An Estimation of the Size of Non-Compact Suffix Trees

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1 Ata Cyernetia 22 (2016) An Estimation of the Size of Non-Compat Suffix Trees Bálint Vásárhelyi Astrat A suffix tree is a data struture used mainly for pattern mathing. It is known that the spae omplexity of simple suffix trees is quadrati in the length of the string. By a slight modifiation of the simple suffix trees one gets the ompat suffix trees, whih have linear spae omplexity. The motivation of this paper is the question whether the spae omplexity of simple suffix trees is quadrati not only in the worst ase, ut also in expetation. 1 Introdution A suffix tree is a powerful data struture whih is used for a large numer of ominatorial prolems involving strings. Suffix tree is a struture for ompat storage of the suffixes of a given string. The ompat suffix tree is a modified version of the suffix tree, and it an e stored in linear spae of the length of the string, while the non-ompat suffix tree is quadrati (see [11, 14, 18, 19]). The notion of suffix trees was first introdued y Weiner [19], though he used the name ompated i-tree. Grossi and Italiano mention that in the sientifi literature, suffix trees have een redisovered many times, sometimes under different names, like ompated i-tree, prefix tree, PAT tree, position tree, repetition finder, suword tree et. [10]. Linear time and spae algorithms for reating the ompat suffix tree were given soon y Weiner [19], MCreight [14], Ukkonen [18], Chen and Siferas [4] and others. The statistial ehaviour of suffix trees has een also studied. Most of the studies onsider improved versions. The average size of ompat suffix trees was examined y Blumer, Ehrenfeuht and Haussler [3]. They proved that the average numer of nodes in the ompat suffix tree is asymptotially the sum of an osillating funtion and a small linear funtion. An important question is the height of suffix trees, whih was answered y Devroye, Szpankowski and Rais [6], who proved that the expeted height is logarithmi in the length of the string. Szegedi Tudományegyetem, TTIK, Szeged, 6720, Hungary. mesti@math.u-szeged.hu DOI: /atay

2 824 Bálint Vásárhelyi The appliation of suffix trees is very wide. We mention ut only a few examples. Apostolio et al. [2] mention that these strutures are used in text searhing, indexing, statistis, ompression. In omputational iology, several algorithms are ased on suffix trees. Just to refer a few of them, we mention the works of Höhl et al. [12], Adeiyi et al. [1] and Kaderali et al. [13]. Suffix trees are also used for deteting plagiarism [2], in ryptography [15, 16], in data ompression [7, 8, 16] or in pattern reognition [17]. For the interested readers further details on suffix trees, their history and their appliations an e found in [2], in [10] and in [11], whih soures we also used for the overview of the history of suffix trees. It is well-known that the non-ompat suffix tree an e quadrati in spae as we referred efore. In our paper we are setting a lower ound on the average size, whih is also quadrati. 2 Preliminaries Before we turn to our results, let us define a few neessary notions. Definition 1. An alphaet Σ is a set of different haraters. The size of an alphaet is the size of this set, whih we denote y σ(σ), or more simply σ. A string S is over the alphaet Σ if eah harater of S is in Σ. Definition 2. Let S e a string. S[i] is its ith harater, while S[i, j] is a sustring of S, from S[i] to S[j], if j i, else S[i, j] is the empty string. Usually n(s) (or n if there is no danger of onfusion) denotes the length of the string. Definition 3. The suffix tree of S is a rooted direted tree with n leaves, where n is the length of S. Its struture is the following: Eah edge e has a lael l(e), and the edges from a node v have different laels (thus, the suffix tree of a string is unique). If we onatenate the edge laels along a path P, we get the path lael L(P). We denote the path from the root to the leaf j y P(j). The edge laels are suh that L(j) = L(P(j)) is S[j, n] and a $ sign at the end. The definition eomes more lear if we hek the example on 1 and 2. A naive algorithm for onstruting the suffix tree is the following: Notie that in 2 a leaf always remain a leaf, as $ (the last edge lael efore a leaf) is not a harater in S. Definition 4. The ompat suffix tree is a modified version of the suffix tree. We get it from the suffix tree y ompressing its long ranhes. The struture of the ompat suffix tree is asially similar to that of the suffix tree, ut an edge lael an e longer than one harater, and eah internal node (i.e. not leaf) must have at least two hildren. For an example see 2. With a regard to suffix trees, we an define further notions for strings.

