We derive a simple efficient algorithm for Abelian periods knowing all
Abelian squares in a string. An efficient algorithm for the latter problem was
given by Cummings and Smyth in 1997. By the way we show an alternative
algorithm for Abelian squares. We also obtain a linear time algorithm finding
all `long' Abelian periods. The aim of the paper is a (new) reduction of the
problem of all Abelian periods to that of (already solved) all Abelian squares
which provides new insight into both connected problems

Crochemore, Maxime

Iliopoulos, Costas

Kociumaka, Tomasz

Kubica, Marcin

Pachocki, Jakub

Radoszewski, Jakub

Rytter, Wojciech

Tyczyński, Wojciech

Waleń, Tomasz

English

arXiv

International audienceWe derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all "long" Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems

S. Iliopoulos, Costas

Tyczynski, Wojciech

Walen, Tomasz

Hal-Diderot

A note on efficient computation of all Abelian periods in a string

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all "long" Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems.</p

Crochemore, M.

Iliopoulos, C. S.

Kociumaka, T.

Kubica, M.

Pachocki, J.

Radoszewski, J.

Rytter, W.

Tyczyński, W.

Waleń, T.

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A note on efficient computation of all Abelian periods ina stringMaxime Crochemore, Costas S. Iliopoulos, Tomasz Kociumaka, MarcinKubica, Jakub Pachocki, Jakub Radoszewski, Wojciech Rytter, WojciechTyczynski, Tomasz WalenTo cite this version:Maxime Crochemore, Costas S. Iliopoulos, Tomasz Kociumaka, Marcin Kubica, Jakub Pa-chocki, et al.. A note on efficient computation of all Abelian periods in a string. Informa-tion Processing Letters, Elsevier, 2013, 113 (3), pp.74-77. <10.1016/j.ipl.2012.11.001>. <hal-00837005>HAL Id: hal-00837005https://hal-upec-upem.archives-ouvertes.fr/hal-00837005Submitted on 26 Jun 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s. A Note on Efficient Computation of All Abelian Periodsin a StringM. Crochemorea,b, C. S. Iliopoulosa,c, T. Kociumakad, M. Kubicad,J. Pachockid, J. Radoszewskid,∗, W. Rytterd,e,1, W. Tyczyńskid, T. Waleńf,daKing’s College London, London WC2R 2LS, UKbUniversité Paris-Est, FrancecDigital Ecosystems & Business Intelligence Institute, Curtin University of Technology,Perth WA 6845, AustraliadFaculty of Mathematics, Informatics and Mechanics, University of Warsaw,ul. Banacha 2, 02-097 Warsaw, PolandeDept. of Math. and Informatics, Copernicus University, ul. Chopina 12/18,87-100 Toruń, PolandfLaboratory of Bioinformatics and Protein Engineering, International Institute of Molecularand Cell Biology in Warsaw, PolandAbstractWe derive a simple eﬃcient algorithm for Abelian periods knowing all Abeliansquares in a string. An eﬃcient algorithm for the latter problem was given byCummings and Smyth in 1997. By the way we show an alternative algorithmfor Abelian squares. We also obtain a linear time algorithm ﬁnding all “long”Abelian periods. The aim of the paper is a (new) reduction of the problem of allAbelian periods to that of (already solved) all Abelian squares which providesnew insight into both connected problems.Keywords: algorithms, Abelian period, Abelian square1. IntroductionWe present an eﬃcient reduction of the Abelian period problem to theAbelian square problem. For a string of length n the latter problem was solvedin O(n2) by Cummings and Smyth [7]. The best previously known algorithmsfor the Abelian periods, see [12], worked in O(n2m) time (where m is the al-phabet size) which for large m is O(n3). Our algorithm works in O(n2) time,independently of the alphabet size. As a by-product we obtain an alternative∗Corresponding author. Tel.: +48-22-55-44-484, fax: +48-22-55-44-400.Email addresses: maxime.crochemore@kcl.ac.uk (M. Crochemore),c.iliopoulos@kcl.ac.uk (C. S. Iliopoulos), kociumaka@mimuw.edu.pl (T. Kociumaka),kubica@mimuw.edu.pl (M. Kubica), pachocki@mimuw.edu.pl (J. Pachocki),jrad@mimuw.edu.pl (J. Radoszewski), rytter@mimuw.edu.pl (W. Rytter),w.tyczynski@mimuw.edu.pl (W. Tyczyński), walen@mimuw.edu.pl (T. Waleń)1The author is supported by grant no. N206 566740 of the National Science Centre.O(n2) time algorithm ﬁnding all Abelian squares and an O(n) time algorithmﬁnding a compact representation of all Abelian periods of length greater thann/2, in particular, the shortest such period.Abelian squares were ﬁrst studied by Erdös [11], who posed a question onthe smallest alphabet size for which there exists an inﬁnite Abelian-square-free string. An example of such a string over ﬁve-letter alphabet was given byPleasants [16] and afterwards the best possible example over four-letter alphabetwas shown by Keränen [13].Quite recently there have been several results on Abelian complexity in words[1, 4, 8, 9, 10] and partial words [2, 3] and on Abelian pattern matching [5,14, 15]. Abelian periods were ﬁrst deﬁned and studied by Constantinescu andIlie [6].We say that two strings are (commutatively) equivalent, and write x ≡ y, ifone can be obtained from the other by permuting its symbols. In other words,the Parikh vectors P(x), P(y) are equal, where the Parikh vector gives frequencyof each symbol of the alphabet in a given string. Parikh vectors were introducedalready in [6] for this problem.A string w is an Abelian k-power if w = x1x2 . . . xk, wherex1 ≡ x2 ≡ . . . ≡ xkThe size of x1 is called the base of the k-power. In particular w is an Abeliansquare if and only if it is an Abelian 2-power.A string x is an Abelian factor of y if P(x) ≤ P(y), that is, each element ofP(x) is smaller than the corresponding element of P(y). The pair (i, p) is anAbelian period of w = w[1, n] if and only if w[i+1, j] is an Abelian k-power withbase p (for some k) and w[1, i] and w[j+1, n] are Abelian factors of w[i+1, i+p],see Fig. 1. Here p is called the length of the period.c a a a b a c b b b a a b c b b a b c a b b c b aFigure 1: A word of length 25 with an Abelian period (i = 3, p = 6). This period impliestwo Abelian squares: abacbbbaabcb and baabcbbabcab.In Section 2 we introduce two auxiliary tables that we use in computingAbelian squares and powers. Next in Section 3 we show new O(n2) time algo-rithms for all Abelian squares and all Abelian periods in a string and a reductionbetween these problems.Finally in Section 4 we present an O(n) time algorithm ﬁnding a compactrepresentation of all “long” Abelian periods. DeﬁneMinLong(i) = min{p > n/2 : (i, p) is an Abelian period of w}.If no such p exists, we set MinLong(i) =∞. All long Abelian periods are of theform (i, p) where p ≥ MinLong(i), the table MinLong is a compact O(n) spacerepresentation of potentially quadratic set of long Abelian periods.22. Auxiliary tablesLet w be a string of length n. Assume its positions are numbered from 1 to n,w = w1w2 . . . wn. By w[i, j] we denote the factor of w of the form wiwi+1 . . . wj .Factors of the form w[1, i] are called preﬁxes of w and factors of the form w[i, n]are called suﬃxes of w.We introduce the following table:head(i, j) = minimum k such that P(w[i, j]) ≤ P(w[j + 1, j + k]).If no such k exists, we set head(i, j) =∞, and if j < i, we set head(i, j) = 0. Inthe algorithm below we actually compute a slightly modiﬁed table head ′(i, j) =j + head(i, j).Example 1. For the infinite Fibonacci word F = abaababaabaababaababaa . . .the first several values of the table head(1, i) are:i 1 2 3 4 5 6 7 8 9 10 11 . . .F [i] a b a a b a b a a b a . . .head(1, i) 2 3 3 5 5 6 8 8 10 10 11 . . .We have here Abelian square prefixes of lengths 6, 10, 12, 16, 20, 22.We show how to compute