioning. G is partitioned

into two subgraphs G

4,{3}

and G

4,{5}

at node 4 that

does not belong to any cycle as shown in Fig. 7(b).

If a node to be partitioned does not belong to any cy-

cle, only the node can be a new terminal node. Then,

production rules of (R2) and (R3) can be constructed,

and G is generated from G

4,{3}

and G

4,{5}

by series

compositions. In Fig. 7(c), we cannot partition G

4,{3}

at node 1 or 2 because two connected components are

not generated. G

4,{3}

is partitioned into G

3,{1,2}

and

3,{4},4,{3}

at node 3. Then, production rules of (R2)

and (R3) can be constructed, and G

4,{3}

is generated

from G

3,{1,2}

and G

3,{4},4,{3}

by series compositions.

On the other hand, if a node to be partitioned belongs

to only a cycle, another node belonging to the cycle is

needed. In Fig. 7(d), G is partitioned into G

1,{2},3,{2}

and G

1,{3},3,{1,4}

at nodes 1 and 3. Then, a production

rule of (R4) can be constructed, and G is generated

from G

1,{2},3,{2}

and G

1,{3},3,{1,4}

by parallel com-

position. In Fig. 7(e), G

1,{3},3,{1,4}

cannot be parti-

tioned at node 4 because only subgraphs with at most

two terminal nodes are allowed. Hence, G

1,{3},3,{1,4}

is partitioned into G

1,{3},3,{1}

and G

3,{4}

at node 3.

Then, a production rule of (R3) can be constructed,

and G

1,{3},3,{1,4}

is generated from G

1,{3},3,{1}

and

3,{4}

by generalized series composition.

Suppose that I (G) is a set of indices (i,S, j,T)

of all subgraphs G

i,S, j,T

of G obtained by re-

peatedly partitioning, S (G) is a set of all distinct

subgraphs G

i,S, j,T

, and E(u) is a set of all sub-

Grammar-based Compression for Directed and Undirected Generalized Series-parallel Graphs using Integer Linear Programming

109

graphs G

i,S, j,T

that are isomorphic to u. Consider

the case that G

i,S, j,T

) is correctly

partitioned into G

′

, j

′

, j

′

, j

′

)

and G

′′

, j

′′

, j

′′

, j

′′

). Let

C (G

i,S, j,T

) be a set of all index com-

binations (i

′

, j

′

′′

, j

′′

) that

′

, j

′

∪ V

′′

, j

′′

= V

i,S, j,T

, V

′

, j

′

∩

′′

, j

′′

= {i, j}, E

′

, j

′

∪ E

′′

, j

′′

= E

i,S, j,T

′

, j

′

∩ E

′′

, j

′′

0, E

′

, j

′

0, and

′′

, j

′′

0 in such cases. Then, we propose the

following integer linear programming formulation

for ﬁnding the minimum directed and undirected

GSPGGs that produce only a given generalized

series-parallel graph G.

Minimize

∑

u∈S ( G)

Subject to

ε,

0,ε,

= 1, (1)

i,S, j,T

= 1

for all (i,S, j,T) ∈ I (G) s.t. |E

i,S, j,T

| = 1, (2)

i,S, j,T

≤

∑

′

, j

′

′′

, j

′′

)∈C (G

i,S, j,T

)

′

, j

′

′′

, j

′′

for all (i,S, j,T) ∈ I (G) s.t. |E

i,S, j,T

| ≥ 2, (3)

′

, j

′

′′

, j

′′

≤

′

, j

′

+ x

′′

, j

′′

)

for all (i

′

, j

′

′′

, j

′′

) ∈ C (G

i,S, j,T

),(4)

≥

|E|

∑

i,S, j,T

∈E (u)

i,S, j,T

for all u ∈ S (G), (5)

< 1+

|E|

∑

i,S, j,T

∈E (u)

i,S, j,T

for all u ∈ S (G),(6)

i,S, j,T

′

, j

′

′′

, j

′′

, p

∈ {0,1}.

In this formulation, x

i,S, j,T

= 1 if G

i,S, j,T

is gen-

erated by the minimum GSPGG, otherwise 0. In the

minimum SEO(U)TG, the Euler string is used to de-

termine whether or not partitioned subtrees are iso-

morphic. However, it cannot be used for GSPGs,

and we must investigate whether or not G

i,S, j,T

isomorphic to G

′

, j

′

. Eqs. (5) and (6) represent

that p

= 1 for u ∈ S (G) if and only if a subgraph

i,S, j,T

isomorphic to u is generated by the mini-

mum GSPGG, otherwise 0. The objective function

indicates the number of nonterminal symbols in the

grammar, and the integer linear programming prob-

lem ﬁnds the minimum GSPGG. Eq. (1) represents

that G is constructed by the grammar. Eq. (2) repre-

sents that each edge in G is constructed by the gram-

mar. Eq. (3) represents that G

i,S, j,T

is constructed

by some production rule. Eq. (4) represents that a

production rule can be candidate in the grammar if

both of G

′

, j

′

and G

′′

, j

′′

are constructed by

the grammar. Since the problem of ﬁnding the mini-

mum directed and undirected GSPGGs that produce

only a given GSPG is NP-hard, it is reasonable to

solve it by utilising integer linear programs.

3 CONCLUSION

In this paper, we proposed the deﬁnition of di-

rected and undirected generalized series-parallel

graph (GSPG) grammars, and an integer linear

programming-based method for ﬁnding the minimum

GSPG grammar that produces only a given GSPG.

It has been proved that any outerplanar graph is a

GSPG. We can ﬁnd the minimum grammar for trees,

outerplanar graphs, and GSPGs. As future work, we

would like to apply our method to biological struc-

tured data, and extract production rules to construct

the structure. Our integer linear programming for-

mulation can take exponential time of the size of a

GSPG. Hence, we would like to analyze the time

complexity for the case that the degree of every node

is less than a constant value. Furthermore, we would

like to uncover what kind of graphs other than trees

and outerplanar graphs can be handled by directed

and undirected GSPGGs.

ACKNOWLEDGEMENTS

This work was partially supported by Grants-in-Aid

#16K00392, #16KT0020, and #26240034 from JSPS,

Japan.

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