Segment Tree (Range Maximum Query with Node Update)Segment Trees are an important data structure in competitive programming and algorithmic problem-solving. We'll go deeper into Segment Trees in this extensive talk, concentrating notably on Range Maximum Query (RMQ) and node update procedures. These procedures enable quick querying and updating of data inside a certain range, providing a valuable tool for effectively tackling a wide range of issues.
Maximum Range Query (RMQ)A typical issue is RMQ, which requires determining the greatest value inside a certain range of array members. Consider the following array A = [5, 7, 2, 4, 9, 1]. In this scenario, an RMQ query for the range [1, 4] would return the largest value inside that range, which is 9 (found at index 4). Update Operations on NodesNode update activities include changing the values of array members and then efficiently updating the Segment Tree to reflect these changes. In the context of RMQ, this may involve changing the value of a specific array element and then updating the Segment Tree to preserve consistency. Segment Tree Structure for RMQ with Node UpdateA Segment Tree optimized for RMQ with node updates has a structure that is similar to the one used for other segment-based queries. Each node in the tree keeps the maximum value in its related array segment. This hierarchical structure allows for the quick traversal and retrieval of maximum values within specified ranges. Segment Tree RepresentationA Segment Tree can be represented as an array. The tree's nodes each represent a part of the original array. In the case of RMQ with node updates, each node stores the maximum value inside its associated segment. The root node represents the complete array, whereas the leaf nodes are individual elements. An array is frequently used to implement the tree, where:
Segment Tree constructionSegment Tree building is a recursive technique that constructs the tree from the bottom up. Here are the steps for building the Segment Tree:
ImplementationOutput:
Next TopicTree Vertex Splitting in data structure
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