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# «Abstract. A factor u of a word w is a cover of w if every position in w lies within some occurrence of u in w. A factor u is a seed of w if it is a ...»

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Eﬃcient Algorithms

for Shortest Partial Seeds in Words

Tomasz Kociumaka1, Solon P. Pissis2, Jakub Radoszewski1,

Wojciech Rytter1,3, and Tomasz Wale´1n

Faculty of Mathematics, Informatics and Mechanics,

University of Warsaw, Warsaw, Poland

Department of Informatics, King’s College London,

London WC2R 2LS, UK

solon.pissis@kcl.ac.uk

Faculty of Mathematics and Computer Science,

Copernicus University, Toru´, Poland

n

Abstract. A factor u of a word w is a cover of w if every position in w lies within some occurrence of u in w. A factor u is a seed of w if it is a cover of a superstring of w. Covers and seeds extend the classical notions of periodicity. We introduce a new notion of α-partial seed, that is, a factor covering as a seed at least α positions in a given word. We use the Cover Suﬃx Tree, introduced recently in the context of α-partial covers (Kociumaka et al, CPM 2013); an O(n log n)-time algorithm constructing such a tree is known. However it appears that partial seeds are more complicated than partial covers—our algorithms require algebraic manipulations of special functions related to edges of the modiﬁed Cover Suﬃx Tree and the border array. We present an algorithm for computing shortest α-partial seeds that works in O(n) time if the Cover Suﬃx Tree is already given.

1 Introduction Periodicity in words is a fundamental topic in combinatorics on words and string algorithms (see [5]). The concept of quasiperiodicity is a generalization of the notion of periodicity [1]. Quasiperiodicity enables detecting repetitive structure of words when it cannot be found using the classical characterizations of periods.

Several types of quasiperiods have already been introduced. It depends on the type of quasiperiod what kinds of repetitive structure it allows to detect.

The best-known type of quasiperiodicity is the cover of word. A factor u of a word w is said to be a cover of w if every position in w lies within some occurrence of u in w, we also say that w is covered by u. An extension of the notion of cover is the notion of seed, in this case the positions covered by a seed Supported by Polish budget funds for science in 2013-2017 as a research project under the ‘Diamond Grant’ program.

u are also positions within overhanging occurrences of u. More formally, u is a seed of w if w is a factor of a word y covered by u.

Several algorithms are known for computation of covers and seeds. A lineartime algorithm for computing the shortest cover of a word was proposed by Apostolico et al. [2], and a linear-time algorithm for computing all the covers was proposed by Moore & Smyth [12]. Linear-time algorithms providing yet more complete characterizations of covers by so-called cover arrays were given in [3, 11]. Seeds were ﬁrst introduced by Iliopoulos, Moore, and Park [7] who presented an O(n log n)-time algorithm computing seeds. This result was improved recently by Kociumaka et al. [8] who gave a complex linear-time algorithm.

It remains unlikely that an arbitrary word has a cover or a seed shorter than the word itself. Due to this reason, relaxed variants of quasiperiodicity have been introduced. One of the ideas are approximate covers [13] and approximate seeds [4] that require each position to lie within an approximate occurrence of the corresponding quasiperiod. Another idea, introduced recently in [9], was the notion of partial cover that is required to cover a certain number of positions of a word. We extend the ideas of [9] and introduce the notion of partial seed.

Let C(u, w) denote the number of positions in w covered by (full) occurrences of the word u in w. The word u is called an α-partial cover of w if C(u, w) ≥ α.

We call a non-empty preﬁx of w that is also a suﬃx of u a left-overhanging occurrence of u in w. Symmetrically, a non-empty suﬃx of w which is a preﬁx of u is called a right-overhanging occurrence. Let S(u, w) denote the number of positions in w covered by full, left-overhanging, or right-overhanging occurrences of u in w. We call u an α-partial seed of w if S(u, w) ≥ α. If the word w is clear from the context, we use the simpler notations of C(u) and S(u).

Example 1. If w = aaaabaabaaaaaba, see also Fig.

1, then S(abaa) = 12, S(aba) = 10, S(ab) = 7, S(a) = 12.

–  –  –

Fig. 1. The positions covered by abaa as a partial seed are underlined. The word abaa is a 12-partial seed of w, it has four overhanging occurrences and two full occurrences.

Note that a is the shortest 12-partial seed of w.

We study the following two related problems.