3 An Estimation of the Size of Non-Compat Suffix Trees 825 a $ 5 $ 4 $ 3 $ 6 $ 2 a $ 1 Growth of the string Figure 1: Suffix tree of string aa Definition 5. Let S e a string, and T e its (non-ompat) suffix tree. A natural diretion of T is that all edges are direted from the root towards the leaves. If there is a direted path from u to v, then v is a desendant of u and u is an anestor of v. We say that the growth of S (denoted y γ(s)) is one less than the shortest distane of leaf 1 from an internal node v whih has at least two hildren (inluding leaf 1), that is, we ount the internal nodes on the path different from v. If leaf j is a desendant of v, then the ommon prefix of S[j, n] and S[1, n] is the longest among all j s. 1. If we onsider the string S = aa, the growth of S is 5, as it an e seen on An important notion is the following one. Definition 6. Let Ω(n, k, σ) e the numer of strings of length n with growth k over an alphaet of size σ. Oserve that the onnetion etween the growth and the numer of nodes in a suffix tree is the following: Oservation 1. If we onstrut the suffix tree of S y using 2, we get that the sum of the growths of S[n 1, n], S[n 2, n],..., S[1, n] is a lower ound to the numer of nodes in the final suffix tree. In fat, there are only two more internal nodes, the root vertex, the only node on the path to leaf n, and we have the leaves. In the proofs we will need the notion of period and of aperiodi strings.

4 826 Bálint Vásárhelyi Let S e a string of length n. Let j = 1 and T e a tree of one vertex r (the root of the suffix tree). Step 1: Consider X = S[j, n] + $. Set i = 0, and v = r. Step 2: If there is an edge vu laelled X[i + 1], then set v = u and i = i + 1. Step 3: Repeat Step 2 while it is possile. Step 4: If there is no suh an edge, add a path of n j i + 2 edges from v, with laels orresponding to S[j + i, n] + $, onseutively on the edges. At the end of the path, numer the leaf with j. Step 5: Set j = j + 1, and if j n, go to Step 1. a $ $ $ $ $ a$ Figure 2: Compat tree of string aa Definition 7. Let S e a string of length n. We say that S is periodi with period d, if there is a d n for whih S[i] = S[i + d] for all i n d. Otherwise, S is aperiodi. The minimal period of S is the smallest d with the property aove. Definition 8. µ(j, σ) is the numer of j-length aperiodi strings over an alphaet of size σ. A few examples for the numer of aperiodi strings are given in 1. σ µ(1, σ) µ(2, σ) µ(3, σ) µ(4, σ) µ(5, σ) µ(6, σ) µ(7, σ) µ(8, σ) Tale 1: Numer of aperiodi strings for small alphaets. σ is the size of the alphaet, and µ(j, σ) is the numer of aperiodi strings of length j

5 An Estimation of the Size of Non-Compat Suffix Trees Main results Our main results are formulated in the following theorems. Theorem 2. On an alphaet of size σ for all n 2k, Ω(n, k, σ) φ(k, σ) for some funtion φ. Theorem 3. There is a > 0 and an n 0 suh that for any n > n 0 the following is true. Let S e a string of length n 1, and S e a string otained from S y adding a harater to its eginning hosen uniformly random from the alphaet. Then the expeted growth of S is at least n. Theorem 4. There is a d > 0 that for any n > n 0 (where n 0 is the same as in 3) the following holds. On an alphaet of size σ the simple suffix tree of a random string S of length n has at least d n 2 nodes in expetation. 4 Proofs Proof. (4) Considering 1 we have that the expeted size of the simple suffix tree of a random string S is at least E (γ(s[n m, n])) m=1 E(γ(S[n m, n])). (1) m=1 If m n 0, 3 is ovious. If m > n 0, we an divide the sum into two parts: E(γ(S[n m, n])) = m=1 n 0 m=1 E(γ(S[n m, n])) + m=n 0+1 E(γ(S[n m, n])). (2) The first part of the sum is a onstant, while the seond part an e estimated with 3: E(γ(S[n m, n])) n = d n 2. (3) This proves 4. m=n 0+1 m=n 0+1 First, we show a few lemmas aout the numer of aperiodi strings. 1 an e found in [9] or in [5], ut we give a short proof also here. Lemma 1. For all j > 0 integer and for all alphaet of size σ the numer of aperiodi strings is µ(j, σ) = σ j µ(d, σ). d j d j (4)

