Mapping sequences by parts

1748-7188-2-11 1748-7188 Research Mapping sequences by parts Didier Gilles didier@iml.univ-mrs.fr Guziolowski Carito cvargas@irisa.fr

Institut de Mathématiques de Luminy, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France.

Projet Symbiose, IRISA – campus de Beaulieu, 35042 Rennes Cedex, France.

Algorithms for Molecular Biology 1748-7188 2007 2 1 11 http://www.almob.org/content/2/1/11 17880695 10.1186/1748-7188-2-11 02 2 2007 19 9 2007 19 9 2007 2007 Didier and Guziolowski; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background:

We present the N-map method, a pairwise and asymmetrical approach which allows us to compare sequences by taking into account evolutionary events that produce shuffled, reversed or repeated elements. Basically, the optimal N-map of a sequence s over a sequence t is the best way of partitioning the first sequence into N parts and placing them, possibly complementary reversed, over the second sequence in order to maximize the sum of their gapless alignment scores.

Results:

We introduce an algorithm computing an optimal N-map with time complexity O (|s| × |t| × N) using O (|s| × |t| × N) memory space. Among all the numbers of parts taken in a reasonable range, we select the value N for which the optimal N-map has the most significant score. To evaluate this significance, we study the empirical distributions of the scores of optimal N-maps and show that they can be approximated by normal distributions with a reasonable accuracy. We test the functionality of the approach over random sequences on which we apply artificial evolutionary events.

Practical Application:

The method is illustrated with four case studies of pairs of sequences involving non-standard evolutionary events.

Background

Classic alignments methods are unable to extract homologies involving shuffled, reverse-complemented or repeated elements between sequences, despite the fact that there are identified mechanisms of evolution of sequences which lead to such types of homologies. This can happen on large scale with genome rearrangements but it can also occur on a smaller scale, for instance within genes, with domain recombinations, duplications, exon shufflings, etc.

On the other hand, there are few methods allowing us to compare sequences with relaxed assumptions about conservation of linear order and one-to-one association of positions between sequences 123. In particular, as it is pairwise and asymmetrical, the approach proposed in 1 is similar to the work presented here. The authors introduce the transformation distance, similar to the Levenstein distance between sequences, which includes editing operations like transposition, duplication, etc. The algorithmic complexity of the computation of this distance, which was initially high, has been improved in 4.

However, the transformation distance has some drawbacks; mainly, it does not take into account mutations. In 2, the authors introduce the Glocal alignment method which allows one to compare sequences with shuffled or inverted elements. The main idea of their work is to combine local and global alignments. During the first stage, the method selects conserved segments (local) and during the second stage, it chains an optimal subset of the pairs of segments previously selected (global). Many specialized approaches have been developed to model the specific evolutions "by blocks" of certain elements of sequences: minisatellites 5 or swaps in proteins sequences 6. In the latter, the method is based on selection of common segments by local alignments scores and can be applied in a more general framework. The approach proposed in 3, mostly applied to more than two sequences, proceeds in similar manner in its first stage, then performs post-treatments and a graph representation of common elements of sequences.

For simplicity, we present the N-maps without taking into account inversions. Some hints will be given about how to extend the definitions and the algorithms in order to handle this type of evolution. Under this restriction, the (optimal) N-map of a sequence s over a sequence t is basically the way of cutting s into N parts that maximizes the sums of the scores of the gapless alignments of all the N parts against t. The gapless alignments can be local or global and so can be the N-map. This approach can be seen as a generalization of the "alignment with a fixed number of gaps" method initially introduced in 7 and recently studied in 89. As this method, our approach is an attempt to avoid the introduction of some arbitrary costs on the transformations between sequences (like gap penalties in the case of alignment). For this purpose, we need a concrete way to determine the "best" number N of parts for mapping a sequence s over a sequence t. As in 9, we define this problem from a probabilistic point of view. Practically, we choose the number of parts leading to the most significant optimal score. The significance is empirically evaluated among pairs of independent identically distributed (iid) random sequences of same lengths and symbols distributions as s and t.

The rest of this paper is organized as follows. Section 1 is devoted to formal definitions and basic properties of the N-maps. We present the algorithms computing the optimal scores and correponding N-maps of a sequence s over a sequence t in Section 2. The algorithmic complexities of these computations are O (|s| × |t| × N) in time and O (|s| + |t| × N) in memory space. These complexities have an extra factor N with regard to the classical pairwise alignment algorithms. However typical values of interest of N are small compared to the lengths of the sequences: choosing a number of parts of the same order as the lengths of the sequences does not make any sense. The choice of the number of parts is discussed in Section 3, in which we investigate the distributions of the scores of the optimal N-maps of random sequences. In particular, empirical evidences lead us to approximate these distributions by normal ones and to measure the significance of optimal scores in terms of Z-values. The approach is evaluated in Section 4 by applying artificial evolutionary events over random sequences and by measuring the ability of the approach to retrieve the corresponding homologous segments. Section 5 shows four case studies of sequences (two pairs of proteins, a pair of DNA sequences of transposon elements and a pair of sequences of genes of microbial genomes) in which the homologies cannot be reported by a classic alignment. Finally in Section 6, we discuss the approach and present some research directions we plan to explore.