PartialSeeds Input: a word w of length n and a positive integer α ≤ n Output: all shortest factors u of w such that S(u, w) ≥ α LimitedLengthPartialSeeds Input: a word w of length n and an interval [, r], 0 ≤ r ≤ n Output: a factor u of w, |u| ∈ [, r], which maximizes S(u, w) In [9] a data structure called the Cover Suﬃx Tree and denoted by CST (w) was introduced. For a word w of length n the size of CST (w) is O(n) and the construction time is O(n log n). In this article, we obtain the following results.

Theorem 2. Given CST (w), the LimitedLengthPartialSeeds problem can be solved in O(n) time.

By applying binary search, Theorem 2 implies an O(n log n)-time solution to the PartialSeeds problem. However, this solution can be improved to an O(n)time algorithm, provided that CST (w) is known.

Theorem 3. [Main result] Given CST (w), the PartialSeeds problem can be solved in O(n) time.

Structure of the paper. In Section 2 we introduce basic notation related to words and suﬃx trees and recall the Cover Suﬃx Tree. Next in Section 3 we extend CST to obtain its counterpart suitable for computation of partial seeds, which we call the Seed Suﬃx Tree (SST ). In Section 4 we introduce two abstract problems formulated in terms of simple functions which encapsulate the intrinsic diﬃculty of the two types of PartialSeeds problems. We present the solutions of the

Abstract

problems in Section 5; this section is essentially the most involved part of our contribution. We summarize our results in the Conclusions section.

2 Preliminaries

Let us ﬁx a word w of length n over a totally ordered alphabet Σ. For a factor v of w, by Occ(v) we denote the set of positions where occurrences of v in w start.

By ﬁrst(v) and last(v) we denote min Occ(v) and max Occ(v), respectively.

By w[i.. j] we denote the factor starting at the position i and ending at the position j. Factors w[1.. i] are called preﬁxes of w, and factors w[i.. n] are called suﬃxes of w. Words shorter than w that are both preﬁxes and suﬃxes of w are called borders of w. By β(w) we denote the length of the longest border of w.

The border array β[1.. n] and reverse border array β R [1.. n] of w are deﬁned as follows: β[i] = β(w[1.. i]) and β R [i] = β(w[i.. n]). The arrays β, β R can be constructed in O(n) time [6].

The suﬃx tree of w, denoted by ST (w), is the compacted suﬃx trie of w in which only branching nodes and suﬃxes are explicit. We identify the nodes of ST (w) with the factors of w that they represent. An augmented suﬃx tree may contain some additional explicit nodes, called extra nodes. For an explicit node v = ε, we set path(v) = (v0, v1,..., vk ) where v0 = v and v1,..., vk are the implicit nodes on the path going upwards from v to its nearest explicit ancestor.

E.g., in the right tree in Fig. 2 we have path(v) = (v, v1, v2, v3, v4, v5 ). We deﬁne the locus of a factor v of w as a pair (v, j) such that v = vj where vj ∈ path(v).

The Cover Suﬃx Tree is an augmented version of a suﬃx tree introduced in [9] that allows to eﬃciently compute C(v) for any explicit or implicit node, as shown in the following theorem.

Theorem 4 ([9]). Let w be a word of length n. There exists an augmented suﬃx tree of size O(n), such that for each edge path(v) we have C(vj ) = c(v) − j∆(v) for some positive integers c(v), ∆(v). Such a tree together with values c(v), ∆(v), denoted as CST (w), can be constructed in O(n log n) time and O(n) space.

Actually [9] provides explicit formulas for c(v), ∆(v) in terms of Occ(v). Their form is not important here; the only property which we use is that 1 ≤ ∆(v) ≤ |Occ(v)|.

3 Seed Suﬃx Tree CST introduces some extra nodes to ST thanks to which the cover index C(vj ) on each edge becomes a linear function: C(vj ) = c(v) − j∆(v). With seed index S(vj ), the situation is more complex. However, if we make some more nodes explicit, then S(vj ) becomes a relatively simple function. We call the resulting tree the Seed Suﬃx Tree, denoted by SST (w).

Lemma 5. Let w be a word of length n.

We can construct an augmented suﬃx tree, denoted by SST (w), of size O(n) such that for each node v there exists a function φv (x) = av x + bv + min(cv, β[x]) and a range Rv = ( v, rv ] such that for all vj ∈ path(v) we have S(vj ) = φv (rv − j). Additionally, 0 ≤ av ≤ |Occ(v)|.

The tree SST (w), together with the border array β and tuples (av, bv, cv, v, rv ) representing φv, can be constructed in O(n) time given CST (w).

