Sage-Tips

What is centroid decomposition?

Centroid Decomposition is a divide and conquer technique for trees. Centroid Decomposition works by repeated splitting the tree and each of the resulting subgraphs at the centroid, producing O ( log ⁡ N ) \mathcal{O}(\log N) O(logN) layers of subgraphs.

What is heavy light decomposition used for?

Heavy-light decomposition is a fairly general technique that allows us to effectively solve many problems that come down to queries on a tree .

What is a centroid in a tree?

Centroid of a Tree is a node which if removed from the tree would split it into a ‘forest’, such that any tree in the forest would have at most half the number of vertices in the original tree.

What is HLD in competitive programming?

Heavy Light decomposition (HLD) is one of the most used techniques in competitive programming.

What is Mo’s algorithm?

Mo’s algorithm is a generic idea. It applies to the following class of problems: You are given array Arr of length N and Q queries. Each query is represented by two numbers L and R, and it asks you to compute some function Func with subarray Arr[L..

What is binary lifting?

Binary Lifting is a technique used to find the k-th ancestor of any node in a tree in O(logn). This also leads to a faster algorithm in finding the lowest common ancestor (LCA) between two nodes in a tree. The technique requires preprocessing the tree in O(N log N) using dynamic programming.

How many centroid can a tree have?

A tree may have one centroid or may have two centroids. If it has two centroids, they are always connected (otherwise, the tree can’t have n vertices).

What is lazy propagation?

Lazy propagation is a range update and query optimized implementation of a segment tree that performs both operation O(logN) time. *In short, as the name suggests, the algorithm works on laziness for what is not important at the time. Don’t update a node until needed, which will avoid the repeated sharing.

Is centroid and Circumcenter the same?

The centroid of a triangle is the point at which the three medians meet. The circumcenter is also the center of the circle passing through the three vertices, which circumscribes the triangle. This circle is sometimes called the circumcircle.

What is heavy-light decomposition?

Heavy-light decomposition – Competitive Programming Algorithms CP-AlgorithmsPage Authors Heavy-light decomposition Heavy-light decompositionis a fairly general technique that allows us to effectively solve many problems that come down to queries on a tree. Description Let there be a tree \\(G\\) of \\(n\\) vertices, with an arbitrary root.

What is heavy light decomposition in DBMS?

The main idea of Heavy Light Decomposition of a unbalanced tree is to club vertices of long (or heavy) paths together in chains so that these linear chains can be queried in O (Log n) time using a data structure like Segment Tree.

What is the difference between centroid and center of mass?

A node u is a centroid iff the size of each subtree after removing u is not greater than n 2. Note that the center of a tree is different from the centroid of a tree, which is usually related with height of the tree. Intuitively, centroid can be viewed as the “center of mass” of T.

What is HLD (heavy-light distribution)?

HLD can also be seen as ‘coloring ’ of the tree’s edges. The ‘Heavy-Light’ comes from the way we segregate edges. We use size of the subtrees rooted at the nodes as our criteria.