Zkw线段树 — ((top))

Abstract The segment tree is a fundamental data structure for range queries and point updates. While recursive implementations are intuitive, they suffer from high constant factors due to function call overhead and conditional branching. This paper describes the zkw segment tree , a non‑recursive alternative introduced by Zhang Kunwei (zkw). By storing data in a perfect binary tree indexed from the bottom layer, it eliminates recursion entirely. The resulting implementation is shorter, faster, and particularly well‑suited for competitive programming and low‑latency systems. 1. Introduction A standard segment tree is usually built as an array tree[] of size about 4*N . Recursive functions traverse the tree to answer range sum/min/max queries or apply point updates. Despite its asymptotic $O(\log N)$ performance, recursion overhead and repeated bounds checking slow down execution.

On a sum tree, find smallest p such that sum[0..p] >= k . zkw线段树

template<typename T> class ZkwSegTree int N; vector<T> tree; public: ZkwSegTree(int n, const vector<T>& init) N = 1; while (N < n) N <<= 1; tree.assign(2*N, 0); for (int i = 0; i < n; i++) tree[N+i] = init[i]; for (int i = N-1; i; i--) tree[i] = tree[2*i] + tree[2*i+1]; void update(int p, T val) p += N; tree[p] = val; while (p >>= 1) tree[p] = tree[2*p] + tree[2*p+1]; T query(int l, int r) // inclusive l += N, r += N; T res = 0; while (l <= r) if (l & 1) res += tree[l++]; if (!(r & 1)) res += tree[r--]; l >>= 1; r >>= 1; return res; ; Abstract The segment tree is a fundamental data

int lower_bound(int k) int pos = 1; while (pos < N) if (tree[pos<<1] < k) k -= tree[pos<<1]; pos = pos<<1 else pos = pos<<1; return pos - N; By storing data in a perfect binary tree

Prefix sum [0, r] :