The sources of the software computing N-maps are available at 10. We also provide additional utilities to estimate Z-values, represent N-maps as pictures (see Section 5), filter, merge and extract common segments.

1 Notations and Definitions

We consider sequences (or strings) over some finite alphabet A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqaaa@3820@ of elements called letters or symbols. In practical applications, symbols can represent nucleotides, amino acids or genes. The elements of a sequence s are indexed from 1 to |s|, where |s| denotes the length of s, i.e. s = s₁s₂... s_|s|. For 1 ≤ i ≤ j ≤ |s|, the notation s_{[i, j]}designates the substring s_is_i+1... s_j. We note s^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaqcaaaa@2E2B@ the reverse sequence of s, i.e.

s^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaqcaaaa@2E2B@ = s_|s|s_|s|-1 ...s₁. The set of all sequences of length l over A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqaaa@3820@ is noted Al MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqdaahaaWcbeqaaiabdYgaSbaaaaa@39AE@. Let s and t be two sequences. A pair of intervals of positions ([a, b], [c, d]) is a diagonal of (s, t) if 1 ≤ a ≤ b ≤ |s|, 1 ≤ c ≤ d ≤ |t| and b - a = d - c. The first (resp. the second) interval of a diagonal D of (s, t) will be designated as the s-interval (resp. the t-interval) of D. In order to avoid to deal specifically with some "pathological cases", we allow diagonals to be empty (of length 0).

Definition 1 Let s and t be two sequences. A N-map of s over t is a N-tuple of diagonals of (s, t): [([a₁, b₁], [c₁, d₁]), ([a₂, b₂], [c₂, d₂]), ...,([a_N, b_N], [c_N, d_N])] such that [a_i, b_i] ∩ [a_j, b_j] = ∅ for all 1 ≤ i, j ≤ N with j ≠ i.

Without loss of generality, we assume in the following that the diagonals of a N-map of s over t are indexed according to the positions of their s-intervals. In particular, the first diagonal (resp. the last diagonal) is the one with the smallest (resp. the greatest) start position of s-interval. Notation Ω(s,t)N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHPoWvdaqhaaWcbaGaeiikaGIaem4CamNaeiilaWIaemiDaqNaeiykaKcabaGaemOta4eaaaaa@34FE@ denotes the set of all the N-maps of s over t.

A N-map of s over t is nothing but a peculiar type of map from a subset of positions of s to the set of positions of t. In other words, it associates at most one position of t to a position of s; and none, one or several positions of s to a position of t (See Figure 1 or Figure 2 for dotplot representation).

Figure 1

Representation of the 3-map [([2, 3], [1,2]), ([4,8], [7,11]), ([11,13], [5,7])] of s over t

Representation of the 3-map [([2, 3], [1,2]), ([4,8], [7,11]), ([11,13], [5,7])] of s over t. The positions associated in a diagonal are connected by a line.

Figure 2

Dotplot representations of two 4-maps

Dotplot representations of two 4-maps. a) Position m is inside a diagonal. b) Position m is not inside a diagonal.

A given classical alignment (which is also a map between positions) can be seen, for a certain positive integer N, as a N-map, both of s over t and of t over s. More precisely, an alignment with a fixed number K of gaps, like studied in 789, is a (K + 1)-map which, with notations of Definition 1, verifies the additional conditions: [c_i, d_i] ∩ [c_j, d_j] = ∅ and (a_i- a_j) × (c_i- c_j) > 0 for all 1 ≤ i, j ≤ K + 1 with j ≠ i. For 0 <N ≤ K ≤ |s| and a given N-map of s over t, there is at least one K-map defining the same map from positions of s to positions of t.

Let S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=uaaa@3844@ be a scoring scheme, i.e. a map from ∪l∈ℕAl×Al MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWeqaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=bq8bnaaCaaaleqabaGaemiBaWgaaOGaey41aqRae8haXh0aaWbaaSqabeaacqWGSbaBaaaabaGaemiBaWMaeyicI48efv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39gaiuaacqGFveItaeqaniablQIivbaaaa@4E0E@ to ℝ. The score associated to a N-map following S MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=uaaa@3844@ is:

S [ ( [ a 1 , b 1 ] , [ c 1 , d 1 ] ) , ... , ( [ a N , b N ] , [ c N , d N ] ) ] = ∑ k = 1 N S ( s [ a k , b k ] , t [ c k , d k ] ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=ucqGGBbWwcqGGOaakcqGGBbWwcqWGHbqydaWgaaWcbaGaeGymaedabeaakiabcYcaSiabdkgaInaaBaaaleaacqaIXaqmaeqaaOGaeiyxa0LaeiilaWIaei4waSLaem4yam2aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWGKbazdaWgaaWcbaGaeGymaedabeaakiabc2faDjabcMcaPiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabcIcaOiabcUfaBjabdggaHnaaBaaaleaacqWGobGtaeqaaOGaeiilaWIaemOyai2aaSbaaSqaaiabd6eaobqabaGccqGGDbqxcqGGSaalcqGGBbWwcqWGJbWydaWgaaWcbaGaemOta4eabeaakiabcYcaSiabdsgaKnaaBaaaleaacqWGobGtaeqaaOGaeiyxa0LaeiykaKIaeiyxa0Laeyypa0ZaaabCaeaacqWFse=ucqGGOaakcqWGZbWCdaWgaaWcbaGaei4waSLaemyyae2aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGIbGydaWgaaadbaGaem4AaSgabeaaliabc2faDbqabaGccqGGSaalcqWG0baDdaWgaaWcbaGaei4waSLaem4yam2aaSbaaWqaaiabdUgaRbqabaWccqGGSaalcqWGKbazdaWgaaadbaGaem4AaSgabeaaliabc2faDbqabaGccqGGPaqkaSqaaiabdUgaRjabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aaaa@8817@

