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Problems

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Range queries

Fenwick tree (BIT)

Point update + prefix sum in O(log n) with i & -i.

Visual guide Quiz (8) Spot the pattern
Notes

Recognise it when: you need point updates and prefix or range sums, both in O(log n). Also for counting inversions and "smaller elements after self" over compressed values.

def add(i, v):            # 1-indexed
    while i <= n: t[i] += v; i += i & -i
def prefix(i):
    s = 0
    while i > 0: s += t[i]; i -= i & -i
    return s

A range sum is prefix(r) - prefix(l - 1).

Gotchas: it must be 1-indexed. It only handles invertible operations (sum, xor), not min or max.

5 problems

Advanced & competitive

Range queries Fenwick trees, segment trees, sparse tables.

Fenwick tree (BIT) guide

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