How does a Morris approximate counter work?

If you want to count to a very big number, then your standard 32-bit or 64-bit types will limit you. The ordinary solution is to use more memory: switch 128 bits, or to a “bignum” type. But what if you don’t have the space to store this number, and you actually don’t care about the exact value of the counter, only an approximation? In this case, you could use an “approximate counter”, invented by Robert Morris:

#include <math.h>
#include <stdio.h>
#include <stdint.h>
#include <limits.h>
#include <stdlib.h>

typedef uint8_t morris_t;
morris_t morris_new(uint64_t v) { return round(log2(v + 2)); }
uint64_t morris_estimate(morris_t c) { return (2 << c) - 2; }
morris_t morris_increment(morris_t c) { return (rand() % (2<<c)) ? c : c+1; }

Here’s an example of usage which re-creates the unix tool wc -c, counting the number of characters from stdin:

#include "morris.c"
#include <unistd.h>
int guard(int n, char* err) { if (n == -1) { perror(err); exit(1); } return n; }
int main(void) {
  sranddev();
  char buf[1024];
  morris_t bytes_seen = morris_new(0);
  for (;;) {
    ssize_t bytes_read = guard(read(0, buf, sizeof(buf)), "could not read stdin");
    if (bytes_read == 0) break;
    while (bytes_read--) bytes_seen = morris_increment(bytes_seen);
  }
  printf("     %llu\n", morris_estimate(bytes_seen));
  return 0;
}
$ cc wc.c
$ cat wc.c | wc -c    # first, the exact answer ...
     471
$ cat wc.c | ./a.out  # the estimated answer - a bit off, but in the ballpark
     254
$ cat wc.c | ./a.out  # try it again - a different answer, also in the ballpark
     510

But how does the Morris counter work? Forget counting for the moment, and concentrate on just storing large numbers. The floating-point standard IEEE 754 shows how to do this: it stores numbers in the form sig * 2 ^ exp. The 32-bit float type dedicates 8 bits to the exponent, exp. A float can represent much larger numbers than an int; the price it pays for this is accuracy. Our Morris counter is very similar: its 8 bits are just like the 8-bit exponent part of a float. Just like the float, the Morris counter can represent very large numbers, but it sacrifices accuracy.

To initialize a Morris counter with a value, we find the exponent, using round(log2(v)). For example, instead of storing the exact value 32, instead store the exponent 5, since 2 ^ 5 = 32. To get the actual counter, calculate pow(2, 5), or 2 << 5 in C.

Then, to increment the counter, do so probabilistically. If the counter currently stores 5, this represents a range of 2 ^ 32 possible values. The counter should only be incremented if it represents the maximum value in that range. The probability of this is 1/32, and so we increment with this probability.

Actually, the implementation I showed is slightly different: the implementation has a mysterious + 2 in the initialization, and a mysterious - 2 in the estimate function. Where does this constant come from? Honestly, I don’t know. Here’s a comprehensive analysis by Flajolet.

(I learned about this at Redis Day London 2018, in a talk that Elena Kolevska gave about Redis cache eviction. You can see the Redis implementation here.)

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