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$ curl repovive.com/roadmaps/data-structures/fenwick-trees/finding-first-position

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$ curl repovive.com/roadmaps/data-structures/fenwick-trees/finding-first-position

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Finding First Position - Fenwick Trees | Data Structures | Repovive

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Data Structures19 sections · 729 units49h total

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Finding First Position

Binary search on BIT

Given a prefix sum value, find the smallest index $i$ such that $\text{prefix}(i) \geq k$ .

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

This walks down the BIT structure, jumping to larger indices when the sum is still too small. Time: $O(\log n)$ .

Use case: finding the $k$ -th positive element in a 0/1 array, implementing an ordered multiset.