the head ′ table in O(n2) time. The computationis performed in row-order of the table using the fact that it is non-decreasing:Observation 2. For any 1 ≤ i ≤ j < n, head ′(i, j) ≤ head ′(i, j + 1).We assume that the alphabet of w is Σ = {1, 2, . . . ,m} where m ≤ n. For aParikh vector Q, by Q[i] for i = 1, 2, . . . ,m we denote the number of occurrencesof the letter i. For two Parikh vectorsQ and R, we deﬁne their Parikh difference,denoted as Q−R, as a Parikh vector: (Q −R)[i] = Q[i]−R[i].In the algorithm we store the diﬀerence∆j = P(yj)−P(xj) of Parikh vectorsofxj = w[i, j] and yj = w[j + 1, k]where k = head ′(i, j). Note that ∆j [a] ≥ 0 for any a = 1, 2, . . . ,m.Assume we have computed head ′(i, j − 1) and ∆j−1. When we proceed toj, we move the letter w[j] from y to x and update ∆ accordingly. Thus at mostone element of ∆ might have dropped below 0. If there is no such element, weconclude that head ′(i, j) = head ′(i, j − 1) and that we have obtained ∆j = ∆.Otherwise we keep extending y to the right with new letters and updating ∆until all its elements become non-negative. We obtain the following algorithmCompute-head .Lemma 3. The head table can be computed in O(n2) time.Proof. The time complexity of the algorithm Compute-head is O(n2). Indeed,the total number of steps of the while-loop for a ﬁxed value of i is O(n), sinceeach step increases the variable k. ✷We also use the following tail table that is analogical to the head table:tail(i, j) = minimum k such that P(w[i, j]) ≤ P(w[i− k, i− 1]).3Algorithm Compute-head(w)for i := 1 to n do∆ := (0, 0, . . . , 0);∆[w[i]] := 1; {Boundary condition}k := i;for j := i to n do∆[w[j]] := ∆[w[j]] − 2;while (k < n) and (∆[w[j]] < 0) dok := k + 1;∆[w[k]] := ∆[w[k]] + 1;if ∆[w[j]] < 0 then k :=∞;head ′(i, j) := k; head(i, j) := head ′(i, j) − j;3. Abelian squares and Abelian periodsIn this section we show how Abelian periods can be inferred from Abeliansquares in a string.Deﬁne by maxpower (i, p) the maximal size of a preﬁx of w[i, n] which is anAbelian k-power with base p (for some k). Deﬁne square(i, p) = 1 if and onlyif maxpower (i, p) ≥ 2p. Cummings and Smyth [7] compute an alternative tablesquare ′(i, p), such that square ′(i, p) = 1 if and only if w[i − p + 1, i + p] is anAbelian square. These tables are clearly equivalent:square(i, p) = 1 ⇔ square ′(i + p− 1, p) = 1.The maxpower(i, p) table can be computed from the square(i, p) table in lineartime using a simple dynamic programming recurrence:maxpower (i, p) ={0 if n− i < p− 1p+ square(i, p) ·maxpower (i+ p, p) otherwise.(1)An alternative O(n2) time algorithm for computing the table square(i, p) fora string w of length n is a consequence of the following observation, see alsoExample 1.Observation 4. square(i, p) = 1 ⇔ head(i, i+ p− 1) = p.Theorem 5. All Abelian squares in a string of length n can be computed inO(n2) time.The following observation provides a constant-time condition for checkingan Abelian period.4Observation 6. (i, p) is an Abelian period of w if and only ifp ≥ head(1, i), tail(j, n)where j = i+ 1 +maxpower (i+ 1, p).We conclude with the following algorithm for computing Abelian periods. Inthe algorithm we use our alternative version of computing the table square fromhead , since the latter table is computed anyway (instead of that Cummings andSmyth’s algorithm can be used for Abelian squares).Algorithm Compute-Abelian-PeriodsCompute head(i, j), tail(i, j) using algorithm Compute-head ;Initialize the table maxpower to zero table;for p := 1 to n dofor i := n downto 1 doif i ≤ n− p+ 1 thenmaxpower (i, p) := p;if head(i, i+ p− 1) = p thenmaxpower (i, p) := p+maxpower(i+ p, p);for i := 0 to n− 1 dofor p := 1 to n− i doj := i+ 1 +maxpower (i+ 1, p);if (p ≥ head(1, i)) and (p ≥ tail(j, n)) thenReport an Abelian period (i, p);Theorem 