As in classical alignment methods, we will consider in the following only additive scoring schemes, i.e. defined from a A×A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqcqGHxdaTcqWFaeFqaaa@3BEE@ substitution matrix π, for all lengthes l and all pairs of sequences (u, v) ∈ Al MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqdaahaaWcbeqaaiabdYgaSbaaaaa@39AE@ × Al MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFaeFqdaahaaWcbeqaaiabdYgaSbaaaaa@39AE@, by:

S ( u , v ) = ∑ j = 1 l π [ u j , v j ] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=ucqGGOaakcqWG1bqDcqGGSaalcqWG2bGDcqGGPaqkcqGH9aqpdaaeWbqaaGGaciab+b8aWnaaBaaaleaacqGGBbWwcqWG1bqDdaWgaaadbaGaemOAaOgabeaaliabcYcaSiabdAha2naaBaaameaacqWGQbGAaeqaaSGaeiyxa0fabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGSbaBa0GaeyyeIuoaaaa@5110@

The score of an empty diagonal (l = 0) is 0.

The maximum of the scores over all the N-maps of s over t is noted ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t):

ℳ N ( s , t ) = max ⁡ Γ ∈ Ω ( s , t ) N S ( Γ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaakiabcIcaOiabdohaZjabcYcaSiabdsha0jabcMcaPiabg2da9maaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqqHtoWrcqGHiiIZcqqHPoWvdaqhaaadbaGaeiikaGIaem4CamNaeiilaWIaemiDaqNaeiykaKcabaGaemOta4eaaaWcbeaakiab=jr8tjabcIcaOiabfo5ahjabcMcaPaaa@540B@

An optimal N-map of s over t is a N-map with score ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t). By convention, a 0-map is the empty set and ℳ0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabicdaWaaaaaa@38AA@ (s, t) = 0.

Depending on whether the substitution matrix contains negative values or not, the optimal N-map is said to be local or global. These concepts are used by analogy with the case of alignment. When the matrix contains only non-negative values (global case), a corresponding optimal N-map of s over t will attempt to associate each position of s with a position of t, as in a global alignment. When the matrix contains some negative values (local case), the optimal N-map will be reached by considering only subparts of s which lead to a positive contribution of the total score when associated with a segment of t, once more as in local alignment. Basically, a global N-map of s over t spans the entire length of s (except possibly some boundary positions) while a local N-map identifies N non-overlapping segments of s with maximum scores against t.

Some pathological situations could arise in the local case. In particular there could be some positions i of s such that π [s_i, t_j] is negative for all positions j of t. Without considering empty diagonals, ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t) would be not always growing with N.

The current software implementation incorporates various substitution matrices in particular for amino acids (PAM, BLOSUM, etc.).

For handling inversions, which is not allowed by Definition 1, it is first needed to extend this definition by adding a sign to the pair of intervals:

• (+, [a, b], [c, d]) means that the positions [(a, c), (a + 1, c + 1),...] are associated (normal case),

• (-, [a, b], [c, d]) means that the positions [(a, d), (a + 1, d - 1),...] are associated (inversion case).

Another required extension concerns the way of calculating the score for "reverse diagonals". This point depends on the nature of sequences. For instance in the case of DNA sequences, the score of (-, [a, b], [c, d]) is computed by summing the individual substitution scores of s_{[a, b]}against the complementary-reverse of t_{[c, d]}. If s and t are sequences of genes, this score is obtained by considering s_{[a, b]}against the reverse of t_{[c, d]}.

2 Algorithms

Given two sequences s, t and a positive integer N, we address two problems:

• Problem 1: computing the optimal scores ℳK MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabdUealbaaaaa@38DB@ (s, t) with K running from 1 to N.

• Problem 2: outputting the diagonals of an optimal N-map.

Computing the optimal scores

Let Best_{[i, j, K]}be the maximal score obtained by a K-map of s_{[1, i]}over t ending at (i, j), i.e. such that its last diagonal ([a_K, b_K], [c_K, d_K]) verifies b_K= i and d_K= j. By setting Best_{[0, j, K]}, Best_{[i, 0 , K]}and Best_{[i, j, 0]}to 0 for all integers i, j and K, we have the following recurrence relation:

Best [ i + 1 , j + 1 , K ] = π [ s i + 1 , t j + 1 ] + max ⁡ { Best [ i , j , K ] , max ⁡ k ≤ i , l ≤ | t | Best [ k , l , K − 1 ] } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqcqqGLbqzcqqGZbWCcqqG0baDdaWgaaWcbaGaei4waSLaemyAaKMaey4kaSIaeGymaeJaeiilaWIaemOAaOMaey4kaSIaeGymaeJaeiilaWIaem4saSKaeiyxa0fabeaakiabg2da9GGaciab=b8aWnaaBaaaleaacqGGBbWwcqWGZbWCdaWgaaadbaGaemyAaKMaey4kaSIaeGymaedabeaaliabcYcaSiabdsha0naaBaaameaacqWGQbGAcqGHRaWkcqaIXaqmaeqaaSGaeiyxa0fabeaakiabgUcaRiGbc2gaTjabcggaHjabcIha4naacmaabaGaeeOqaiKaeeyzauMaee4CamNaeeiDaqNaaCjaVpaaBaaaleaacqGGBbWwcqWGPbqAcqGGSaalcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaOGaeiilaWYaaCbeaeaacyGGTbqBcqGGHbqycqGG4baEaSqaaiabdUgaRjabgsMiJkabdMgaPjabcYcaSiabdYgaSjabgsMiJoaaemaabaGaemiDaqhacaGLhWUaayjcSdaabeaakiabbkeacjabbwgaLjabbohaZjabbsha0naaBaaaleaacqGGBbWwcqWGRbWAcqGGSaalcqWGSbaBcqGGSaalcqWGlbWscqGHsislcqaIXaqmcqGGDbqxaeqaaaGccaGL7bGaayzFaaGaaCjaVdaa@875A@

The correctness of this relation is straightforwardly proved by induction. Let us consider the maximum involved in the right part of this equation. It is equal to:

• Best_{[i, j, K]}, if the greatest score is obtained by incrementing the length of the last diagonal of an optimal K-map of s_{[1, i]}over t ending at (i, j) – then the last diagonal will end at (i + 1, j + 1).

• max⁡k≤i,l≤jBest[k,l,K−1] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfqaqaaiGbc2gaTjabcggaHjabcIha4bWcbaGaem4AaSMaeyizImQaemyAaKMaeiilaWIaemiBaWMaeyizImQaemOAaOgabeaakiabbkeacjabbwgaLjabbohaZjabbsha0naaBaaaleaacqGGBbWwcqWGRbWAcqGGSaalcqWGSbaBcqGGSaalcqWGlbWscqGHsislcqaIXaqmcqGGDbqxaeqaaaaa@4A37@, if the greatest score is obtained by adding the diagonal of length 1 ([i + 1, i + 1], [j + 1, j + 1]) to an optimal (K - 1)-map of s_{[1, i]}over t.

To compute the entries of Best referred to the index (i + 1), we only need to know the entries referred to the index i. Thus computing the optimal scores of all the K-maps of s over t, with K from 1 to N, can be done in O (|s| × |t| × N) time using O (|t| × N) memory space to store the dynamic programming variables. We can introduce now the formal algorithm Alg_1 which solves the problem of computing the optimal scores for global or local N-maps without inversion.

Algorithm Alg_1 takes as input two sequences s, t and a number of parts N and returns:

• B_sand B_l, two |t| × N matrices where the entry B_{s[j, K]}contains the maximal score of a K-map of s over t ending at (|s|, j) – with the preceding notations B_{s[j, K]}= Best_{[|s|, j, K]}– and the entry B_{l[j, K]}stores the length of the last diagonal of a K-map ending at (|s|, j) with score B_{s[j, K]};

• M_sand M_d, two arrays of size N where the entry M_s[K]stores the optimal score of a K-map of s over t – with the preceding notations M_s[K]= ℳK MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabdUealbaaaaa@38DB@ (s, t) = max_{q ≤ |s|, p ≤ |t|}Best_{[q, p, K]}– and the entry M_d[K]stores the last diagonal of a K-map with score M_s[K].

The correctness of Algorithm Alg_1 is proved by induction over the positions of s. The time and memory space complexities are straightforwardly analyzed.

The variables B_land M_dare not involved in the computation of the maximal scores of K-maps for 1 ≤ K ≤ N (they will be used by the algorithm in charge of outputting the diagonals). If we are only interested in solving Problem 1, these variables as well as the lines 7, 10, 14 and 19 can be deleted. Algorithm Alg_1 will still return the optimal scores of K-maps with 0 ≤ K ≤ N in the array M_s.

Algorithm 1 Alg_1 (s, t, N, B_s, B_l, M_s, M_d)

1: B_{s[j, K]}← 0 ; B_{l[j, K]}← 0 ; M_s[K]← 0 ; M_d[K]← NULL ; (j = 0 ... |t|, K = 0 ... N)

2: for i = 1 to |s| do

3: for j = 1 to |t| do

4: for K = N to 1 do

5: if B_{s[j-1, K]}≥ M_s[k-1]then

6: B′s[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGZbWCcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@3538@ ← π_{[s_i, t_j]}+ B_{s[j-1, K]};

7: B′l[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGSbaBcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@352A@ ← B_{l[j-1, K]}+ 1 ;

8: else

9: B′s[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGZbWCcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@3538@ ← π_{[s_i, t_j]}+ M_s[K-1];

10: B′l[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGSbaBcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@352A@ ← 1;

11: end if

12: if B′s[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGZbWCcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@3538@ ≥ M_s[K]then

13: M_s[K]← B′s[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGZbWCcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@3538@ ;

14: M_d[K]← ([i - B′l[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGSbaBcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@352A@ + 1, i], [j - B′l[j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGSbaBcqGGBbWwcqWGQbGAcqGGSaalcqWGlbWscqGGDbqxaeqaaaaa@352A@ + 1, j]) ;

15: end if

16: end for

17: end for

18: swap (B_s, B′s MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGZbWCaeqaaaaa@2F5C@) ;

19: swap (B_l, B′l MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGcbGqgaqbamaaBaaaleaacqqGSbaBaeqaaaaa@2F4E@) ;

20: end for

Theorem 1 Algorithm Alg_1 computes the optimal score of the K-maps of a sequence s over a sequence t, for K from 1 to N, in time O (|s| × |t| × N) using O (|s| + |t| × N) memory space.