–  –  –

Fig. 2. In CST (w) there is a constant-space description of a linear function σv associated with each explicit node v, which gives the values of C(vj ) for implicit nodes on the edge from v upwards. In SST (w) there is a corresponding function ψv which is a combination of the linear function σv and two functions depending on border arrays.

After suitable linear transformation of variable j, the function ψv (j) is converted to a more convenient function φv (x). When transforming CST (w) to SST (w), some implicit nodes are made explicit to guarantee that φv has a simple form (v on the ﬁgure).

Proof. For any factor v of w we deﬁne:

–  –  –

The following observation relates these values to S(v), see also Fig. 3.

Claim. S(v) = C(v) + LeftS(v) + RightS(v).

Proof (of the claim). C(v) counts all positions covered by full occurrences of v.

We claim that the remaining positions covered by left-overhanging occurrences are counted by LeftS(v). Let p = ﬁrst(v) + |v| − 1. Note that w[1.. p] has v as a suﬃx, which means that β[p] is the length of the longest left-overhanging occurrence of v. It covers β[p] positions, but, among them, positions greater than or equal to ﬁrst(v) are already covered by a full occurrence of v. RightS(v) has a symmetric interpretation.

–  –  –

Fig. 3. The positions contained in S(v) are marked gray. In this case LeftS(v) = β[ﬁrst(v) + |v| − 1], and RightS(v) = n − |v| + 1 − last(v).

–  –  –

We consider the function S(vj ) = C(vj ) + LeftS(vj ) + RightS(vj ). Note that only C(vj ) is a linear function of j. Also, RightS(vj ) is relatively simple. It either already is a linear function in the whole {0,..., k}, or it becomes one in both parts of {0,..., k} if we split it at j ∈ {0,..., k} such that β R [last(v)] = n − |v| + j + 1 − last(v). We subdivide each path(v) at vj if such j exists. Note that we can easily update values c and ∆ for newly created edges. Also, we make explicit at most O(n) nodes (at most one per edge of CST (w)), so the resulting tree SST (w) has O(n) explicit nodes in total.

It remains to show that after these transformations S(vj ) = φv (rv − j) for rv = ﬁrst(v) + |v| − 1 and that the coeﬃcients of φv can be eﬃciently computed.

We omit the explicit formulae in this version, they can be obtained with just a few simple algebraic transformations. The additional inequality 0 ≤ av ≤ |Occ(v)| follows from the property that 1 ≤ ∆(v) ≤ |Occ(v)|.

The following observation is a direct consequence of Lemma 5.

Observation 6. Given SST (w) and a locus of u, a factor of w, one can compute S(u) in constant time.

4 Reduction to two abstract problems

–  –  –

Observation 7. Any border array is a linear-oscillation array.

To solve the LimitedLengthPartialSeeds problem, we make explicit all nodes corresponding to factors of length − 1 and r. This way each edge either contains only nodes at tree levels in {,..., r} or none of them. Note that the functions φv on the subdivided edges stay the same, only the ranges shrink.

Consider the following abstract problem.

Problem A1 Input: a linear-oscillation array B of size n and m pairs (φi, Ri ), where φi is a function φi (x) = ai x+bi +min(ci, B[x]) and Ri = ( i, ri ] ⊆ [1, n] is a non-empty range Output: the values xi = argmax{φi (x) : x ∈ Ri }.3 Applying Problem A1 for B = β and a query for each edge path(v), we obtain vj ∈ path(v) maximizing S(vj ). Taking a global maximum over all edges containing factors of lengths within {,..., r}, we get the sought factor u, which maximizes S(u) among all factors of w with |u| ∈ {,..., r}. This results in the following lemma.

Lemma 8. Given SST (w) and an O(n + m)-time oﬀ-line solution to Problem A1, the LimitedLengthPartialSeeds problem can be solved in O(n) time.

To solve the PartialSeeds problem, we also apply Problem A1 to compute max S(vj ) for each edge path(v) of SST (w) (this time we do not introduce any additional extra nodes). We say that an edge path(v) is feasible, if max S(vj ) ≥ α, and important if it is feasible and no ancestor edge is feasible. It is easy to see For a set X and a function f : X → R, we deﬁne argmax{f (x) : x ∈ X} as the

largest argument for which f attains its maximum value, that is, max{x ∈ X :

∀x ∈X f (x) ≥ f (x )} (we use the maximality of x later on). We assume max ∅ = −∞ and min ∅ = ∞.

that all shortest α-partial seeds lie on important edges. Also, by Lemma 5, av summed over all feasible edges ev is at most n. Consider the following abstract problem.

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