In this article, an approach to convert an N-ary rooted tree( a tree with more than 2 children) into a segment tree is discussed which is used to perform a range update queries.
Why do we need a segment tree when we already have an n-ary rooted tree?
Many times, a situation occurs where the same operation has to be performed on multiple nodes and their subtrees along with query operations multiple times.
Let’s say that we have to perform N updates on different subtrees. Every operation can take up to O(N) time as it is an N-ary tree so overall complexity will be O(N^2) which is too slow to process more than 10^3 updates queries. So we have to go the other way around and we will build a segment tree for the same.
Approach: A depth first search is performed to walk through all the nodes and keep the track of the indexes of the subtree of every node in a converted array using two arrays tin and tout(which will be the range to do updates and queries). The DFS will perform a Euler walk. The idea is to create an array and add nodes to it in the order they get visited to the converted array.
Let’s see how the tin and tout arrays help in determining the range in the converted array.
real values on nodes: 1 2 2 1 4 3 6 converted arr(indexes): 1 2 3 5 6 7 4 Node 3 has three children 5, 6, 7. Therefore, the range of node 3 is index 3-6. NODE: RANGE(tin-tout) NODE 1: 1 - 7 NODE 2: 2 - 2 NODE 3: 3 - 6 NODE 5: 7 - 7 NODE 6: 4 - 4 NODE 7: 5 - 5 NODE 4: 6 - 6
Here, Node 1 has a range from 1-7 (all nodes) so the update and query will be performed on all the nodes. Leaf nodes like 2 which have no children will only update range 2-2(only itself) this proves that our range arrays tin and tout are correct. Similarly, tin and tout for all the nodes determine the range for query and update in the segment tree.
The following is the implementation of the approach:
sum at node 1: 19 sum at node 2: 2 sum at node 3: 15 sum at node 4: 1 sum at node 5: 4 sum at node 6: 3 sum at node 7: 6 After Update sum at node 1: 20 sum at node 2: 2 sum at node 3: 16 sum at node 4: 1 sum at node 5: 4 sum at node 6: 4 sum at node 7: 4
- Overview of Data Structures | Set 3 (Graph, Trie, Segment Tree and Suffix Tree)
- Cartesian tree from inorder traversal | Segment Tree
- LIS using Segment Tree
- Segment Tree | (XOR of a given range )
- Segment Tree | Set 3 (XOR of given range)
- Segment Tree | Set 1 (Sum of given range)
- Reconstructing Segment Tree
- Segment tree | Efficient implementation
- Number of subarrays with GCD = 1 | Segment tree
- Smallest subarray with GCD as 1 | Segment Tree
- Persistent Segment Tree | Set 1 (Introduction)
- Lazy Propagation in Segment Tree
- Two Dimensional Segment Tree | Sub-Matrix Sum
- Lazy Propagation in Segment Tree | Set 2
- Euler Tour | Subtree Sum using Segment Tree
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