7. All Abelian periods of a string of length n can be computed inO(n2) time.4. Long Abelian periodsIn this section we show how to compute the table MinLong(i), see the ex-ample in the table below.i 0 1 2 3 4 5 6 7 8 9 10 11 12 13w[i] c a a b b c a b b c a a aMinLong(i) 7 7 9 8 7 7 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞For a non-decreasing function f : {1, 2, . . . , n+1} → {−∞}∪{1, 2, . . . , n+1}deﬁne the functionfˆ(i) = min{j : f(j) > i}.If the minimum is undeﬁned then we set fˆ(i) =∞.5Observation 8. Let f be a function non-decreasing and computable in constanttime. Then all the values of fˆ can be computed in linear time.Theorem 9. A compact representation of all long Abelian periods can be com-puted in linear time.Proof. Let us take f(j) = j − tail(j, n). This function is non-decreasing, seealso Observation 2. Then for i < n2we have:MinLong(i) = max{⌊n2⌋+ 1, head(1, i), fˆ(i)− i− 1}and otherwise MinLong(i) =∞, see also Fig. 2.w[1, i] w[i+ 1, j − 1] w[j, n]p = j − i− 1head(1, i)tail(j, n)Figure 2: A schematic view of a long Abelian period: p > n2, p ≥ head(1, i), tail(j, n).Hence the computation of MinLong table is reduced to linear time algorithmfor fˆ and the conclusion of the theorem follows from Observation 8. ✷References[1] S. V. Avgustinovich, A. Glen, B. V. Halldórsson, and S. Kitaev. On short-est crucial words avoiding Abelian powers. Discrete Applied Mathematics,158(6):605–607, 2010.[2] F. Blanchet-Sadri, J. I. Kim, R. Mercas, W. Severa, and S. Simmons.Abelian square-free partial words. In A. H. Dediu, H. Fernau, andC. Martín-Vide, editors, LATA, volume 6031 of Lecture Notes in ComputerScience, pages 94–105. Springer, 2010.[3] F. Blanchet-Sadri and S. Simmons. Avoiding Abelian powers in partialwords. In Mauri and Leporati [14], pages 70–81.[4] J. Cassaigne, G. Richomme, K. Saari, and L. Q. Zamboni. Avoiding Abelianpowers in binary words with bounded Abelian complexity. Int. J. Found.Comput. Sci., 22(4):905–920, 2011.6[5] F. Cicalese, G. Fici, and Z. Lipták. Searching for jumbled patterns instrings. In J. Holub and J. Žďárek, editors, Proceedings of the PragueStringology Conference 2009, pages 105–117, Czech Technical University inPrague, Czech Republic, 2009.[6] S. Constantinescu and L. Ilie. Fine and Wilf’s theorem for Abelian periods.Bulletin of the EATCS, 89:167–170, 2006.[7] L. J. Cummings and W. F. Smyth. Weak repetitions in strings. J. Combi-natorial Math. and Combinatorial Computing, 24:33–48, 1997.[8] J. D. Currie and A. Aberkane. A cyclic binary morphism avoiding Abelianfourth powers. Theor. Comput. Sci., 410(1):44–52, 2009.[9] J. D. Currie and T. I. Visentin. Long binary patterns are Abelian 2-avoidable. Theor. Comput. Sci., 409(3):432–437, 2008.[10] M. Domaratzki and N. Rampersad. Abelian primitive words. In Mauri andLeporati [14], pages 204–215.[11] P. Erdös. Some unsolved problems. Hungarian Academy of Sciences Mat.Kutató Intézet Közl., 6:221–254, 1961.[12] G. Fici, T. Lecroq, A. Lefebvre, and É. Prieur-Gaston. Computing Abelianperiods in words. In J. Holub and J. Žďárek, editors, Proceedings of thePrague Stringology Conference 2011, pages 184–196, Czech Technical Uni-versity in Prague, Czech Republic, 2011.[13] V. Keränen. Abelian squares are avoidable on 4 letters. In W. Kuich,editor, ICALP, volume 623 of Lecture Notes in Computer Science, pages41–52. Springer, 1992.[14] G. Mauri and A. Leporati, editors. Developments in Language Theory -15th International Conference, DLT 2011, Milan, Italy, July 19-22, 2011.Proceedings, volume 6795 of Lecture Notes in Computer Science. Springer,2011.[15] T. M. Moosa and M. S. Rahman. Indexing permutations for binary strings.Inf. Process. Lett., 110(18-19):795–798, 2010.[16] P. A. Pleasants. Non-repetitive sequences. Proc. Cambridge Phil. Soc.,68:267–274, 1970.7