6 828 Bálint Vásárhelyi Proof. µ(1, σ) = σ is trivial. There are σ j strings of length j. Suppose that a string is periodi with minimal period d. This implies that its first d haraters form an aperiodi string of length d, and there are µ(d, σ) suh strings. This finishes the proof. Speially, if p is prime, then µ(p, σ) = σ p σ. Corollary 1. If p is prime and t N, then µ (p t, σ) = σ pt σ pt 1 for all alphaet of size σ. Proof. We ount the aperiodi strings of length p t. There are σ pt strings. Consider the minimal period of the string, i.e. the period whih is aperiodi. If we exlude all minimal periods of length k, we exlude µ(k, σ) strings. This yields the following equality: µ ( p t, σ ) = σ pt µ (p s, σ). (5) 1 s<t With a few transformations and using 1, we have that (5) is equal to σ pt µ ( p t 1, σ ) µ (p s, σ) = σ pt σ pt 1 + µ (p s, σ) whih is 1 s<t 1 1 s<t 1 1 s<t 1 µ (p s, σ), (6) σ pt σ pt 1. (7) Lemma 2. For all j > 1 and for all alphaet of size σ, µ(j, σ) σ j σ. Proof. From 1 we have µ(j, σ) = σ j µ(d, σ). Considering µ(d, σ) 0 and d j d j µ(1, σ) = σ, we get the laim of the lemma. Lemma 3. For all j 1, and for all alphaet of size σ µ(j, σ) σ(σ 1) j 1. (8) Proof. We prove y indution. For j = 1 the laim is ovious, as µ(1, σ) = σ. Suppose we know the laim for j 1. Consider σ(σ 1) j 2 aperiodi strings of length j 1. Now, for any of these strings there is at most one harater y appending that to the end of the string we reeive a periodi string of length j. Therefore we an append at least σ 1 haraters to get an aperiodi string, whih gives the desired result. Oservation 5. Oserve that if the growth of S is k, then there is a j suh that S[1, n k] = S[j +1, j +n k]. For example, if the string is adefada (n = 12), one an hek that the growth is 8 (the new ranh in the suffix tree whih ends in leaf 1 starts after ad), and with j = 6 we have S[1, 4] = S[7, 10] = ad.

7 An Estimation of the Size of Non-Compat Suffix Trees 829 The reverse of this oservation is that if there is a j < n suh that S[1, n k] = S[j + 1, j + n k], then the growth is at most k, as S[j + 1, n] and S[1, n] shares a ommon prefix of length n k, thus, the paths to the leaves j + 1 and n share n k internal nodes, and at most k new internal nodes are reated. Proof. (2) For proving the theorem we ount the numer of strings with growth k for n 2k. First, we fix j, and then ount the numer of possile strings where the growth ours suh that S[1, n k] = S[j + 1, j + n k] for that fixed j. Note that y this way, we only have an upper ound for this numer, as we might found an l suh that S[1, n k + 1] = S[l + 1, l + n k + 1]. We know that j k, otherwise S[j + 1, j + n k] does not exist. If j = k, then we know S[1, n k] = S[k + 1, n]. S[1, k] must e aperiodi. Suppose the opposite and let S[1, k] = p... p, where p is the minimal period, and its length is d. Then S[k + 1, n] = p... p. Oviously, in this ase S[1, n d] = S[d + 1, n], whih y 5 means that the growth would e at most d. See also 3. Therefore this ase gives us at most µ(k) strings of growth k. p p p p p p 1 k Figure 3: Proof of 2, ase j = k n If j < k, then we have S[1, n k] = S[j + 1, j + n k]. First, we note that S[1, j] must e aperiodi. Suppose the opposite and let S[1, j] = p... p, where p is the minimal period, and its length is d. Then whih means that S S[j + 1, 2j] = S[2j + 1, 3j] =... = p... p, (9) [ 1, ] [ k j = S j + 1, j + j ] k j = p... p. (10) j This implies that S[1, j + n k] = p... pp, where p is a prefix of p. However, S[1, j + n k d] = S[d, j + n k] is true, and using 5, we have that γ(s) n (j + n k) + d = k j + d < k, whih is a ontradition. Further, S[j + n k + 1] must not e the same as S[k + 1], whih means that this harater an e hosen σ 1 ways. Therefore this ase gives us at most µ(j)(σ 1)σ k j 1 strings of growth k for eah j. By summing up for eah j, we have

