7 research outputs found
Succinct Partial Sums and Fenwick Trees
We consider the well-studied partial sums problem in succint space where one
is to maintain an array of n k-bit integers subject to updates such that
partial sums queries can be efficiently answered. We present two succint
versions of the Fenwick Tree - which is known for its simplicity and
practicality. Our results hold in the encoding model where one is allowed to
reuse the space from the input data. Our main result is the first that only
requires nk + o(n) bits of space while still supporting sum/update in O(log_b
n) / O(b log_b n) time where 2 <= b <= log^O(1) n. The second result shows how
optimal time for sum/update can be achieved while only slightly increasing the
space usage to nk + o(nk) bits. Beyond Fenwick Trees, the results are primarily
based on bit-packing and sampling - making them very practical - and they also
allow for simple optimal parallelization
Partial Sums on the Ultra-Wide Word RAM
We consider the classic partial sums problem on the ultra-wide word RAM model
of computation. This model extends the classic -bit word RAM model with
special ultrawords of length bits that support standard arithmetic and
boolean operation and scattered memory access operations that can access
(non-contiguous) locations in memory. The ultra-wide word RAM model captures
(and idealizes) modern vector processor architectures.
Our main result is a new in-place data structure for the partial sum problem
that only stores a constant number of ultraword in addition to the input and
supports operations in doubly logarithmic time. This matches the best known
time bounds for the problem (among polynomial space data structures) while
improving the space from superlinear to a constant number of ultrawords. Our
results are based on a simple and elegant in-place word RAM data structure,
known as the Fenwick tree. Our main technical contribution is a new efficient
parallel ultra-wide word RAM implementation of the Fenwick tree, which is
likely of independent interest.Comment: Extended abstract appeared at TAMC 202