n k <. If an even palindrome subsequence consists 2

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1 Fidig all Palidrome Subsequeces o a Strig K.R. Chuag 1, R.C.T. Lee 2 ad C.H. Huag 3* 1, 2 Departmet of Computer Sciece, Natioal Chi-Na Uiversity, Puli, Natou Hsieh, Taiwa Departmet of Computer Sciece ad Iformatio Egieerig, Natioal Formosa Uiversity, 64, We-Hwa Road, Hu-wei, Yu-Li, Taiwa 632 * Correspodig author: chhuag@suws.fu.edu.tw Abstract GATCTAG are odd palidromes. Palidromes are strigs of symbols that read the same forward ad backward. I DNA sequeces, palidromes appear frequetly ad are widespread i huma cacers ad idetifyig them could help advace the uderstadig of geomic istability. The palidrome detectio problem is therefore a importat issue i computatioal biology. I this paper, we propose the fidig all palidrome subsequeces problem ad give a algorithm to fid all palidrome subsequeces. Keywords: Palidrome, Palidrome Subsequece Sectio 1 Itroductio I this paper, the followig otatios are used. A strig is a sequece of symbols from a alphabet set. For a strig S = s 1 s 2 s of legth, let s i deote the ith symbol i S. A subsequece of S is obtaied by deletig zero or more (ot ecessarily cosecutive) symbols from S. Palidromes are strigs of the form ww R or waw R where w is a o-empty substrig, w R is the reverse of w R ad a is o-empty symbol. If we have strigs i the form ww R, we call these strigs eve palidromes. If we have strigs i the form waw R, we call these thigs odd palidromes. For example, GG ad TAGGAT are both eve palidromes. ATA ad I computatioal molecular biology, explorig DNA fuctio is very importat. Whe explorig DNA fuctio, special subsequeces such as palidromes may represet importat messages. I DNA sequeces, palidromes appear frequetly ad are widespread i huma cacers. Idetifyig palidromes of DNA sequeces could help advace the uderstadig of geomic istability [5, 6]. The fidig palidromes problem is therefore a importat issue i computatioal biology. There have bee may researches o the palidromes fidig problem ad there are may various classic computig problems o the palidromes fidig problem. Maacher discovered a o-lie sequetial algorithm that fids all iitial palidromes i a strig [3]. Da Gusfield gave a liear-time algorithm to solve the fidig all maximal palidromes problem i a strig [2]. Porto ad Barbosa gave a algorithm to fid all approximate palidromes i a strig [1]. The palidromes which the above algorithms foud are substrigs of a give strig. I this paper, we pay attetio to the palidrome subsequece. The palidrome subsequece is defied as followig. Give a strig S = s 1 s 2 s, a palidrome subsequece of S is a subsequeces of S which is a palidrome. For example, we suppose that S = ACGATGTAC. AGGA is a palidrome 1