Outputting the diagonals of an optimal N-map

Before presenting the formal algorithm, we need to introduce some additional notations and results about "dividing maps".

We say that a position m of s is inside a diagonal of a N-map Γ if there is a diagonal ([a, b], [c, d]) ∈ Γ such that a ≤ m <b (Figure 2a). This notion excludes two cases:

1. when m is not contained by any diagonal (this is usual with local N-maps),

2. when a diagonal is exactly ending at m in its first interval.

We denote as Best¯[i,j,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabbkeacjabbwgaLjabbohaZjabbsha0baadaWgaaWcbaGaei4waSLaemyAaKMaeiilaWIaemOAaOMaeiilaWIaem4saSKaeiyxa0fabeaaaaa@3A38@ the maximal score obtained by a K-map of s_[i,|s|]over t_[j,|t|]starting at (i, j), i.e. such that its first diagonal ([a₁, b₁], [c₁, d₁]) verifies a₁= i and c₁= j.

Lemma 1 Let s and t be two sequences, m a position of s and N a positive integer. The optimal score of a N-map ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t) is equal to the maximum of the two following quantities:

• Q1=max⁡1≤K≤N;1≤p<|t|{Best[m,p,K]+Best¯[m+1,p+1,N−K+1]} MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@76C3@

• Q2=max⁡1≤K≤N{ℳK(s[1,m],t)+ℳN−K(s[m+1,|s|],t)} MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGOmaidabeaakiabg2da9maaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqaIXaqmcqGHKjYOcqWGlbWscqGHKjYOcqWGobGtaeqaaOGaei4EaSNae83mH00aaWbaaSqabeaaieGacqGFlbWsaaGccqGGOaakcqWGZbWCdaWgaaWcbaGaei4waSLaeGymaeJaeiilaWIaemyBa0Maeiyxa0fabeaakiabcYcaSiabdsha0jabcMcaPiabgUcaRiab=ntinnaaCaaaleqabaGaemOta4KaeyOeI0Iaem4saSeaaOGaeiikaGIaem4Cam3aaSbaaSqaaiabcUfaBjabd2gaTjabgUcaRiabigdaXiabcYcaSiabcYha8jabdohaZjabcYha8jabc2faDbqabaGccqGGSaalcqWG0baDcqGGPaqkcqGG9bqFaaa@6C6E@

Proof: Let Γ = [([a₁, b₁], [c₁, d₁]), ..., ([a_N, b_N], [c_N, d_N])] be a N-map of s over t with score ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t). There are two possibilities: either the position m is inside a diagonal K of Γ, or not. In the first case, there are a K-map Γ' ending at [m, c_K+ m - a_K] and a (N - K + 1)-map Γ" starting at position [m + 1, c_K+ m - a_K+ 1] such that S(Γ′)+S(Γ″)=ℳN(s,t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=ucqGGOaakcuqHtoWrgaqbaiabcMcaPiabgUcaRiab=jr8tjabcIcaOiqbfo5ahzaagaGaeiykaKIaeyypa0Jae83mH00aaWbaaSqabeaacqWGobGtaaGccqGGOaakcqWGZbWCcqGGSaalcqWG0baDcqGGPaqkaaa@4A48@, which implies ℳN(s,t)≤Q1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaakiabcIcaOiabdohaZjabcYcaSiabdsha0jabcMcaPiabgsMiJkab=br8rnaaBaaaleaacqaIXaqmaeqaaaaa@4305@. In the second case, let K be such that m = b_Kor b_K<m <a_K+1. There is a K-map Γ' of s_{[1, m]}over t and a (N - K)-map Γ" of s_[m+1,|s|]over t such that S(Γ′)+S(Γ″)=ℳN(s,t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=ucqGGOaakcuqHtoWrgaqbaiabcMcaPiabgUcaRiab=jr8tjabcIcaOiqbfo5ahzaagaGaeiykaKIaeyypa0Jae83mH00aaWbaaSqabeaacqWGobGtaaGccqGGOaakcqWGZbWCcqGGSaalcqWG0baDcqGGPaqkaaa@4A48@, which implies ℳN(s,t)≤Q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaakiabcIcaOiabdohaZjabcYcaSiabdsha0jabcMcaPiabgsMiJkab=br8rnaaBaaaleaacqaIYaGmaeqaaaaa@4307@. In both cases, ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t) is smaller than max{Q1,Q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=br8rnaaBaaaleaacqaIYaGmaeqaaaaa@3D3B@}.

