How do Segment Trees facilitate range querying and updating?
Instruction: Discuss the structure of Segment Trees and how they are used for range querying and update operations.
Updated
Official answer
A segment tree stores an aggregate for nested intervals of an array. Leaves represent elements; each internal node combines its two children's aggregates. For a sum tree, [2, 1, 4, 3] has total 10, and querying the half-open interval [1,3) returns 1+4=5.
A range query combines disjoint stored intervals covering the requested range. A point assignment changes one leaf and recomputes its ancestors. With constant-cost associative combination, the standard tree uses O(n) space/build time and supports these operations in O(log n) time. The aggregate need not be commutative if left-to-right combination order is preserved.
A plain array already has O(1) point assignment; the tree's benefit is supporting efficient queries while values change. Static prefix sums may be simpler when updates are absent. General range updates are not automatically logarithmic: they need a compatible lazy-update representation or another method.
Define interval conventions and the identity for an empty contribution, such as zero for sum. Test empty/full/singleton ranges, boundary indices, negative values and updates that affect queried and unqueried segments. For averages, retain enough state such as sum and count rather than averaging unequal-sized child averages.