2 subsequece of S. We propose the fidig all palidrome subsequeces problem ad give a algorithm to solve it. Sectio 2 The Method To begi with, we itroduce the property of palidrome. Let P = p 1 p 2 p m-1 p m be a palidrome ad (p i, p j ) be a matched pair where p i ad p j are idetical character ad 1 i < j m. If P is a m eve palidrome of legth m, P cosists of 2 matched pairs such as (p 1, p m ) (p 2, p m-1 ) ( p, m p m ). For example, P = ATTA is a eve palidrome which cosists of 2 matched pairs such as (p 1, p 4 ) (p 2, p 3 ), show i Figure 2-1. If P is a odd m palidrome of legth m, P cosists of 2 matched pairs ad a cetral symbol p c. For example, P = ATGTA is a odd palidrome which cosists 2 matched pairs such as (p 1, p 5 ) (p 2, p 4 ) ad p 3 is the cetral symbol, show i Figure 2-2. Figure 2-1 Palidrome subsequeces also have the same property with palidromes, because palidrome subsequeces are palidromes. Give a strig S = s 1 s 2 s -1 s of legth, let (i, j) be the match pair where i ad j deote that s i is matched with s j ad 1 i < j. Let (i 1, j 1 )-(i 2, j 2 )- -(i k, j k ) deote eve palidrome subsequeces of S with k matched pairs where 1 i1 < i2 <... ik < jk <... < j2 < j1 ad k <. If a eve palidrome subsequece cosists 2 k matched pair, we call it a k-pair palidrome subsequece. For example, give a strig S = CGATGTAC, ATTA, (3, 7)-(4, 6), is a 2-pair palidrome subsequece of S. Let (i 1, j 1 )-(i 2, j 2 )- -(i k, j k )-c deote odd palidrome subsequeces of S with k matched pairs ad oe cetral symbol s c where c is the positio of s c o S ad i k < c < j k. The odd palidrome subsequeces with k matched pairs ca be obtaied from k-pair palidrome subsequeces with oe cetral symbol. Odd palidrome subsequeces could be foud easily, whe all eve palidrome subsequeces are foud. For example, give a strg S = CGATGTAC, ATGTA, (3, 7)-(4, 6)-5, composes of 2-pair palidrome subsequece, (3, 7)-(4, 6) ad oe cetral symbol s 5. There may be too may odd palidrome subsequeces based o a eve palidrome subsequeces, so we oly fid the eve palidrome subsequeces i this paper. The k-pair palidrome subsequece has a property. The k-pair palidrome subsequece composes of a k-1-pair palidrome subsequece ad 1-pair palidrome subsequece. Let k-1-palidrome be (i 1, j 1 )- -(i k-1, j k-1 ) ad 1-pair palidrome subsequece be (i, j ). The k-pair palidrome subsequece, (i 1, j 1 )- -(i k-1, j k-1 )-(i, j ), ca be foud from k-1-pair palidrome subsequece ad 1-pair palidrome subsequece where i > i k-1 ad j < j k-1. For example, give a strig S = ACGATGTAC, CC, CAAC ad CATTAC are palidrome subsequeces of S. CC is a 1-pair palidrome subsequece, (2, 9), show i Figure 2-2 (a), AA is also a 1-pair palidrome subsequece, (4, 8) ad TT is also a 1-pair palidrome subsequece, (5, 7). CAAC is a 2-pair palidrome subsequece, (2, 9)-(4, 8) which composes of two 1-pair palidrome 2

3 subsequeces (2, 9) ad (4, 8), show i Figure 2-2 (b). CATTAC is a 3-pair palidrome subsequece, (2, 9)-(4, 8)-(5, 7), which composes of a 2-pair palidrome subsequece, (2, 9)-(4, 8), ad a 1-pair palidrome subsequece, (5, 7), show i Figure 2-2 (c). A C G A T G T A C First, we fid all matched pairs of S ad each matched pair is a 1-pair palidrome subsequece. (1, 4) AA (1, 8) AA (2, 9) CC (3, 6) GG (a) The matched pair of CC (4, 8) AA (5, 7) TT After all 1-palidrome subsequeces of S are foud, we ca fid all 2-palidrome subsequeces based upo them. (1, 8)-(3, 6) AGGA (b) The matched pairs of CAAC (1, 8)-(5, 7) ATTA (2, 9)-(3, 6) CGGC (2, 9)-(4, 8) CAAC (2, 9)-(5, 7) CTTC (c) The matched pairs of CATTAC Figure 2-2 Accordig to the above property of k-pair palidrome subsequece, we ca use it to fid all palidrome subsequeces. For example, give a strig S = ACGATGTAC, we ca use it to fid all palidrome subsequeces of S as follows: S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 (4, 8)-(5, 7) ATTA After fidig all 2-palidrome subsequeces, we ca fid all 3-palidrome subsequeces based upo 2-palidrome subsequece ad 1-palidrome subsequece. (2, 9)-(4, 8)-(5, 7) CATTAC The recursive process cotiues util all palidrome subsequece are foud out. 3

