Freaks City

The following C++ implementation demonstrates a complete Heavy-Light Decomposition (HLD) framework integrated with a lazy propagation segment tree to support efficient path updates and queries on trees. It passes the standard template problem on Luogu. #include <bits/stdc++.h> using namespace std; using i64 = long long; int MOD; struct ...

Definition of Degree The degree of a vertex (v) in a graph (G), denoted by (d_G(v)), is the number of edges of (G) incident with (v), with each loop counting as two edges. (J.A. Bondy, Graph Theory) Two fundamental results follow from this definition: Handshaking Theorem: For any graph (G), [ \sum_{v \in V} d(v) = 2m ] where (m) is the number ...

Sorting and Monotonicity When a problem does not enforce a specific elemant order, applying a sort operation often introduces monotonicity. This property simplifies constraint checking and enables efficient querying through prefix sums combined with binary search. Processing Cumulative Constraints By sorting the input array and computing its pr ...

Prim's Algorithm for Minimum Spanning Trees Prim's algorithm utilizes a distance array where dist[j] represents the shortest distance from node j to the current connected component. The process begins by selecting an arbitrary starting node and initializing distances to all other nodes. Algorithm Steps: Initialize all distances to infinity exc ...