CC BY
8
ISSN 2079-3316 PROGRAM SYSTEMS: THEORY AND APPLICATIONS no.1(28), 2016, pp. 201-208
S. V. Znamenskij
A picture of common subsequence length for two random strings over an alphabet of 4 symbols
Abstract. The maximal length of longest common subsequence (LCS) for a couple of random finite sequences over an alphabet of 4 characters was considered as a random function of the sequences lengths m and n. Exact probability distributions tables are presented for all couples of length in a range 2 <m + n < 19.
The graphs of expected value and standard deviation as a functions of length are shown in linear perspective which presents the behaviour of large lengths at the horizon. In order to illustrate behaviour on large lengths, the results of numeric simulation for m + n = 32, 512, 8192 and 131072 are also shown on the same graphs. The presented graph of expected value dependency of m and n looks to have asymptotic right circular cone. The variance looks alike growing as (n + m.) 4 .
Key words and phrases: similarity of strings, sequence alignment, edit distance, LCS, Leven-shtein metric.
2010 Mathematics Subject Classification: 68T37; 68P10, 68W32.
The need for a theoretical estimation of occasional wrong match probability in fussy search with longest common subsequence (LCS) related algorithm raises the difficult problem of estimation the LCS length of two random symbol strings.
For practically important for applications in bioinfiormatics case of alphabet E4 = A, C, G, T [1] it was found in [2] that natural DNA-sequences from a database and true random sequences show the same statistical behavior in terms of similarity scores. The aim of this paper is to get a brief overall picture of functional dependency from both length of strings instead in spite of know for binary alphabet results [3].
Let's random variable Lm,n notes the length of the longest common subsequence of random strings over the alphabet E with lengths m and n. The behaviour of its expected value E(Lm,n) and variance V(Lm,n~) as a function of m and n remains dark and unclear for many decades. After [4] investigations for this alphabet are mostly concerned on bounding of
© S. V. Znamenskij, 2016
© Ailamazyan Program System Institute of RAS, 2016 © Program systems: Theory and Applications, 2016
Chvatal-Sankoff constant 74 = lim E(Lrt-") and detection of convergence
n^TO n
rate for the limit value [5].
1. Exact probability mass function
We start from calculation of the exact probability mass function
(k) = P (Lm,n = k),
where P is probability for random variable Lm,n to get a value k = 0, 1, . . . , min(m, n).
Theorem 1. Let m < n are natural numbers. In the following simple cases the probability mass function fLm n (k) can be calculated by formulae:
Pi,n(0) = (f )n; """
pi,n(i) = i -(4)n;
P2,n(0) = 1 (f )n + ^ for n > 1;
P2,n(1) = (4)n+1 + f (4)n-1 - ^ for n > 1;
P2,n(2) = 1 -(4)n+1 - 1 (4)n - n (f r1 for n> 1;
Pf,n(0) = 116 (4)n + 2^+4 + 4^ for n > 2;
The proof can be obtained in a standard way by the law of total probability application to following conditions
• shorter string consists of identical symbols,
• shorter string contains exactly two different symbols,
• shorter string contains three different symbols
and counting all the cases with the use of geometric series formula. Unfortunately for larger n values calculations became too complicated and huge to get through.
