We are given 2*N – 1 integers. We need to check whether it is possible to construct a Range Minimum Query segment tree for an array of N distinct integers from these integers. If so, we must output the segment tree array. N is given to be a power of 2.
An RMQ segment tree is a binary tree where each node is equal to the minimum value of its children. This type of tree is used to efficiently find the minimum value of elements in a given range.
Input : 1 1 1 1 2 2 3 3 3 4 4 5 6 7 8 Output : 1 1 3 1 2 3 4 1 5 2 6 3 7 4 8 The segment tree is shown below
Input : -381 -460 -381 95 -460 855 -242 405 -460 982 -381 -460 95 981 855 Output : -460 -460 -381 -460 95 -381 855 -460 -242 95 405 -381 981 855 982 By constructing a segment tree from the output, we can see that it a valid tree for RMQ and the leaves are all distinct integers.
What we first do is iterate through the given integers counting the number of occurrences of each number, then sorting them by value. In C++, we can use the data structure map, which stores elements in sorted order.
Now we maintain a queue for each possible level of the segment tree. We put the initial root of the tree (array index 0) into the queue for the max level. We then insert the smallest element into the leftmost nodes. We then detach these nodes from the main tree. As we detach a node, we create a new tree of height h – 1, where h is the height of the current node. We can see this in figure 2. We insert the root node of this new tree into the appropriate queue based on its height.
We go through each element, getting a tree of appropriate height based on the number of occurrences of that element. If at any point such a tree does not exist, then it is not possible to create a segment tree.
YES 1 1 3 1 2 3 4 1 5 2 6 3 7 4 8
Main time complexity is caused by sorting elements.
Time Complexity: O(N log N)
Space Complexity: O(N)
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- Overview of Data Structures | Set 3 (Graph, Trie, Segment Tree and Suffix Tree)
- Cartesian tree from inorder traversal | Segment Tree
- Build a segment tree for N-ary rooted tree
- Segment Tree | Set 1 (Sum of given range)
- Segment Tree | Set 2 (Range Minimum Query)
- Lazy Propagation in Segment Tree
- Longest Common Extension / LCE | Set 3 (Segment Tree Method)
- Segment Tree | Set 3 (XOR of given range)
- Segment tree | Efficient implementation
- LIS using Segment Tree
- Segment Tree | (XOR of a given range )
- Levelwise Alternating OR and XOR operations in Segment Tree
- Euler Tour | Subtree Sum using Segment Tree
- Two Dimensional Segment Tree | Sub-Matrix Sum
- Segment Tree | Set 2 (Range Maximum Query with Node Update)
- Levelwise Alternating GCD and LCM of nodes in Segment Tree
- Iterative Segment Tree (Range Maximum Query with Node Update)
- Iterative Segment Tree (Range Minimum Query)
- Counting inversions in an array using segment tree
- Lazy Propagation in Segment Tree | Set 2