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# Segment Tree
A segment tree is a data structure designed to perform certain array operations efficiently - especially those involving range queries. A common application is the [Range Minimum Query](https://en.wikipedia.org/wiki/Range_minimum_query) (RMQ) problem, where we are given an array of numbers and need to support operations of updating values of the array and finding the minimum of a contiguous subarray. A segment tree implementation for the RMQ problem takes O(n) to initialize, and O(log n) per query or update. The "minimum" operation can be replaced by any array operation (such as sum).
A segment tree is a data structure designed to perform
certain array operations efficiently - especially those
involving range queries.
A segment tree is a binary tree with contiguous subarrays as nodes. The root of the tree represents the whole array. The two children of the root represent the first and second halves of the array. Similarly, the children of each node corresponds to the two halves of the array corresponding to the node. If the array has size n, we can prove that the segment tree has size at most 4n. Each node stores the minimum of its corresponding subarray.
A common application is the [Range Minimum Query](https://en.wikipedia.org/wiki/Range_minimum_query) (RMQ) problem,
where we are given an array of numbers and need to
support operations of updating values of the array and
finding the minimum of a contiguous subarray.
A segment tree implementation for the RMQ problem
takes `O(n)` to initialize, and `O(log n)` per query or
update. The "minimum" operation can be replaced by any
array operation (such as sum).
In the implementation, we do not explicity store this tree structure, but represent it using a $4n$ sized array. The left child of node i is 2i and the right child is 2i+1. This is a standard way to represent segment trees, and lends itself to an efficient implementation.
A segment tree is a binary tree with contiguous
sub-arrays as nodes. The root of the tree represents the
whole array. The two children of the root represent the
first and second halves of the array. Similarly, the
children of each node corresponds to the two halves of
the array corresponding to the node. If the array has
size `n`, we can prove that the segment tree has size at
most `4n`. Each node stores the minimum of its
corresponding sub-array.
We build the tree bottom up, with the value of each node being the minimum of its children's values. This will take time O(n), with one operation for each node. Updates are also done bottom up, with values being recomputed starting from the leaf, and up to the root. The number of operations done is the height of the tree, which is O(log n) To answer queries, each node splits the query into two parts, one subquery for each child. If a query contains the whole subarray of a node, we can use the precomputed value at the node. Using this optimisation, we can prove that only O(log n) minimum operations are done.
In the implementation, we do not explicitly store this
tree structure, but represent it using a `4n` sized array.
The left child of node i is `2i+1` and the right child
is `2i+2`. This is a standard way to represent segment
trees, and lends itself to an efficient implementation.
We build the tree bottom up, with the value of each node
being the minimum of its children's values. This will
take time `O(n)`, with one operation for each node. Updates
are also done bottom up, with values being recomputed
starting from the leaf, and up to the root. The number
of operations done is the height of the tree, which
is `O(log n)`. To answer queries, each node splits the
query into two parts, one sub-query for each child.
If a query contains the whole subarray of a node, we
can use the precomputed value at the node. Using this
optimisation, we can prove that only `O(log n)` minimum
operations are done.
![Segment Tree](https://www.geeksforgeeks.org/wp-content/uploads/segment-tree1.png)
## References
- [Wikipedia](https://en.wikipedia.org/wiki/Segment_tree)
- [YouTube](https://www.youtube.com/watch?v=ZBHKZF5w4YU&index=65&list=PLLXdhg_r2hKA7DPDsunoDZ-Z769jWn4R8)
- [GeeksForGeeks](https://www.geeksforgeeks.org/segment-tree-set-1-sum-of-given-range/)