#####   ######  #####    ###    #   #  ###  #   #  ######
##  ##  ##      ##  ##  ## ##   #   #   #   #   #  ##
#####   ####    #####   #   #   #   #   #   #   #  ####
##  #   ##      ##      ## ##    # #    #    # #   ##
##   #  ######  ##       ###      #    ###    #    ######

$ curl repovive.com/roadmaps/data-structures/wavelet-trees/efficient-rank-with-prefix-sums

░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░██████████████████████████████████████████████████████████████████████████████████████████

#####   ######  #####    ###    #   #  ###  #   #  ######
##  ##  ##      ##  ##  ## ##   #   #   #   #   #  ##
#####   ####    #####   #   #   #   #   #   #   #  ####
##  #   ##      ##      ## ##    # #    #    # #   ##
##   #  ######  ##       ###      #    ###    #    ######

$ curl repovive.com/roadmaps/data-structures/wavelet-trees/efficient-rank-with-prefix-sums

Repovive

Efficient Rank with Prefix Sums - Wavelet Trees | Data Structures | Repovive

Repovive.

Data Structures19 sections · 729 units49h total

Open in Course

Efficient Rank with Prefix Sums

O(1) rank queries

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.