Computable Bounds and Monte Carlo Estimates of the Expected Edit Distance

Abstract: The edit distance is a metric of dissimilarity between strings, widely applied in computational biology, speech recognition, and machine learning. Let

e_{k} (n)

denote the average edit distance between random, independent strings of

n

characters from an alphabet of size

k

. For

k g e q 2

, it is an open problem how to efficiently compute the exact value of

a l p h a_{k} (n) = e_{k} (n) / n

as well as of

a l p h a_{k} = l i m_{n o i n f t y} a l p h a_{k} (n)

, a limit known to exist. This paper shows that

a l p h a_{k} (n) - Q (n) l e q a l p h a_{k} l e q a l p h a_{k} (n)

, for a specific

Q (n) = T h e t a (s q r t l o g n / n)

, a result which implies that

a l p h a_{k}

is computable. The exact computation of

a l p h a_{k} (n)

is explored, leading to an algorithm running in time

T = m a t h c a l O (n^{2} k m i n (3^{n}, k^{n}))

, a complexity that makes it of limited practical use. An analysis of statistical estimates is proposed, based on McDiarmid's inequality, showing how

a l p h a_{k} (n)

can be evaluated with good accuracy, high confidence level, and reasonable computation time, for values of

n

say up to a quarter million. Correspondingly, 99.9% confidence intervals of width approximately

1 0^{- 2}

are obtained for

a l p h a_{k}

. Combinatorial arguments on edit scripts are exploited to analytically characterize an efficiently computable lower bound

to

a l p h a_{k}

, such that

. In general,

; for

k

greater than a few dozens, computing

is much faster than generating good statistical estimates with confidence intervals of width

. The techniques developed in the paper yield improvements on most previously published numerical values as well as results for alphabet sizes and string lengths not reported before.

Mathematics Subject Classification ID

Combinatorics in computer science (68R05) Rate of convergence, degree of approximation (41A25) Algorithms on strings (68W32)

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