Approximating General Metric Distances Between a Pattern and a Text

Let $T=t_0 ... t_{n-1}$ be a text and $P = p_0 ... p_{m-1}$ a pattern taken from some finite alphabet set $\Sigma$, and let $\dist$ be a metric on $\Sigma$. We consider the problem of calculating the sum of distances between the symbols of $P$ and the symbols of substrings of $T$ of length $m$ for all possible offsets. We present an $\epsilon$-approximation algorithm for this problem which runs in time $O(\frac{1}{\epsilon^2}n\cdot \mathrm{polylog}(n,\abs{\Sigma}))$