4 Sectio 3 The Algorithm We proposed a algorithm to solve the fidig all palidrome subsequeces problem. I this algorithm, we fid all palidrome subsequeces from oe palidrome subsequece to the logest palidrome subsequece. Give a strig S of legth, let U k be the set of k-pair palidrome where 1 k 2. Step 1: 1 i /* Fidig out matched pair for < */ U 1 := {} for i = 1 to do for j = i +1 to do if s i = s j the j Step 1: We use icidece matrix to fid all matched pairs (i, j) where 1 i < j ad add them ito U 1, because each matched pair is 1-pair palidrome subsequece. w := (i, j) U 1 := U 1 U {w} 1 Step 2: We geerate U k from U k-1 ad U 1 where k 2. For all k-1-pair palidrome subsequeces i U k-1, we take a k-1-pair palidrome subsequece (i 1, j 1 )- -(i k-1, j k-1 ) from U k-1 ad we check all 1-pair palidromes from U 1 whether there is a 1-pair palidrome (i, j ) which satisfies the rule i > i k-1 ad j < j k-1. If it is satisfied, we combie the k-1-pair palidrome (i 1, j 1 )- -(i k-1, j k-1 ) with the 1-pair palidrome (i, j ) to be k-pair palidrome (i 1, j 1 )- -(i k-1, j k-1 )-(i, j ) ad add it ito the set U k. Util the U /2 is geerated, we ca get the set U = U 1 U U 2 U U U /2 which cotais all palidrome subsequeces of S. I the followig, we preset the algorithm for fidig all palidrome subsequeces. Step 2: /* Fidig all palidrome subsequeces of S */ for k = 2 to /2 do U k := {} for all k-1-pair palidrome (i 1, j 1 ) (i k-1, j k-1 ) from U k-1 do for all 1-pair palidrome (i, j ) from U 1 do if i > i k-1 ad j < j k-1 the i k := i Algorithm fidallpalidromesubsequeces(s) j k := j Iput: A strig S = s 1 s 2 s. w := (i 1, j 1 ) (i k-1, j k-1 ) (i k, j k ) Output: All palidrome subsequeces of S. U k := U k U {w} 4

5 edif 7 T A 0 9 C U := U 1 U U 2 U U U /2 /* U is the set of all palidrome subsequeces of S */ After the icidece matrix is geerated, we ca get the U 1. U 1 = {(1, 4), (1, 8), (2, 9), (3, 6), (4, 8), (5, 7)} Step 2: (1) k = 2, U 1 = {(1, 4), (1, 8), (2, 9), (3, 6), (4, 8), (5, 7)}, U 2 = {} Obviously, the time complexity of this sample algorithm is O( 3 ) where is the legth of the sequece. (1-1) We take the 1-palidrome subsequece (1, 4) from U 1. Sectio 4 A Example Give a strig S = ACGATGTAC, We ow illustrate the whole procedure i detail. S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 A C G A T G T A C For all 1-pair palidrome subsequeces from U 1, there is o 1-pair palidrome subsequece (i, j ) which satisfies that i > 1 ad j < 4. U 2 = {} (1-2) Step 1: We use icidece matrix to fid all matched 1 i pairs (i, j) where <. j We take the 1-palidrome subsequece (1, 8) from U 1. Table 1 The icidece matrix for this sequece S = ACGATGTAC S j S i A C G A T G T A C 1 A C G A T G For all 1-pair palidrome subsequeces from U 1, there is a 1-pair palidrome subsequece (3, 6) which satisfies that 3 > 1 ad 6 < 8. We combie (1, 8) with (3, 6) to be 2-pair palidrome subsequece (1, 8)-(3, 6) ad add it ito the set U 2 = {(1, 8)-(3, 6)} There is aother 1-pair palidrome subsequece (5, 7) which ca satisfy that 5 > 1 ad 7 < 8. We 5

6 combie (1, 8) with (5, 7) to be 2-pair palidrome subsequece (1, 8)-(5, 7) ad add it ito the set U 2 = {(1, 8)-(3, 6), (1, 8)-(5, 7)} There is o 1-pair pair palidrome subsequece which ca be U 2 = {(1, 8)-(3, 6), (1, 8)-(5, 7)} (1-3) 9)-(4, 8), (2, 9)-(5, 7)} (1-4) We take the 1-pair palidrome subsequece (3, 6) from U 1. Check all 1-pair palidromes from U 1. We take the 1-pair palidrome subsequece (2, 9) from U 1. U 2 = {(1, 8) (3, 6), (1, 8) (5, 7), (2, 9) (3, 6), (2, 9) (4, 8), (2, 9) (5, 7)} There is a 1-pair palidrome subsequece (3, 6) which ca be We combie (2, 9) with (3, 6) to be 2-pair palidrome subsequece (2, 9)-(3, 6) ad add it ito the set U 2 = {(1, 8)-(3, 6), (1, 8)-(5, 7), (2, 9)-(3, 6)} There is aother 1-pair palidrome subsequece (4, 8) which ca be We combie (2, 9) with (4, 8) to be 2-pair palidrome subsequece (2, 9)-(4, 8) ad add it ito the set 9)-(4, 8)} There is aother 1-pair palidrome subsequece (5, 7) which ca be We combie (2, 9) with (5, 7) to be 2-pair palidrome subsequece (2, 9)-(5, 7) ad add it ito the set 9)-(4, 8), (2, 9)-(5, 7)} There is o 1-pair palidrome subsequece which ca be (1-5) We take the 1-pair palidrome (4, 8) from U 1. Check all 1-pair palidromes from U 1. There is a 1-pair palidrome (5, 7) which ca be We combie (4, 8) with (5, 7) to be 2-pair palidrome (4, 8)-(5, 7) ad add it ito the set 9)-(4, 8), (2, 9)-(5, 7), (4, 8)-(5, 7)} 9)-(4, 8), (2, 9)-(5, 7), (4, 8)-(5, 7)} (1-6) We take the 1-pair palidrome (5, 7) from U 1. Check all 1-pair palidromes from U 1. 6

