Improved Algorithms for Enumerating Tree-like Chemical Graphs with Given Path Frequency

Transcription

1 Improved Algorithms for Enumerating Tree-like hemical Graphs with Given Path Frequency Yusuke Ishida* Liang Zhao iroshi agamochi Graduate School of Informatics, Kyoto University, Japan Tatsuya Akutsu Institute for hemical Research, Kyoto University, Japan utline. Problem Formulation. anonical Representation and Family tree. Algorithm (Branch and Bound) Detachment-cut 4. Experimental Results for the first formulation 5. -less Single-bond Formulation ydrogen-cut 6. Experimental Results for the second formulation 7. onclusions

2 Background inference chemical compounds given partial structures applications: structure determination using mass-spectrum drug design Definition A feature vector fk(g) : #occurrences of each vertex-labeled path of length 0,,..., K length = 0 4 f(g) (K = ) a chemical compound G length = length = length = 6

3 Enumerating hemical Multitree Problem Input utput label set Σ = {,, } valence val () = val () = val () = 4 level of g K = feature vector g 4 all Σ-labeled multitrees satisfying the feature vector constraint and the valence constraint Previous Work Aringhieri et al. [4R, 00] designed two algorithms to generate all alkane isomers. Fujiwara et al. [J. hem. Inf. Model., 008] proposed a branch and bound algorithm for chemical multitree problem. gave -less single-bond formulation of chemical multitree problem and their algorithm can be also applied to it.

4 anonical Representation for multitrees depth 0 canonical representation T,T,T : isomorphic trees T T T Define the canonical representation as the embedding with largest depth-label sequence. Family tree φ Add a vertex to T = φ. Repeatedly attach a new vertex to T based on canonical representation. family tree(#labels=, #vertices=5 ) 4

5 Parent of multitrees canonical representation rightmost path canonical representation canonical representation T branching P(T) branching P(P(T)) Define the parent P(T) of T as the tree obtained by removing the rightmost leaf from T. If T is canonical, P(T) remains canonical. Branching operation : () time / node Algorithm (Branch and Bound) Add a vertex to T = φ. φ Attach a new vertex to T based on canonical representation. (Branching peration) heck each constraint and decide whether to discard T. (Bounding peration) If T = (#input vertices), output T as a solution. family tree(#labels=, #vertices=5) 5

6 Bounding perations For a current tree T, feature-vector-cut test whether fk(t) g holds. the input the feature vector of T bond-cut test whether deg(v;t) val(l (v)) for v T holds. #edges incident to v in T the valence of the label of v detachment-cut test whether T can be extended to multitrees satisfying degree constraint. T Input T Detachment-cut cut Test whether T can be extended to multitrees satisfying degree constraint. input residual 4 6 A A rightmost 0 ( E path G=(V, 5 A A shrink a current tree T r() = 4 r() = r() = r() = r(a) = ρ : = A (,,, ) = ρ() ( ) = ρ() ( ) = ρ() ( 4,, ) = ρ() ( ) = ρ(a) 6

7 Detachment : A Reverse peration of ontraction The number r(v) of copies of v and the degree ρ(v i ) of each copy v i are specified. b r(b)= ( ρ(b)=(,, Wa Wb Wd a r(a)= ( ρ(a)=( c a multigraph G d r(d)= ( ρ(d)=(, r(c)=4 ( ρ(c)=(,,, Wc a ρ-detachment of G Detachment-cut cut Test whether G has a connected and loopless ρ-detachment. (,,, ) = ρ() = 9 8 for v = 4++ ( ) = ρ() ρ(v i ) deg(v;g) v V i r(v) ( ) = ρ() #edges incident to v in G ( 4,, ) = ρ() ( ) = ρ(a) condition for degree constraint 7 5 d(x,v;g) r(v) + c(g-x) - φ X V v X #edges joining X and V condition for connectivity #connected components in G - X v A r() = 4 r() = r() = r() = r(a) = X ( E G=(V, 7

8 Experiment omparing the running time of our algorithm with Fujiwara et al. s [008] Instances from KEGG LIGAD database (Replacing each benzene ring by a virtual atom of valence 6) Pentium4.00Gz T.. : time over 800 (sec) Experimental Results for the first formulation.f. : not found Formula #atoms K Fujiwara et al. s algorithm PU time (sec) T....5 T T T.. T.. T.. #solutions.f. 9.F. 55.F. 6.F..F..F. our algorithm PU time (sec) T #solutions 570,77 9 7, ,70 6.F.,98 8 8

