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Segment Tree - Sum of given Range

Let us consider the following problem to understand Segment Trees.

We have an array arr[0... n-1]. We should be able to

  • Find the sum of elements from index l to r where 0 <= l <= r <= n-1
  • Change the value of a specified element of the array to a new value x. We need to do arr[i] = x where 0 <= i <= n-1.

Range sum with nested loops:

Running a loop from l to r and computing the sum of the elements in the provided range is a straightforward solution. Just enter arr[i] = x to change a value. While the first operation requires O(n) time, the second only needs O(1) time.

Range total using Prefix Sum:

An alternative is to make a new array and place the total from start to I at the ith index. The update procedure now requires O(n) time, while the total of a given range can be calculated in O(1) time. If there are many query operations and little updates, this works nicely.

Range total using a segment tree:

The most effective method is to utilise a segment tree; we can do both operations using a segment tree in O(log(N)) time.

Segment tree representation

  • The elements in the input array are Leaf Nodes.
  • Each internal node shows how the leaf nodes have merged in some way. For certain issues, there can be a different merging. Summarizing the leaf nodes under a node is merging for this issue.
  • Segment Trees are shown as an array representation of a tree. The left child of each node at index I is at index (2*i+1), the right child is at index (2*i+2), and the parent is at index I - 1) / 2).
Segment Tree - Sum of given Range

Building a segment tree from the supplied array

We begin by examining the segment arr[0,... n-1]. And each time, we split the current segment in half (assuming it hasn't already become a segment of length 1), run the identical operation on both halves, and then store the result for each such segment in the appropriate node.

Except for the last level, every level of the created segment tree will be fully filled. Additionally, since we always break each segment into two at every level, the tree will be a Full Binary Tree. There will always be n-1 internal nodes because the created tree is always a full binary tree with n leaves. Consequently, there will be 2*n - 1 nodes overall.

How tall is the segment tree for the supplied array?

The segment tree will be log2N in height. The amount of memory allotted for the segment tree will be (2 * 2log2n-1) since the tree must be represented using an array and the relationship between the parent and child indexes must be preserved.

Search for the Sum of a Range

How to calculate the sum using the segment tree once it has been built. The algorithm to calculate the sum of the elements is as follows.

There are three situations in the aforementioned implementation that we need to address.

  • When traversing the tree, if the current node's range does not fall within the specified range, the value of that node was not added to the answer.
  • If the supplied range and the range of the node partially overlap, then move left or right in accordance with the overlap.
  • Add the range to the answer if the given range entirely overlaps it.

Update a value:

The update can be carried out recursively, just like tree building and query operations can. An outdated index has been provided to us. Let diff represent the new value. Beginning at the segment tree's root, we append diff to each node that has the specified index within its range. We don't modify a node if it doesn't have the specified index inside its range.

C++ Code

The application of the aforementioned strategy is seen below:


Sum of values in given range = 15
Updated sum of values in given range = 22
  • Time complexity: O(N*log(N))
  • Auxiliary Space: O(N)

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