On the other hand, for all integers 1 ≤ K ≤ N, for all positions 1 ≤ p < |t|, for all K-maps Γ′=[([a′1,b′1],[c′1,d′1]),...,([a′K,m],[c′K,p])] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHtoWrgaqbaiabg2da9iabcUfaBjabcIcaOiabcUfaBjqbdggaHzaafaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcuWGIbGygaqbamaaBaaaleaacqaIXaqmaeqaaOGaeiyxa0LaeiilaWIaei4waSLafm4yamMbauaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiqbdsgaKzaafaWaaSbaaSqaaiabigdaXaqabaGccqGGDbqxcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqGGOaakcqGGBbWwcuWGHbqygaqbamaaBaaaleaacqWGlbWsaeqaaOGaeiilaWIaemyBa0Maeiyxa0LaeiilaWIaei4waSLafm4yamMbauaadaWgaaWcbaGaem4saSeabeaakiabcYcaSiabdchaWjabc2faDjabcMcaPiabc2faDbaa@5ADE@ and for all (N - K + 1)-maps Γ″=[([m+1,b″1],[p+1,d″1]),...,([a″N−K+1,b″N−K+1],[c″N−K+1,d″N−K+1])] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6E7B@ the N-map Γ=[([a′1,b′1],[c′1,d′1]),...,([a′K,b″1],[c′K,d″1]),...,([a″N−K+1,b″N−K+1],[c″N−K+1,d″N−K+1])] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHtoWrcqGH9aqpcqGGBbWwcqGGOaakcqGGBbWwcuWGHbqygaqbamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIafmOyaiMbauaadaWgaaWcbaGaeGymaedabeaakiabc2faDjabcYcaSiabcUfaBjqbdogaJzaafaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcuWGKbazgaqbamaaBaaaleaacqaIXaqmaeqaaOGaeiyxa0LaeiykaKIaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaeiikaGIaei4waSLafmyyaeMbauaadaWgaaWcbaGaem4saSeabeaakiabcYcaSiqbdkgaIzaagaWaaSbaaSqaaiabigdaXaqabaGccqGGDbqxcqGGSaalcqGGBbWwcuWGJbWygaqbamaaBaaaleaacqWGlbWsaeqaaOGaeiilaWIafmizaqMbayaadaWgaaWcbaGaeGymaedabeaakiabc2faDjabcMcaPiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabcIcaOiabcUfaBjqbdggaHzaagaWaaSbaaSqaaiabd6eaojabgkHiTiabdUealjabgUcaRiabigdaXaqabaGccqGGSaalcuWGIbGygaGbamaaBaaaleaacqWGobGtcqGHsislcqWGlbWscqGHRaWkcqaIXaqmaeqaaOGaeiyxa0LaeiilaWIaei4waSLafm4yamMbayaadaWgaaWcbaGaemOta4KaeyOeI0Iaem4saSKaey4kaSIaeGymaedabeaakiabcYcaSiqbdsgaKzaagaWaaSbaaSqaaiabd6eaojabgkHiTiabdUealjabgUcaRiabigdaXaqabaGccqGGDbqxcqGGPaqkcqGGDbqxaaa@8518@ has score S(Γ′)+S(Γ″) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFse=ucqGGOaakcuqHtoWrgaqbaiabcMcaPiabgUcaRiab=jr8tjabcIcaOiqbfo5ahzaagaGaeiykaKcaaa@414E@, which is by definition smaller than ℳN MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaaaaa@38E1@ (s, t). It implies that Q1≤ℳN(s,t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaakiabgsMiJkab=ntinnaaCaaaleqabaGaemOta4eaaOGaeiikaGIaem4CamNaeiilaWIaemiDaqNaeiykaKcaaa@430F@. A similar argument establishes that Q2≤ℳN(s,t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGOmaidabeaakiabgsMiJkab=ntinnaaCaaaleqabaGaemOta4eaaOGaeiikaGIaem4CamNaeiilaWIaemiDaqNaeiykaKcaaa@4311@ and ends the proof.

Remark 1 Let s and t be two sequences, N a positive integer, and [D₁, ..., D_N] an optimal N-map of s over t with diagonals D₁, ..., D_Nindexed following the increasing order of their s-intervals.

1. For all 1 ≤ K <N, [D₁, ... , D_K] is an optimal K-map of s[1,aK+1−1] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaWgaaWcbaGaei4waSLaeGymaeJaeiilaWIaemyyae2aaSbaaWqaaiabdUealjabgUcaRiabigdaXaqabaWccqGHsislcqaIXaqmcqGGDbqxaeqaaaaa@38E8@over t. Reciprocally, if [D′1,...,D′K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGBbWwcuWGebargaqbamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIafmiraqKbauaadaWgaaWcbaGaem4saSeabeaakiabc2faDbaa@384D@is an optimal K-map of s[1,aK+1−1] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaWgaaWcbaGaei4waSLaeGymaeJaeiilaWIaemyyae2aaSbaaWqaaiabdUealjabgUcaRiabigdaXaqabaWccqGHsislcqaIXaqmcqGGDbqxaeqaaaaa@38E8@over t then [D′1,...,D′K,DK+1,...,DN] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGBbWwcuWGebargaqbamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIafmiraqKbauaadaWgaaWcbaGaem4saSeabeaakiabcYcaSiabdseaenaaBaaaleaacqWGlbWscqGHRaWkcqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemiraq0aaSbaaSqaaiabd6eaobqabaGccqGGDbqxaaa@443D@is an optimal N-map of s over t.

