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For given two lists $[s_1, s_2, ... s_n]$ and $[t_1, t_2, ..., t_m]$

I need to implement DTW algorithm with one extra constraint:

If $s_i$ is matched with $t_j$ then the next element $s_{i+1}$ has to be matched with some $t_{j+k}$ close enough to $t_j$ (i.e. $0 \le k \le W$ for some fixed window size $W$).

This constraint has to hold for each $s_i$.

1) Does there exist $O(nm)$ solution to such problem?

2) If an efficient algorithm does not exist, what would be a good practical solution?

(Note that locality constraint mentioned in the DTW link above is different from the constraint I need).

mercury0114
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2 Answers2

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There is a $O(nmW)$-time algorithm using dynamic programming. Let $A[i,j] = $ the cost of the best matching of $[s_1,\dots,s_i]$ to $[t_1,\dots,t_j]$ such that $s_i$ is matched to $t_j$. Then

$$A[i,j] = \min\{c(s_i,t_j) + A[i-1,j-k] : k=0,1,\dots,W\}.$$

If you consider $W$ a constant, then you obtain a $O(nm)$-time algorithm.

I don't know if the factor of $W$ can be eliminated.

If $W$ is large, this can be improved to $O(nm \lg W)$ time. The basic primitive we need is:

Given an array $B[1..m]$ and $W$, preprocess $B$ so that we can efficiently answer queries "compute $\min\{B[i],B[i+2],\dots,B[i+W-1]\}$".

Here's how to do that. Assume first for simplicity that $W$ is a power of two. You construct secondary arrays $M_2,M_4,M_8,\dots,M_W$ such that $M_w[i] = \min\{B[i],B[i+1],\dots,B[i+w-1]\}$. You can build them up in $O(m \lg W)$ time, as each $M_{2w}$ can be constructed from $M_w$ in $O(m)$ time. Thus if $W$ is a power of two you can answer subsequent queries in $O(1)$ time per query.

If $W$ isn't a power of two, you can still express the interval $i,i+1,i+2,\dots,i+W-1$ as the union of at most $\lg W$ intervals each of whose width is a power of two. Thus, if $W$ isn't a power of 2, you can answer subsequent queries in $O(\lg W)$ time per query.

Finally, apply this to the original problem. You apply this to each row of $A$, i.e., to $A[i,\cdot]$ for each $i$. Then the recurrence can be computed using a single query, and it takes $O(\lg W)$ time instead of $O(W)$ time to compute the min. You'll need to update the secondary arrays each time you update $A[i,j]$, but this takes only $O(\lg W)$ time per update to $A[i,j]$ (as there are only $\lg W$ secondary arrays to update).

In all, you obtain an algorithm whose running time is $O(nm \lg W)$ and with $O(nm \lg W)$ space usage. (I suspect it's also possible to achieve $O(nm \lg W)$ time and $O(nm)$ space, if necessary, by only storing $M_w[i]$ for values of $i$ that are a multiple of $w$, at the cost of about a 2x increase in running time.)

D.W.
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Although the accepted answer is correct, an alternative solution also works well in practise, when one is not very strict about the $0 \leq k \leq W$ constraint, but still wants to have the "locality" property:

1) Let $cost[i][j]$ be the minimum cost to align $[s_1, s_2, ... s_i]$ and $[t_1, t_2, ..., t_j]$. We first build $cost$ matrix in $O(nm)$ time using dynamic programming.

2) We find the minimum index $last$ such that $s_n$ could have been aligned with $t_{last}$:

last = m;
// While ignoring extra element in the sequence (t_n) won't make cost worse
while (cost[n][last] >= cost[n][last - 1])
    // We ignore it
    last--;
}
return last;

3) After we know $last$, we find the maximum index $first$ such that $s_1$ could have been aligned with $t_{first}$:

int first = last;
    int i = n - 1;
    while (i > 0) {
        // if to align i-th element with first is not worse that other options  
        if (cost[i - 1][first] + distance(i, first) <= cost[i][first - 1]) {
            // we align it
            i--;
        } else {
            // otherwise we have to use one more point from sequence (t_n)
            first--;
        }
    }
return first;

4) At the end we have found a minimal interval $[t_{first}, ... t_{last}]$ with which the whole sequence $(s_n)$ was aligned. If the length of this interval is not too big (e.g. less than $\frac{3}{2}n)$ we can assume that an alignment satisfying "locality" property has been found.

The nice thing about this approach is that it can be coded in $O(nm)$ time and the code is relatively short.

mercury0114
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