The Monte Carlo simulation never gives exact information. Nevertheless, if lengths are small then computer can just find LCS for each couple of strings with given lengths and exactly calculate the probabilities by the formula pn,m(k) = ^"T^ where Xn,m(k) is the total number of string couples with lengths n and m and LCS length k. Any modern PC allows to calculate probability mass function for any length with m + n < 16 in a time less then one hour. But the performance time grows very fast, always more then 4 times with each added to total length symbol. Even if the Moore's Law continue its action, we shall never get a computational resource enough to proceed with m + n = 100. So as the table will not grow much we publish it except lines of the values that
Table 1. Exact values xm.n(k) for rn = 3,... , 5
m n xm,n(0) xm,n(l) xm,n(2) xm,n(3) xm,n(4)
3 3 105 570 333 16
3 4 231 1797 1860 208
3 5 537 5436 8715 1696
3 6 1311 16131 36990 11104
3 7 3345 47502 147441 63856
3 8 8871 139713 562920 337072
3 9 24297 411888 2083239 1674880
3 10 68271 1219263 7530450 7959232
3 11 195585 3626226 26726205 36560848
3 12 568311 10834653 93468060 163564432
3 13 1668057 32507028 322953267 716613472
3 14 4930431 97874331 1104629190 3087533344
4 4 453 4800 8742 2325 64
4 5 951 12537 34737 16287 1024
4 6 2109 32688 125919 91572 9856
4 7 4911 85983 431556 452142 73984
4 8 11973 229512 1426380 2049063 477376
4 9 30471 623565 4602219 8740545 2780416
4 10 80589 1726080 14611437 35649990 15040768
4 11 220191 4864899 45895014 140496120 76959232
4 12 617493 13937016 143156802 538909425 377121088
4 13 1766391 40488273 444586053 2022505587 1785620992
xm,n(5)
5 5 1833 28890 118404 98340 14421 256
5 6 3759 67587 372240 478146 121980 4864
5 7 8097 161496 1118523 2048547 803625 54016
5 8 18231 395277 3276435 8083134 4546155 457984
5 9 42873 992934 9470214 30124806 23194581 3283456
5 10 105279 2562903 27226098 107738208 109830936 20972032
5 11 269457 6797844 78276255 373719987 491599113 123079168
5 12 715911 18504897 225874791 1266831732 2105965533 677074432
consists of calculated by the theorem values. Table 1 and Table 2 contains the numbers Xn,™(k) to avoid rounding errors completely.
Table 2. Exact values xm.„(k) for m = 6,..., 9
m n xm,n(0) xm,„(1) xm,n(2) xm,n(3) xm,n(4) xm,n(5) xm,n(6) xm,n(7) xm,n(8) xm,n(9)
6 6 7221 144216 1028931 2022960 907179 82773 1024
6 7 14631 317769 2753415 7587861 5259105 821907 22528
6 8 30909 721704 7267770 26375340 26208204 6221289 283648
6 9 67839 1687911 19132338 87148998 117952467 39763023 2682880
6 10 154821 4067112 50587443 278159046 493461417 226129521 21182464
6 11 368151 10107261 134995695 867001353 1954801602 1180257714 147435520
7 7 28185 653256 6653442 25126110 26901270 7292628 449877 4096
7 8 56751 1393707 16048740 77752320 118644660 49303485 5133393 102400
7 9 118257 3069144 38981253 230646870 474140310 281409102 43935096 1441792
7 10 254391 6959589 95813703 666472293 1767695262 1429288875 313426287 15056896
7 11 566025 16255368 239097306 1896158694 6264656730 6664862241 1968372276 129900544
8 8 109893 2818902 35794041 216361560 465327408 297936519 53028744 2348373 16384
8 9 220551 5910921 81120723 582733095 1662146217 1521603615 410485389 30288033 458752
8 10 454989 12762708 187198638 1542583476 5569004667 6924999531 2646395274 289326477 7143424
9 9 429513 11892570 173097576 1439627910 5342669283 6969837924 2872947741 357417270 11883861 65536
9 10 860559 24709083 378466404 3527391882 16219427187 28529534742 16745716626 3120355458 170983179 2031616
Table 3. Parameters of numeric simulation
m + n step of n step execution time repetitions for step
32 512 8192 131072 2097152
1 8 128 2048 262144
20 sec 20 sec 20 min 200 min 10000 sec
1111275-1173205 49334-161188 15240-185462 535-4391 2-5
Figure 1. Expected value of LCS as a function of the sequence lengths
2. Design and interpretation of expected value and variance graphs
We believe that the function E(Lm,n) of two string length is concave and grows nearly linear with lengths grows. This is a reason to use projective geometry to draw graphs in linear perspective and see the limit values in infinity as the horizon.
The calculated exact values does not represent behaviour on large lengths. So numeric simulation was used to get a full picture. It used fixed total length n + m and a fixed step on n with fixed stepwise execution time. Table 3 shows the given parameters of numeric simulation and number of repetitions for each step. The graph of expected value on the Picture 1 use the projective transform ^2+5z+yy, 2+5^+5y) mapping numbers to centimeters. The point of view was selected close to a vertex (-0.25, -0.25,0.1) of approximate asymptotic cone by a series of iterations aimed to get images of far points proportionally with respect to logarithms of numbers closer to some limit line. Our experiment detects 74 = 0.6542 to be compatible with the published in [5] more precise value 74 = 0.654361 obtained in very smart numeric simulation.