2. For all 1 <K ≤ N, [D_K, ..., D_N] is an optimal (N - K + 1)-map of s[bK−1+1,|s|] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaWgaaWcbaGaei4waSLaemOyai2aaSbaaWqaaiabdUealjabgkHiTiabigdaXaqabaWccqGHRaWkcqaIXaqmcqGGSaaldaabdaqaaiabdohaZbGaay5bSlaawIa7aiabc2faDbqabaaaaa@3C8B@over t. Reciprocally, if [D′K,...,D′N] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGBbWwcuWGebargaqbamaaBaaaleaacqWGlbWsaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIafmiraqKbauaadaWgaaWcbaGaemOta4eabeaakiabc2faDbaa@3882@is an optimal (N - K + 1)-map of s[bK−1+1,|s|] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaWgaaWcbaGaei4waSLaemOyai2aaSbaaWqaaiabdUealjabgkHiTiabigdaXaqabaWccqGHRaWkcqaIXaqmcqGGSaaldaabdaqaaiabdohaZbGaay5bSlaawIa7aiabc2faDbqabaaaaa@3C8B@over t then [D1,...,DK−1,D′K,...,D′N] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGBbWwcqWGebardaWgaaWcbaGaeGymaedabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdseaenaaBaaaleaacqWGlbWscqGHsislcqaIXaqmaeqaaOGaeiilaWIafmiraqKbauaadaWgaaWcbaGaem4saSeabeaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiqbdseaezaafaWaaSbaaSqaaiabd6eaobqabaGccqGGDbqxaaa@4448@is an optimal N-map of s over t.

We are now able to introduce the formal algorithm Alg_2 which solves the problem of outputting the diagonals of an optimal global N-map without inversion.

Algorithm Alg_2 takes as inputs two sequences s and t, two positions i and j bounding a substring of s, and a number of parts N. It outputs the diagonals of an optimal N-map of s_{[i, j]}over t ordered according to their first intervals.

Algorithm 2 Alg_2 (s, i, j, t, N)

1: if N = 0 then

2: return;

3: end if

4: S_max← -∞ ; m=⌊i+j2⌋ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGTbqBcqGH9aqpdaGbdaqaamaalaaabaGaemyAaKMaey4kaSIaemOAaOgabaGaeGOmaidaaaGaayj84laawUp+aaaa@38C9@ ; D′max⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGebargaqbamaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaaa@3219@ ← NULL ;

5: Alg_1 (s_{[i, m]}, t, N, B_s, B_l, M_s, M_d) ;

6: Alg_1 (s[m+1,j]^,t^,N,Bs∗,Bl∗,Ms∗,Md∗) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGOaakdaqiaaqaaiabdohaZnaaBaaaleaacqGGBbWwcqWGTbqBcqGHRaWkcqaIXaqmcqGGSaalcqWGQbGAcqGGDbqxaeqaaaGccaGLcmaacqGGSaalcuWG0baDgaqcaiabcYcaSiabd6eaojabcYcaSiabbkeacnaaDaaaleaacqqGZbWCaeaacqGHxiIkaaGccqGGSaalcqqGcbGqdaqhaaWcbaGaeeiBaWgabaGaey4fIOcaaOGaeiilaWIaeeyta00aa0baaSqaaiabbohaZbqaaiabgEHiQaaakiabcYcaSiabb2eannaaDaaaleaacqqGKbazaeaacqGHxiIkaaGccqGGPaqkaaa@4F15@ ; \* Loop Q1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaaaaa@395C@ *\

7: for K ← 1 to N do

8: L ← N - K + 1 ;

9: for p ← 1 to (|t| - 1) do

10: q ← |t| - p ;

11: if (B_{s[p, K]}+ Bs[q,L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaee4CamNaei4waSLaemyCaeNaeiilaWIaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@362C@) > S_maxthen

12: S_max← B_{s[p, K]}+ Bs[q,L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaee4CamNaei4waSLaemyCaeNaeiilaWIaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@362C@ ;

13: D_max← ([m - B_{l[p, K]}+ 1, m + Bl[q,L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaeeiBaWMaei4waSLaemyCaeNaeiilaWIaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@361E@], [p - B_{l[p, K]}+ l, p + Bl[q,L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaeeiBaWMaei4waSLaemyCaeNaeiilaWIaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@361E@]) ;

14: N_L← K - 1 ; N_R← L - 1 ; j_L← m - B_{l[p, K]}; i_R← m + Bl[q,L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaeeiBaWMaei4waSLaemyCaeNaeiilaWIaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@361E@ + 1;

15: end if

16: end for

17: end for \* Loop Q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGOmaidabeaaaaa@395E@ *\

18: for K ← 0 to N do

19: L ← N - K ;

20: if (M_s[K]+ Ms[L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaqhaaWcbaGaee4CamNaei4waSLaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@33F7@) > S_maxthen

21: S_max← M_s[K]+ Ms[L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaqhaaWcbaGaee4CamNaei4waSLaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@33F7@

