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Code snippets used to generate to generate the binary images from Java.
Heavy items have negative weights to allow a simple predicate to filter
heavy vs light sketch entires.


varopt_sketch_long_sampling.bin:
final VarOptItemsSketch<String> sk = VarOptItemsSketch.newInstance(1024);
for (int i = 1; i <= 200; ++i) {
    sk.update(Integer.toString(i), 1000.0 / i);
}
byte[] bytes = sk.toByteArray(new ArrayOfStringsSerDe());


varopt_sketch_string_exact.bin:
final VarOptItemsSketch<Long> sk = VarOptItemsSketch.newInstance(1024);
for (long i = 0; i < 2000; ++i) {
  sk.update(i, 1.0);
}
sk.update(-1L, 100000.0);
sk.update(-2L, 110000.0);
sk.update(-3L, 120000.0);
byte[] bytes = sk.toByteArray(new ArrayOfLongsSerDe());


varopt_union_double_sampling.bin:
// parallels small samplign sketch test
final int kSmall = 16;
final int n1 = 32;
final int n2 = 64;
final int kMax = 128;

// small k sketch, but sampling
VarOptItemsSketch<Double> sketch = VarOptItemsSketch.newInstance(kSmall);
for (int i = 0; i < n1; ++i) {
  sketch.update(1.0 * i, 1.0);
}
sketch.update(-1.0, n1 * n1); // add a heavy item

final VarOptItemsUnion<Double> union = VarOptItemsUnion.newInstance(kMax);
union.update(sketch);

// another one, but different n to get a different per-item weight
sketch = VarOptItemsSketch.newInstance(kSmall);
for (int i = 0; i < n2; ++i) {
  sketch.update(1.0 * i, 1.0);
}
union.update(sketch);
byte[] bytes = union.toByteArray(new ArrayOfDoublesSerDe());