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A note on efficient computation of all Abelian periods in
a string
Maxime Crochemore, Costas S. Iliopoulos, Tomasz Kociumaka, Marcin
Kubica, Jakub Pachocki, Jakub Radoszewski, Wojciech Rytter, Wojciech
Tyczynski, Tomasz Walen
To cite this version:
Maxime Crochemore, Costas S. Iliopoulos, Tomasz Kociumaka, Marcin Kubica, Jakub Pachocki, et
al.. A note on efficient computation of all Abelian periods in a string. Information Processing Letters,
Elsevier, 2013, 113 (3), pp.74-77. ￿10.1016/j.ipl.2012.11.001￿. ￿hal-00837005￿
 A Note on Efficient Computation of All Abelian Periods
in a String
M. Crochemorea,b, C. S. Iliopoulosa,c, T. Kociumakad, M. Kubicad,
J. Pachockid, J. Radoszewskid,∗, W. Rytterd,e,1, W. Tyczyńskid, T. Waleńf,d
aKing’s College London, London WC2R 2LS, UK
bUniversité Paris-Est, France
cDigital Ecosystems & Business Intelligence Institute, Curtin University of Technology,
Perth WA 6845, Australia
dFaculty of Mathematics, Informatics and Mechanics, University of Warsaw,
ul. Banacha 2, 02-097 Warsaw, Poland
eDept. of Math. and Informatics, Copernicus University, ul. Chopina 12/18,
87-100 Toruń, Poland
fLaboratory of Bioinformatics and Protein Engineering, International Institute of Molecular
and Cell Biology in Warsaw, Poland
Abstract
We derive a simple efficient algorithm for Abelian periods knowing all Abelian
squares in a string. An efficient algorithm for the latter problem was given by
Cummings and Smyth in 1997. By the way we show an alternative algorithm
for Abelian squares. We also obtain a linear time algorithm finding all “long”
Abelian periods. The aim of the paper is a (new) reduction of the problem of all
Abelian periods to that of (already solved) all Abelian squares which provides
new insight into both connected problems.
Keywords: algorithms, Abelian period, Abelian square
1. Introduction
We present an efficient reduction of the Abelian period problem to the
Abelian square problem. For a string of length n the latter problem was solved
in O(n2) by Cummings and Smyth [7]. The best previously known algorithms
for the Abelian periods, see [12], worked in O(n2m) time (where m is the al-
phabet size) which for large m is O(n3). Our algorithm works in O(n2) time,
independently of the alphabet size. As a by-product we obtain an alternative
∗Corresponding author. Tel.: +48-22-55-44-484, fax: +48-22-55-44-400.
Email addresses: maxime.crochemore@kcl.ac.uk (M. Crochemore),
c.iliopoulos@kcl.ac.uk (C. S. Iliopoulos), kociumaka@mimuw.edu.pl (T. Kociumaka),
kubica@mimuw.edu.pl (M. Kubica), pachocki@mimuw.edu.pl (J. Pachocki),
jrad@mimuw.edu.pl (J. Radoszewski), rytter@mimuw.edu.pl (W. Rytter),
w.tyczynski@mimuw.edu.pl (W. Tyczyński), walen@mimuw.edu.pl (T. Waleń)
1The author is supported by grant no. N206 566740 of the National Science Centre.
O(n2) time algorithm finding all Abelian squares and an O(n) time algorithm
finding a compact representation of all Abelian periods of length greater than
n/2, in particular, the shortest such period.
Abelian squares were first studied by Erdös [11], who posed a question on
the smallest alphabet size for which there exists an infinite Abelian-square-
free string. An example of such a string over five-letter alphabet was given by
Pleasants [16] and afterwards the best possible example over four-letter alphabet
was shown by Keränen [13].
Quite recently there have been several results on Abelian complexity in words
[1, 4, 8, 9, 10] and partial words [2, 3] and on Abelian pattern matching [5,
14, 15]. Abelian periods were first defined and studied by Constantinescu and
Ilie [6].
We say that two strings are (commutatively) equivalent, and write x ≡ y, if
one can be obtained from the other by permuting its symbols. In other words,
the Parikh vectors P(x), P(y) are equal, where the Parikh vector gives frequency
of each symbol of the alphabet in a given string. Parikh vectors were introduced