9 -less Single-bond Formulation [Fujiwara et al. 008] removal transformation single-bond transformation new atom {(,),} 4 R R R R = {(,),} ydrogen-cut input R R R R = {(,),} h*(l ) : sum of #hydrogens adjacent to each atom of label l. 0 {(,),} 0 h*() = = 5 h*() = 0 + = 9

10 ydrogen-cut h(l, T) : sum of #hydrogens that must be adjacent to each atom of label l in a current tree T If h(l ;T) > h*(l ) holds for a label l, discard T. Assume h*() = 7. The degrees of these vertices won t be changed. {(,),} T h( ; T) = rightmost path {(,),} : a new vertex p p p p Tc discard h( ; Tc) = 68 > 7 Experimental Results for the second formulation Formula #atoms P 9 97P K 4 4 Fujiwara et al. s algorithm PU time (sec) T T T.. T #solutions 70,70 6 5,05,4, our algorithm PU time (sec) T T #solutions 70, ,57,65,98 8,50,76 0

11 onclusions We proposed a branch and bound algorithm with two new bounding operations. For the first formulation, we can solve the problem with about 5 non-hydrogen atoms for K. For the second formulation, we can solve the problem with about 0 non-hydrogen atoms for K. Future Work Treat more general graphs (e.g., outerplanar graphs). Use other graph structures for representing partial structures of input.

12 Definition A feature vector fk(g) : #occurrences of each vertex-labeled path of length 0,,..., K f(g) a chemical compound G

13 anonical Representation for multitrees left-heavy T,T,T : isomorphic trees DL: depth label sequence T DL(T) = (0,,,,,, (,,,,,,,,, T DL(T) = (0,,,,,, (,,,,,,,,, T DL(T) = (0,,,,,, (,,,,,,,,, The embedding with largest DL is left-heavy. [akano, Uno (00)] We enumerate left-heavy trees. Parent of multitrees left-heavy left-heavy left-heavy T P(T) P(P(T)) parent P(T) : tree obtained by removing the rightmost leaf from T If T is left-heavy, P(T) remains left-heavy. In branching operation, we attach a new vertex to rightmost path.

14 Feature Vector (K=)( and ρ-detachment Σ = {,, } val () = val () = val () = r() = 4 r() = r() = (,,, ) = ρ() (, ) = ρ() ( 4,4 ) = ρ() G Detachment-cut cut Test whether T will generate trees satisfying given constraint. input Which vertices are counted as residual? ew vertices can be attached. a current tree T residual ( E G=(V, r()= 4+ r()= + r()= 0+ r()= + 4

15 Detachment-cut cut Test whether T will generate trees satisfying given constraint. input Which vertices are counted as residual? ew vertices can be attached. a current tree T residual A A A This edge will be counted. A G=(V, ( E r()= 4+ 4 r()= + r()= 0+ r()= + r(a)= Detachment-cut cut Test whether and holds. necessary conditions for G has a connected and loopless ρ-detachment [agamochi 006] A 7 5 #edges joining X and V #connected components in G - X G=(V, ( E condition for connectivity ρ() = =( r((,,, ) 4 ρ() = ( ) r()= 4++ = 9 8 for v = ρ() = ( ) r()= ρ() = ( 4,, ) r()= ρ(a) = ( ) r(a)= #edges incident to v in G condition for degree specification X 5

16 Detachment-cut cut onsider the degree constraint ρ(v i) of each vertex v i for i r(v). For a vertex v i T, ρ (v i ) = val (l (v i )). For a vertex v i T, ρ (v i ) = val(l (v i )) - deg (v i ;T) +. a current tree T ρ : = A ρ : = ρ : - + = A (,,, ) = ρ() ( ) = ρ() ( ) = ρ() ( 4,, ) = ρ() ( ) = ρ(a) r()= 4 r()= r()= r()= r(a)= Test whether there exists multitrees satisfying the degree constraint by considering a detachment of G. ( E G=(V, Detachment-cut cut Test whether T can be extended multitrees satisfying degree constraint. input residual 4 6 rightmost path A A 0 ( E G=(V, r()= = A r()= = + A r()= = 0+ r()= = + fixed a current tree T r(a) = 6

17 Detachment : A Reverse peration of ontraction Wb b Wa Wd a d c A multigraph G Wc A detachment of G 7