7 (2) k = 3, U 1 = {(1, 4), (1, 8), (2, 9), (3, 6), (4, 8), (5, (2-4) We take the 2-pair palidrome (2, 9)-(4, 8) from 7)}, 9)-(4, 8), (2, 9)-(5, 7), (4, 8)-(5, 7)}, U 3 = {} (2-1) We take the 2-pair palidrome (1, 8)-(3, 6) from There is a 1-pair palidrome (5, 7) which ca be We combie (2, 9)-(4, 8) with (5, 7) to be 3-pair palidrome (2, 9)-(4, 8)-(5, 7) ad add it ito the set U 3. U 3 = {(2, 9)-(4, 8)-(5, 7)} U 3 = {} (2-5) We take the 2-pair palidrome (2, 9)-(5, 7) from (2-2) We take the 2-pair palidrome (1, 8)-(5, 7) from U 3 = {(2, 9)-(4, 8)-(5, 7)} U 3 = {} (2-6) We take the 2-pair palidrome (4, 8)-(5, 7) from (2-3) We take the 2-pair palidrome (2, 9)-(3, 6) from (3) k = 4, U 1 = {(1, 4), (1, 8), (2, 9), (3, 6), (4, 8), (5, 7)}, 9)-(4, 8), (2, 9)-(5, 7), (4, 8)-(5, 7)}, U 3 = {(2, 9)-(4, 8)(5, 7)}, U 4 = {} U 3 = {} (3-1) 7

8 We take the 3-palidrome (2, 9)-(4, 8)-(5, 7) from U 3. U 4 = {} Fially, we get the set U = U 1 U U 2 U U U /2 which cotais all palidrome subsequeces of S. U = {(1, 4), (1, 8), (2, 9), (3, 6), (4, 8), (5, 7), (1, 8)-(3, 6), (1, 8)-(5, 7), (2, 9)-(3, 6), (2, 9)-(4, 8), (2, 9)-(5, 7), (4, 8)-(5, 7), (2, 9)-(4, 8)-(5, 7)} The all palidrome subsequeces of S are as follows: (1, 4) AA (1, 8) AA (2, 9) CC (3, 6) GG (2, 9)-(4, 8)-(5, 7) CATTAC Sectio 5 Coclusios ad Future Work I this paper, we proposed a algorithm to solve the fidig all palidrome subsequeces i a strig. Palidrome subsequeces occur frequetly i DNA sequeces ad have bee proved to be critical for some characteristics. Our algorithm provides a effective tool for the related research. Refereces [1] Alexadre H.L. Porto, Valmir C. Barbosa, Fidig Approximate Palidromes i Strigs. Patter Recogitio, [2] D. Gusfied, Algorithms o Strigs, Trees, ad Sequeces: Computer Sciece ad Computatioal Biology, Cambridge Uiversity Press, New York, [3] G. Maacher. A ew Liear-Time O-Lie Algorithm for Fidig the Smallest Iitial Palidrome of a Strig. J. Assoc. Comput (4, 8) AA (5, 7) TT (1, 8)-(3, 6) AGGA [4] Lloyd Alliso, Fidig Approximate Palidromes i Strigs Quickly ad Simply [5] Choi, Charles Q, DNA palidromes foud i cacer, The Scietist (1, 8)-(5, 7) ACCA [6] Taaka, Hisashi; BERGSTROM, Doald A; YAO, (2, 9)-(3, 6) CGGC (2, 9)-(4, 8) CAAC Meg-Chao ad TAPSCOTT, Stephe J, Large DNA palidromes as a commo form of structural chromosome aberratios i huma cacers, Huma Cell, 2006 (2, 9) (5, 7) CTTC (4, 8)-(5, 7) ATTA 8