Freaks City

Introduction Heavy-light decomposition (HLD) is a sophisticated algorithmic technique used to partition tree structures into linear sequences, enabling efficient query and update operations. This method is particularly effective for handling subtree and path queries on trees. Core Definitions Heavy Child: For any node, its heavy child is the c ...

Introduction During the summer vacation, I systematically studied various operations on trees through Teacher Tuo's sharing and discovered many useful techniques, which I now summarize. Teacher Tuo is amazing! One Problem Type: Batch processing queries of the form f(dep(lca(x,y))), where f(x) is a function of x. Approach: First perform tree ...

Problem Luogu P2881 Given \(n\) numbers and \(m\) relations of the form \(a > b\), determine how many additional pairwise comparisons are needed to deduce the total order of all numbers. Solution Two approaches exist: Floyd‑Warshall and topological sort. This article focuses on the latter. If no relations are given, we need to check every pa ...

Binary Search Methodologies Integer binary search typically relies on partitioning a range [left, right] based on a predicate function. Two common partitions are used depending on whether the midpoint belongs to the left or right sub-interval. // Partition: [left, pivot] | [pivot + 1, right] int find_first_valid(int left, int right) { while ...

Problem A - AtCoder Line Straightforward check: determine whether point z lies between x and y on the number line. Simply swap if necessary to ansure x ≤ y, then verify the condition. Click to view code #include <cstdio> #include <algorithm> using namespace std; int main() { int n, p, q, r; scanf("%d%d%d%d", & ...

The problem involves finding the minimum number of vertices to remove from an undirected graph to make it disconnected. Problem Analysis When a graph becomes disconnected, there exist at least two vertices that cannot reach each other. We can enumerate these two vertices as source and sink, then determine the minimum number of other vertices th ...

Overview This contest proved challenging despite seemingly moderate difficulty. The overall rating leans toward green to purple, but the execution was frustrating. T1 cost me significant points due to rushing through it—225 dropped to 175 points. Strategic lesson: even when T1 appears simple, allocating proper time (up to 1.5 hours is reasonabl ...

[CTS2024] The Gate of All Beings This is a constructive problem on tree traversal. Observation of large test cases shows the answer does not exceed 3. It is posssible to traverse the entire tree with paths of length at most 3. The answer is typically 0 or 1, except for small trees or star-shaped graphs. For small n (≤ 8), a brute-force search o ...

Given a base tree and a subset of its nodes, the virtual tree preserves the ancestral relationships among these nodes by including all pairwise LCAs and cnonecting them appropriately. This structure maintains relative node relationships while being linear in size relative to the input set. When nodes are sorted by their Euler tour order, the se ...

Directed Edge Orientation with Out-Degree ConstraintGiven an undirected graph, determine the number of ways to orient all edges such that every vertex has an out-degree of exactly 1. The result should be modulo 998244353.For such an orientation to exist, the number of edges must exactly equal the number of vertices, i.e., m = n. Furthermore, ev ...