Figure 2. Standard deviation of LCS as a function of the sequence lengths
Figure 3. Standard deviation of LCS scaled by 4m + n
The graph show unexpected structure of dotted consisting from central circle arc extended by two line segments. So the graph visualy looks smoothly combined from two plains z = x, z = y and surface z = s - y/2r + 4r2 - s2 where s = x+y, r = (y/2 + 1)(s - e(s))) and the diagonal function e(n) = E(Ln,n) is monotonic, concave and unknown, but for large m and n it is close to linear and therefore surface is close to asymptotic right circular cone.
The graph of variance on the Picture 2 use another projective transform , 17+x+y^ mapping numbers to centimeters to reflect differences from conical form. We see that variance grows more slow then linear. The asymptotic (non-circle) similar to cone surface appears visible on the Picture 3 for magnified variance 4m + n • V(Lm,n) shown under
(, transform.
^7+x+y'7+x+yJ
References
[1] R. Durbin, S. Eddy, A. Krogh, G. Mitchison. Biological sequence analysis: probabilistic models of proteins and nucleic acids, Cambridge University Press, 1998, 370 a t 201
[2] J. G. Reich, H. Drabsch, A. Daumler. «On the statistical assessment of similarities in DNA sequences», Nucleic acids research, 12:13 (1984), c. 5529-5543. t 201
[3] K. Ning, K. P. Choi. Systematic assessment of the expected length, variance and distribution of Longest Common Subsequences, 2013, arXiv: 1306.4253. t 201
[4] V. Chvatal, D. Sankoff. «Longest Common subsequences of two random sequences», J. Appl. Probability, 12:2 (1975), c. 306-315. t 201
[5] R. Bundschuh. «High precision simulations of the longest common subsequence problem», The European Physical Journal B-Condensed Matter and Complex Systems, 22:4 (2001), c. 533-541. t 202,205
Submitted by dr E. P. Kurshev
About the author:
Sergej Vital'evich Znamenskij
Chair of Mathematics in the Ailamazyan Pereslavl University, head of laboratory in Ailamazyan Program Systems Institute of RAS. Research interests migrate from research in Functional Analysis, Complex Analysis and finite-dimensional Projective Geometry (analogues of Convexity) to the foundations of Collaborative Software Development.
e-mail: svz@latex.pereslavl.ru
Sample citation of this publication:
S. V. Znamenskij. "A picture of common subsequence length for two random strings over an alphabet of 4 symbols", Program systems: theory and applications, 2016, 7:1(28), pp. 201-208.
URL: http://psta.psiras.ru/read/psta2016_1_201-208.pdf
удк 004.416
С. В. Знаменский. Картина наибольшей длины общих подпоследовательностей пары случайных строк 4 буквенного алфавита.
Аннотлция. Наибольшая длина (LCS) общей подпоследовательности пары случайных конечных последовательностей из 4 букв рассмотрена как случайная функция от длин m и п этих двух последовательностей. Представлены таблицы точных значений вероятностей для всех пар конкретных длин в диапазоне 2 < m + п < 19.
Графики зависимости математического ожидания и дисперсии показаны в линейной перспективе, позволяющей просматривать на горизонте поведение при растущих длинах. Для иллюстрации поведения при больших значениях длин на этих же графиках показаны результаты численного эксперимента для больших значений m + п = 32, 512, 8192 и 131072. Представленный график зависимости математического ожидания от m и п выглядит имеющим асимптотический прямой
3
круговой конус. Дисперсия выглядит растущей как (п + т) 4 .
Ключевые слова и фразы: сходство строк, выравнивание последовательностей, случайные общие подпоследовательности, LCS, метрика Левенштейна.
Пример ссылки на эту публикацию:
С. В. Знаменский. «Картина наибольшей длины общих подпоследовательностей пары случайных строк 4 буквенного алфавита», Программные системы: теория и приложения, 2016, 7:1(28), с. 201-208. (Англ). URL: http://psta.psiras.ru/read/psta2016_1_201-208.pdf
© © ©
С. В. Знаменский, 2016 Институт программных систем Программные системы: теория
имени А. К. Айламазяна РАН, и приложения, 2016
2016
