1. CS240B—Fall 2018
Task 4.1.
 Express the Flajolet-Martin's distinct_count sketch as a user-defined aggregate mamed dcount_sketch, to be called in the same way as d_count. You can assume that you have available a function LmostbitH(X) that return K, where the K position contains a 1, whereas all the position to its right are zeros, for the value returned by a randomizing hash function H(X).
We will design a window aggregate that e.g., could be called as follows:
SELECT col_name1, dcount_sketch(col_name2)OVER (ROWS PRECEDING) FROM my_stream;

2. FM dcount_sketch WINDOW AGGREGATE dcount_sketch(next Real) : Real
{ TABLE bitarray (bitpos int, bitvalue int); TABLE inwindow(wnext Real);
INITIALIZE : {insert into bitarray VALUES (1,0), …, (64, 0);
update bitarray SET bitvalue=1 WHERE bitpos= LmostbitH(next )}
ITERATE : {/*the system inserts the new tuple in invindow at the end of iterate*/ update bitarray SET bitvalue=1 WHERE bitpos= LmostbitH(next)};
DELETE FROM inwindow WHERE LmostbitH.wnext=LmostbitH(next) ; INSERT INTO RETURN SELECT 2** MAX(bitpos) /*the estimated count*/ FROM bitarray WHERE BITVALUE=1
%we could also delete weak bits---e.g. those that are less than max-8
%DELETE FROM inwindow WHERE bitpos< MAX(bitpos)-8}
EXPIRE: { /*Expire is processed before iterate*/ UPDATE bitarray SET bitvalue= WHERE bitpos=(SELECT LmostbitH(wnext) FROM inwindow WHERE oldest(inwindow) )} }

3. Task 4.2:  Assume that  you have a stream of temperature readings temperature(Celsius Integer)
 that start everyday at time 00:01  and end at  time 23:59.  At the end of each day, we want to have 
10,000 temperature samples stored into a table   tenKsamples(Rowno integer, Celsius Integer).We do not know how many temperature readings are going to arrive every day, except that their number is significantly larger than 10,000. Please write a UDA that uses the reservoir algorithm to populate tenKsamples(Rowno , Celsius) with 10,000 random samples taken from 
temperature(Celsius Integer), which is then processed and reset to empty at midnight. 
 You can assume that the system support a function random(K), which given a  positive integer K returns a random integer between 1 and K.

4. AGGREGATE reservoir(next integer) : integer
{ TABLE tenKsamples(Rowno integer, Celsius Integer) external;
TABLE cntuples (cnt Integer);
INITIALIZE : {insert into cntuples values 1;
insert into tenKsamples values (1, next);
ITERATE : {update cntuples set cnt=cnt+1;
Insert into tenKsamples
select (cnt, Next) from cntuples where cnt<10000;
UPDATE tenKsamples set Celsius=next,
where Rowno= random(10000) and
1= select(random(cnt), from cntuples
where cnt>10000)
Terminate: {%we might want to return the count}