22: if K > 0 and M_d[K]≠ NULL then

23: ([a, b], [c, d]) ← M_d[K]; D_max← ([a + i - 1, b + i - 1], [c, d]) ;

24: N_L← K - 1 ; j_L← a + i - 2 ;

25: else

26: D_max← NULL ; N_L← 0 ;

27: end if

28: if L > 0 and Md[L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaqhaaWcbaGaeeizaqMaei4waSLaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@33D9@ ≠ NULL then

29: ([a, b], [c, d]) ← Md[L]∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaqhaaWcbaGaeeizaqMaei4waSLaemitaWKaeiyxa0fabaGaey4fIOcaaaaa@33D9@ ; D′max⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGebargaqbamaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaaa@3219@ ← ([a + m, b + m], [c, d]) ;

30: N_R← L - 1 ; i_R← b + m + 1 ;

31: else

32: D′max⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGebargaqbamaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaaa@3219@ ← NULL ; N_R← 0 ;

33: end if

34: end if

35: end for

36: Alg_2 (s, i, j_L, t, N_L) ;

37: Output (D_max) ; output (D′max⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGebargaqbamaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaaa@3219@) ;

38: Alg_2 (s, i_R, j, t, N_R) ;

Correctness of Algorithm Alg_2

Let us consider Best and Best¯ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabbkeacjabbwgaLjabbohaZjabbsha0baaaaa@31F5@ defined for s_{[i, j]}as follows. For all r such that i ≤ r ≤ j and v such that 1 ≤ v ≤ |t|, Best_{[r, v, K]}is the maximal score obtained by a K-map of s_{[i, r]}over t_{[1, v]}ending at (r, v). Analogously, Best¯[r,v,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqdaaqaaiabbkeacjabbwgaLjabbohaZjabbsha0baadaWgaaWcbaGaei4waSLaemOCaiNaeiilaWIaemODayNaeiilaWIaem4saSKaeiyxa0fabeaaaaa@3A62@ is the maximal score obtained by a K-map of s_{[r, j]}over t_{[v, |t|]}starting at (r, v). For all positions p of t and all 1 ≤ K ≤ N, we have B_{s[p, K]}= Best_{[m, p, K]}, Bs[|t|−p,K]∗=Best¯[m+1,p+1,K] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGcbGqdaqhaaWcbaGaee4CamNaei4waSLaeiiFaWNaemiDaqNaeiiFaWNaeyOeI0IaemiCaaNaeiilaWIaem4saSKaeiyxa0fabaGaey4fIOcaaOGaeyypa0Zaa0aaaeaacqqGcbGqcqqGLbqzcqqGZbWCcqqG0baDaaWaaSbaaSqaaiabcUfaBjabd2gaTjabgUcaRiabigdaXiabcYcaSiabdchaWjabgUcaRiabigdaXiabcYcaSiabdUealjabc2faDbqabaaaaa@4DDA@ (since it is obtained from s[m+1,j]^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaqiaaqaaiabdohaZnaaBaaaleaacqGGBbWwcqWGTbqBcqGHRaWkcqaIXaqmcqGGSaalcqWGQbGAcqGGDbqxaeqaaaGccaGLcmaaaaa@3705@), Ms[K]=ℳK(s[i,m],t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaWgaaWcbaGaee4CamNaei4waSLaem4saSKaeiyxa0fabeaakiabg2da9mrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=ntinnaaCaaaleqabaGaem4saSeaaOGaeiikaGIaem4Cam3aaSbaaSqaaiabcUfaBjabdMgaPjabcYcaSiabd2gaTjabc2faDjabcYcaSaqabaGccqWG0baDcqGGPaqkaaa@4C14@ and Ms[K]∗=ℳK(sm+1,j],t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGnbqtdaqhaaWcbaGaee4CamNaei4waSLaem4saSKaeiyxa0fabaGaey4fIOcaaOGaeyypa0ZenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83mH00aaWbaaSqabeaacqWGlbWsaaGccqGGOaakcqWGZbWCdaWgaaWcbaGaemyBa0Maey4kaSIaeGymaeJaeiilaWIaemOAaOMaeiyxa0fabeaakiabcYcaSiabdsha0jabcMcaPaaa@4D9A@. Following the notations of Lemma 1, "Loop Q1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaaaaa@395C@" (resp. "Loop Q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGOmaidabeaaaaa@395E@") parses the quantities maximized by Q1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaaaaa@395C@ (resp. by Q2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGOmaidabeaaaaa@395E@). Thus, Lemma 1 ensures that ℳN(s[i,j],t) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFZestdaahaaWcbeqaaiabd6eaobaakiabcIcaOiabdohaZnaaBaaaleaacqGGBbWwcqWGPbqAcqGGSaalcqWGQbGAcqGGDbqxaeqaaOGaeiilaWIaemiDaqNaeiykaKcaaa@44AB@ is stored in the variable S_maxafter the execution of these two loops. If the maximum is reached in "Loop Q1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFqeFudaWgaaWcbaGaeGymaedabeaaaaa@395C@", the variable D′max⁡ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqGebargaqbamaaBaaaleaacyGGTbqBcqGGHbqycqGG4baEaeqaaaaa@3219@ is NULL and the variable D_maxcontains the diagonal including m, let us say the K^th, of a N-map with score