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$ curl repovive.com/roadmaps/data-structures/fenwick-trees/order-statistics-with-bit

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$ curl repovive.com/roadmaps/data-structures/fenwick-trees/order-statistics-with-bit

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Order Statistics with BIT - Fenwick Trees | Data Structures | Repovive

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Data Structures19 sections · 729 units49h total

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Order Statistics with BIT

Finding k-th element

BIT can answer: "what is the $k$ -th smallest element currently in the structure?"

Approach: binary search on the prefix sum.

def kthSmallest(k):
    pos = 0
    for i in range(LOG, -1, -1):  # LOG = log2(n)
        if pos + (1 << i) <= n and tree[pos + (1 << i)] < k:
            pos += (1 << i)
            k -= tree[pos]
    return pos + 1

This descends the BIT structure bit by bit, similar to binary lifting.

Use case: maintain a dynamic multiset, supporting insert, delete, and k-th element queries in $O(\log n)$ each.