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Basic Algorithms for fining Heavy Elements in Databases

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• Algorithms to compute an initial solution set F
  • The following algorithms are the basis for better algorithms to find the solution set F

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  • Scaled-Sampling

    S = uniform random sample of R of size s (e.g.: Concise Sample) for ( each e ∈ S ) { N count[e] = count of e scaled by ---; |S| if ( count[e] ≥ T ) { add e to solution set F; } }

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  • Properties of the algorithm:

    Elements selected are likely to be heavy... However... Scaled-Sampling can generate false positives Scaled-Sampling can generate false negatives Positive trait: The output size can be very small if the selection probability is set low enough

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  • Coarse-Count (probabilistic counting)

    h1(x) = hash function that maps a value t of R to 1..m A[1..m] = m counter variables /* ----------------------------- initialize counters ----------------------------- */ for ( i = 1; i <= m; i++ ) A[i] = 0; /* ------------------------------ Hash-count items in R ------------------------------ */ for ( each e ∈ R ) A[ h1( e ) ] ++; // Note: 2 items e1 and e2 that hashes // to the same value will be counted together... /* ------------------------------ Form solution set F ------------------------------ */ for ( each v ∈ R ) { if ( A[ h1( e ) ] ≥ T ) add e to F; }

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  • Properties of the algorithm:

    Coarse-Counting can generate false positives Scaled-Sampling does not generate false negatives !!!! If an item e occurs at least T times, then count[ h1(e) ] ≥ T However: F generated by Coarse-Counting can be very large....

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• Improving the basic algorithms
  • Next webpage.....