Finding most similar Range in an Array

Question

I am finding A[i..j] that is the most similar to B. Here calcSimilarity is function that returns similarity of two arrays. Similarity is calculated as

Not than brute force search, I want to know what kind of data structure and algorithm is efficient in range search.

SAMPLE input/output

input: A: [(10,1), (20,1), (-200,2), (33,1), (42,1), (58,1)]   B:[(20,1), (30,1), (1000,2)]
output: most similar Range is [1, 3]
        match [20, 33] => [20, 30]

This is brute force search code.

struct object{
    int type, value;
}A[10000],B[100];
int N, M;
int calcSimilarity(object X[], n, object Y[], m){
    if(n > m) return calcSimilarity(Y, m, X, n);

    for(all possible match){//match is (i, link[i])
        int minDif = 0x7ffff;
        int count = 0;
        for( i = 0; i< n; i++){
            int j = link[i];
            int similar = similar(X[i], Y[j]);
            minDif = min(similar, minDif);
        }
    }
    if(count == 0) return 0x7fffff;
    return minDif/pow(count,3);
}
find_most_similar_range(){
    int minSimilar = 0x7fffff, minI, minJ;
    for( i = 0; i < N; i ++){
       for(j = i+1; j < N; j ++){
            int similarity = calcSimilarity(A + i, j-i, B, M);
            if (similarity < minSimilar)
            {
                minSimilar = similarity;
                minI= i;
                minJ = j;
            }
       }
    }
    printf("most similar Range is [%d, %d]", minI, minJ);
}

Answer 1

it will take O((N^M) * (N^2)).

That looks like the Big-O of the find similarity is N^2. With the pairwise comparison of each element.

So it looks more like

The pairwise comparison is M*(M-1). Each list has to be tested against each other list or about M^2.

This is a problem which has been solved for clustering, and there are data structures (e.g. Metric Tree), which allow the distances between similar objects to be stored in a tree.

When looking for the N closest neighbours, the search of this tree limits the number of pairwise comparisons needed and results in a O( ln(M) ) form

The downside of this particular tree, is the similarity measure needs to be metric. Where the distance between A and B, and the distance between B and C allows inferences to be made about the distance range of A and C.

If your similarity measure is not metric, then this can't be done.

Jaccard distance is a metric of distance which allows it to be placed in a Metric tree.