1. Ariel Rosenfeld

2.  Counter ranges from 0 to M requiers log 2 M bits.  For large data log 2 M is still a lot.  Using probability to reduce to log 2 log 2 M bits. ◦ Small probability of errors.

3. Counting of a large number of events using a small amount of memory, while incorporating some probability. 1977 by Robert Morris.Robert Morris 1982 analyzed by Philippe Flajolet.Philippe Flajolet

4.  Gathering statistics on a large number of events  Streaming data frequency  Data compression  Etc..

5. Because we give up accuracy, we use 2 k approximation and only keep the exponent. Representing if the approximate number is M, we only keep 2 k =M in binary form. Log 2 log 2 M How do we know when to increase k?

6.  Generate "c" pseudo-random bits ◦ "c" = current value of the counter  If all are 1 ◦ What is the probability? ◦ How to check it efficiently?  Simply add the result to the counter.

9.  What is the probability of increment? ◦ 2 -C  After N increments (probabilistic explanation in article) ◦ E(2 C ) = n+2 ◦ Var(2C) = n(n+ 1)/2 ◦ Small chance to be “far off”.

10.  Increase was called 1024 times. ◦ Correct value should be 10. ◦ Chance of being more than 1 off is ~8%.