already in [6] for this problem.
A string w is an Abelian k-power if w = x1x2 . . . xk, where
x1 ≡ x2 ≡ . . . ≡ xk
The size of x1 is called the base of the k-power. In particular w is an Abelian
square if and only if it is an Abelian 2-power.
A string x is an Abelian factor of y if P(x) ≤ P(y), that is, each element of
P(x) is smaller than the corresponding element of P(y). The pair (i, p) is an
Abelian period of w = w[1, n] if and only if w[i+1, j] is an Abelian k-power with
base p (for some k) and w[1, i] and w[j+1, n] are Abelian factors of w[i+1, i+p],
see Fig. 1. Here p is called the length of the period.
c a a a b a c b b b a a b c b b a b c a b b c b a
Figure 1: A word of length 25 with an Abelian period (i = 3, p = 6). This period implies
two Abelian squares: abacbbbaabcb and baabcbbabcab.
In Section 2 we introduce two auxiliary tables that we use in computing
Abelian squares and powers. Next in Section 3 we show new O(n2) time algo-
rithms for all Abelian squares and all Abelian periods in a string and a reduction
between these problems.
Finally in Section 4 we present an O(n) time algorithm finding a compact
representation of all “long” Abelian periods. Define
MinLong(i) = min{p > n/2 : (i, p) is an Abelian period of w}.
If no such p exists, we set MinLong(i) =∞. All long Abelian periods are of the
form (i, p) where p ≥ MinLong(i), the table MinLong is a compact O(n) space
representation of potentially quadratic set of long Abelian periods.
2
2. Auxiliary tables
Let w be a string of length n. Assume its positions are numbered from 1 to n,
w = w1w2 . . . wn. By w[i, j] we denote the factor of w of the form wiwi+1 . . . wj .
Factors of the form w[1, i] are called prefixes of w and factors of the form w[i, n]
are called suffixes of w.
We introduce the following table:
head(i, j) = minimum k such that P(w[i, j]) ≤ P(w[j + 1, j + k]).
If no such k exists, we set head(i, j) =∞, and if j < i, we set head(i, j) = 0. In
the algorithm below we actually compute a slightly modified table head ′(i, j) =
j + head(i, j).
Example 1. For the infinite Fibonacci word F = abaababaabaababaababaa . . .
the first several values of the table head(1, i) are:
i 1 2 3 4 5 6 7 8 9 10 11 . . .
F [i] a b a a b a b a a b a . . .
head(1, i) 2 3 3 5 5 6 8 8 10 10 11 . . .
We have here Abelian square prefixes of lengths 6, 10, 12, 16, 20, 22.
We show how to compute the head ′ table in O(n2) time. The computation
is performed in row-order of the table using the fact that it is non-decreasing:
Observation 2. For any 1 ≤ i ≤ j < n, head ′(i, j) ≤ head ′(i, j + 1).
We assume that the alphabet of w is Σ = {1, 2, . . . ,m} where m ≤ n. For a
Parikh vector Q, by Q[i] for i = 1, 2, . . . ,m we denote the number of occurrences
of the letter i. For two Parikh vectorsQ and R, we define their Parikh difference,
denoted as Q−R, as a Parikh vector: (Q −R)[i] = Q[i]−R[i].
In the algorithm we store the difference∆j = P(yj)−P(xj) of Parikh vectors
of
xj = w[i, j] and yj = w[j + 1, k]
where k = head ′(i, j). Note that ∆j [a] ≥ 0 for any a = 1, 2, . . . ,m.
Assume we have computed head ′(i, j − 1) and ∆j−1. When we proceed to
j, we move the letter w[j] from y to x and update ∆ accordingly. Thus at most
one element of ∆ might have dropped below 0. If there is no such element, we
conclude that head ′(i, j) = head ′(i, j − 1) and that we have obtained ∆j = ∆.
Otherwise we keep extending y to the right with new letters and updating ∆
until all its elements become non-negative. We obtain the following algorithm
Compute-head .
Lemma 3. The head table can be computed in O(n2) time.
Proof. The time complexity of the algorithm Compute-head is O(n2). Indeed,
the total number of steps of the while-loop for a fixed value of i is O(n), since
each step increases the variable k. ✷
We also use the following tail table that is analogical to the head table:
tail(i, j) = minimum k such that P(w[i, j]) ≤ P(w[i− k, i− 1]).
3
Algorithm Compute-head(w)
for i := 1 to n do
∆ := (0, 0, . . . , 0);
∆[w[i]] := 1; {Boundary condition}
k := i;
for j := i to n do
∆[w[j]] := ∆[w[j]] − 2;
while (k < n) and (∆[w[j]] < 0) do
k := k + 1;
∆[w[k]] := ∆[w[k]] + 1;
if ∆[w[j]] < 0 then k :=∞;
head ′(i, j) := k; head(i, j) := head ′(i, j) − j;
3. Abelian squares and Abelian periods
In this section we show how Abelian periods can be inferred from Abelian
squares in a string.
Define by maxpower (i, p) the maximal size of a prefix of w[i, n] which is an
Abelian k-power with base p (for some k). Define square(i, p) = 1 if and only
if maxpower (i, p) ≥ 2p. Cummings and Smyth [7] compute an alternative table
square ′(i, p), such that square ′(i, p) = 1 if and only if w[i − p + 1, i + p] is an
Abelian square. These tables are clearly equivalent:
square(i, p) = 1 ⇔ square ′(i + p− 1, p) = 1.
The maxpower(i, p) table can be computed from the square(i, p) table in linear
time using a simple dynamic programming recurrence:
maxpower (i, p) =
{
0 if n− i < p− 1
p+ square(i, p) ·maxpower (i+ p, p) otherwise.
(1)
An alternative O(n2) time algorithm for computing the table square(i, p) for
a string w of length n is a consequence of the following observation, see also
Example 1.
Observation 4. square(i, p) = 1 ⇔ head(i, i+ p− 1) = p.
Theorem 5. All Abelian squares in a string of length n can be computed in
O(n2) time.
The following observation provides a constant-time condition for checking
an Abelian period.
4
Observation 6. (i, p) is an Abelian period of w if and only if
p ≥ head(1, i), tail(j, n)
where j = i+ 1 +maxpower (i+ 1, p).
We conclude with the following algorithm for computing Abelian periods. In
the algorithm we use our alternative version of computing the table square from
head , since the latter table is computed anyway (instead of that Cummings and
Smyth’s algorithm can be used for Abelian squares).
Algorithm Compute-Abelian-Periods
Compute head(i, j), tail(i, j) using algorithm Compute-head ;
Initialize the table maxpower to zero table;
for p := 1 to n do
for i := n downto 1 do
if i ≤ n− p+ 1 then
maxpower (i, p) := p;
if head(i, i+ p− 1) = p then
maxpower (i, p) := p+maxpower(i+ p, p);
for i := 0 to n− 1 do
for p := 1 to n− i do
j := i+ 1 +maxpower (i+ 1, p);
if (p ≥ head(1, i)) and (p ≥ tail(j, n)) then
Report an Abelian period (i, p);
Theorem 7. All Abelian periods of a string of length n can be computed in
O(n2) time.
4. Long Abelian periods
In this section we show how to compute the table MinLong(i), see the ex-
ample in the table below.
i 0 1 2 3 4 5 6 7 8 9 10 11 12 13
w[i] c a a b b c a b b c a a a
MinLong(i) 7 7 9 8 7 7 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
For a non-decreasing function f : {1, 2, . . . , n+1} → {−∞}∪{1, 2, . . . , n+1}
define the function
fˆ(i) = min{j : f(j) > i}.
If the minimum is undefined then we set fˆ(i) =∞.
5
Observation 8. Let f be a function non-decreasing and computable in constant
time. Then all the values of fˆ can be computed in linear time.
Theorem 9. A compact representation of all long Abelian periods can be com-
puted in linear time.
Proof. Let us take f(j) = j − tail(j, n). This function is non-decreasing, see
also Observation 2. Then for i < n
2
we have:
MinLong(i) = max
{⌊n
2
⌋
+ 1, head(1, i), fˆ(i)− i− 1
}
and otherwise MinLong(i) =∞, see also Fig. 2.
w[1, i] w[i+ 1, j − 1] w[j, n]
p = j − i− 1
head(1, i)
tail(j, n)
Figure 2: A schematic view of a long Abelian period: p > n
2
, p ≥ head(1, i), tail(j, n).
Hence the computation of MinLong table is reduced to linear time algorithm
for fˆ and the conclusion of the theorem follows from Observation 8. ✷
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A Note on Efficient Computation of All Abelian Periods in a String

A Note on Efficient Computation of All Abelian Periods in a String

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