Popcountn

Popcountn is a generalized form of the population count operation in computer science. It computes the number of set bits (bits equal to 1) in the first n bits of an integer, where n is a nonnegative parameter. In some contexts popcountn may refer to counting within an n‑bit word or within a subrange of a word, as defined by a mask. The standard popcount counts all set bits in a word; popcountn narrows the scope to n bits.

Formally, for an integer x and 0 ≤ n ≤ w (the word width) or defined by the mask, popcountn(x, n) = sum_{i=0}^{n-1} bit_i(x). If n > w, the value is equivalent to popcount(x) (treating missing bits as zero). A common implementation is popcountn(x, n) = popcount(x & ((1 << n) - 1)), valid for n < w, with unsigned arithmetic. Variants exist when counting within arbitrary subsets defined by a bitmask rather than a simple prefix of bits.

Hardware and software: On modern CPUs, a popcnt instruction returns the popcount of a full word; popcountn can often be computed in constant time by masking first. In languages with fixed width integers, care is needed to avoid overflow when constructing the mask (using unsigned types and appropriate shifts). SIMD and parallel processing can count bits in blocks to handle larger datasets efficiently.

Applications: Popcountn is useful in algorithms that require the density of set bits within a limited range, such as bitset operations on subranges, compressed data structures, or performance-critical code where partial population counts influence control flow or analytics.

See also: population count, bit count, bit masking, bit operations, hardware popcnt.