Why does Google sparsehash open-source library has two implementations: a dense hashtable and a sparse one?
The dense hashtable is your ordinary textbook hashtable implementation.
The sparse hashtable stores only the elements that have actually been set, divided over a number of arrays. To quote from the comments in the implementation of sparse tables:
// The idea is that a table with (logically) t buckets is divided
// into t/M *groups* of M buckets each. (M is a constant set in
// GROUP_SIZE for efficiency.) Each group is stored sparsely.
// Thus, inserting into the table causes some array to grow, which is
// slow but still constant time. Lookup involves doing a
// logical-position-to-sparse-position lookup, which is also slow but
// constant time. The larger M is, the slower these operations are
// but the less overhead (slightly).
To know which elements of the arrays are set, a sparse table includes a bitmap:
// To store the sparse array, we store a bitmap B, where B[i] = 1 iff
// bucket i is non-empty. Then to look up bucket i we really look up
// array[# of 1s before i in B]. This is constant time for fixed M.
so that each element incurs an overhead of only 1 bit (in the limit).