8 830 Bálint Vásárhelyi Figure 4: Proof of 2, ase j < k This ompletes the proof. k 1 φ(k, σ) = µ(j, σ)(σ 1)σ k j 1 + µ(k, σ) (11) j=1 Proof. (3) Aording to 2, µ(j, σ) σ j σ (if j > 1). In the proof of 2 at (11) we saw for k 1 and n 2k 1 that k 1 φ(k, σ) = µ(k, σ) + µ(j, σ)(σ 1)σ k j 1. (12) We an ound the right hand side of (12) from aove as it follows: µ(k, σ) + k 1 j=1 whih is y 2 at most j=1 µ(j, σ)(σ 1)σ k j 1 = µ(k, σ) + µ(1, σ)(σ 1)σ k 2 + k 1 j=2 µ(j, σ)(σ 1)σ k j 1, (13) k 1 k 1 σ k σ+σ(σ 1)σ k 2 + (σ j σ)(σ 1)σ k j 1 σ k +σ k + σ j σσ k j 1 kσ k. n 2 j=2 Thus, φ(k, σ) kσ k, whih means k=1 j=2 (14) m m φ(k, σ) kσ k (m + 1)σ m+1. (15) k=1 The left hand side of 15 is an upper ound for the strings of growth at most m. Let m = n 2. As σ n n 2 σ n 2, this implies that in most ases the suffix tree of S has at least more nodes than the suffix tree of S[1, n 1]. Thus, a lower ound on the expetation of the growth of S is whih is E (γ(s)) 1 ( n ( σ n 2 σ n 2 + σ n n ) ( 2 σ n n )) , (16) ( ( ) ) 1 n + 2 n σ n σ n n(n + 2) + σ n 2 = n, (17)

9 An Estimation of the Size of Non-Compat Suffix Trees 831 with some, if n is large enough. With this, we have finished the proof and gave a quadrati lower ound on the average size of suffix trees. Referenes [1] E.F. Adeiyi, T. Jiang, and M. Kaufmann. An effiient algorithm for finding short approximate non-tandem repeats. Bioinformatis, 17:5S 12S, [2] A. Apostolio, M. Crohemore, M. Farah-Colton, Z. Galil, and S. Muthukrishnan. 40 years of suffix trees. Communiations of the ACM, 59:66 73, [3] A. Blumer, A. Ehrenfeuht, and D. Haussler. Average sizes of suffix trees and DAWGs. Disrete Applied Mathematis, 24:37 45, [4] M.T. Chen and J. Siferas. Effiient and elegant suword tree onstrution. In Cominatorial algorithms on words, pages Springer-Verlag, [5] J.D. Cook. Counting primitve it strings /12/23/ounting-primitive-it-strings/, [Online; aessed 02-May-2016]. [6] L. Devroye, W. Szpankowski, and B. Rais. A note on the height of suffix trees. SIAM Journal on Computing, 21:48 53, [7] E. R. Fiala and D. H. Greene. Data ompression with finite windows. Communiations of the ACM, 32: , [8] C. Fraser, A. Wendt, and E.W. Myers. Analyzing and ompressing assemly ode. In Proeedings SIGPLAN Symposium on Compiler Constrution, pages , [9] E.N. Gilert and J. Riordan. Symmetry types of periodi sequenes. Illinois Journal of Mathematis, 5: , [10] R. Grossi and G.F. Italiano. Suffix trees and their appliations in string algorithms. In Proeedings of the 1st South Amerian Workshop on String Proessing, pages 57 76, [11] D. Gusfield. Algorithms on Strings, Trees and Sequenes. Camridge University Press, [12] M. Höhl, S. Kurtz, and E. Ohleush. Effiient multiple genome alignment. Bioinformatis, 18:312S 320S, [13] L. Kaderali and A. Shliep. Seleting signature oligonuleotides to identify organisms using DNA arrays. Bioinformatis, 18: , 2002.

10 832 Bálint Vásárhelyi [14] E. M. MCreight. A spae-eonomial suffix tree onstrution algorithm. Journal of the ACM, 23: , [15] L. O Connor and T. Snider. Suffix trees and string omplexity. In Advanes in Cryptology: Proeedings of EUROCRYPT, LNCS 658, pages Springer-Verlag, [16] M. Rodeh. A fast test for unique deipheraility ased on suffix trees,. IEEE Transations on Information Theory, 28(4): , [17] S.L. Tanimoto. A method for deteting struture in polygons. Pattern Reognition, 13: , [18] E. Ukkonen. On-line onstrution of suffix trees. Algorithmia, 14: , [19] P. Weiner. Linear pattern mathing algorithms. In Proeedings of the 14th IEEE Symposium on Swithing and Automata Theory, pages 1 11, Reeived 13th July 2015