Efficient Rank with Prefix Sums

Precompute prefix sums for each bitvector:

function preprocess(bitvector)
    prefix := array of size length + 1
    prefix[0] := 0
    for i from 0 to length - 1
        prefix[i + 1] := prefix[i] + bitvector[i]
    return prefix

function rank1(prefix, i)
    return prefix[i]

function rank0(prefix, i)
    return i - prefix[i]

$\text{prefix}[i]$ stores the count of 1s in the first $i$ positions.

Time: $O(n)$ preprocessing per bitvector, $O(1)$ per rank query.

Space: $O(n)$ extra per bitvector.