Ariel Rosenfeld 1. 2 1 0 1 1 1 0 1 0 0 1 1 Network Traffic Engineering. Call Record Analysis. Sensor...
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Transcript of Ariel Rosenfeld 1. 2 1 0 1 1 1 0 1 0 0 1 1 Network Traffic Engineering. Call Record Analysis. Sensor...
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Network Traffic Engineering.
Call Record Analysis.
Sensor Data Analysis.
Medical, Financial Monitoring.
Etc, etc, etc.
Count the number of ones in N size window. Exact Solution: Θ(N) memory. Approximate Solution: ?
◦ Good approx with o(N) memory?
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Exponential Histogram (EH):◦ 1 + ε approximation. (k = 1/ε)◦ Space: O(1/ε(log2N)) bits.◦ Time: O(log N) worst case, O(1) amortized.
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Bucket sizes = 4,2,2,2,1.Bucket sizes = 4,4,2,1.Bucket sizes = 4,2,2,1,1,1.Bucket sizes = 4,2,2,1,1.Bucket sizes = 4,2,2,1.
….1 1 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1…
k/2 = 1.
Element arrived this step.
Future
Error in last (leftmost) bucket. Bucket Sizes (left to right): Cm,Cm-1, …,C2,C1
Absolute Error <= Cm/2. Answer >= Cm-1+…+C2+C1+1. Error <= Cm/2(Cm-1+…+C2+C1+1). Maintain: Cm/2(Cm-1+…+C2+C1+1) <= 1/k.
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Every Bucket will become last bucket in future.
New elements may be all zeros. Bucket Sizes (left to right): Cm,Cm-1, …,C2,C1
For every bucket i,◦ Ci/2(Ci-1+…+C2+C1+1) <= 1/k.
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Maintain Ci/2(Ci-1+…+C2+C1+1) <= 1/k. Exponentially increasing bucket sizes from
right to left. At least k/2 buckets (at most k/2 +1)of each
size(1,2,4,8,…,2i,...).
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Error Guarantee: ◦ Error <= Cm/2(Cm-1+…+C2+C1) <= 1/k.
Number of buckets: O(k log N). Buckets require O(log N) bits. Total memory: O(k log2 N) bits.
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