Welcome to this explanation of segment trees. A segment tree is a powerful data structure used for efficiently querying and updating intervals or segments of an array. It allows operations like finding the sum, minimum, or maximum over a range in logarithmic time. This makes segment trees ideal for problems requiring frequent range queries and updates. The structure takes linear space and can be built in linear time. On the right, you can see an example of a segment tree built from an array of six numbers. Each node in the tree represents a range of the original array and stores the sum of elements in that range. The root covers the entire array, while leaf nodes represent individual elements.
Let's look at how to build a segment tree. The process starts with the original array and builds the tree bottom-up. First, we create leaf nodes for each element in the array. These form the bottom level of our tree. Next, we build the internal nodes level by level. Each internal node stores the aggregate value of its children - in this case, the sum. For example, the node covering range 0-1 contains the sum of elements at indices 0 and 1, which is 1+3=4. We continue this process until we reach the root node, which represents the entire array range and contains the sum of all elements. The building process takes O(n) time, where n is the size of the array. This is efficient because we process each element a constant number of times.
Now let's see how to perform a range query on a segment tree. In this example, we want to find the sum of elements from index 1 to 4 in our array. The query process starts at the root and follows a recursive approach. For each node, we check if its range is fully within our query range, partially overlapping, or completely outside. If a node's range is fully within our query range, we use its value directly. If it partially overlaps, we check its children. If it's completely outside, we ignore it. For our query of range 1 to 4, we first check the root node which partially overlaps, so we check its children. The left child [0-2] partially overlaps, so we check its children. The right child [3-5] also partially overlaps. At the leaf level, nodes representing indices 1, 2, 3, and 4 are fully within our query range, so we use their values. Nodes for indices 0 and 5 are outside our range, so we ignore them. Finally, we combine the values from the relevant nodes: 3 + 5 + 7 + 9 = 24. This query operation takes O(log n) time, which is much more efficient than the O(n) time needed to scan the array.
Let's examine how to update a value in a segment tree. Suppose we want to change the value at index 2 from 5 to 10. The update process starts by locating the leaf node corresponding to index 2. We update its value from 5 to 10. Then, we need to recalculate the values of all ancestor nodes that are affected by this change. We move up the tree, updating each node along the path to the root. The parent node covering range 0-2 is updated from 9 to 14, as the sum of its children is now 1 + 3 + 10 = 14. Finally, the root node is updated from 36 to 41. This update operation takes O(log n) time, which is much more efficient than updating a precomputed array of sums, which would require O(n) time to recalculate all affected ranges. This efficiency makes segment trees particularly useful for applications requiring frequent updates and range queries.
Let's conclude by exploring the applications and variations of segment trees. Segment trees are widely used for various range query problems, including sum queries, minimum or maximum queries, GCD or LCM queries, and finding frequencies in ranges. They're also valuable in computational geometry and database query optimization. Several variations of segment trees exist to handle specific scenarios. Lazy propagation is an optimization technique that efficiently handles multiple updates over a range by delaying the updates until necessary. Persistent segment trees maintain previous versions of the tree, allowing queries on historical states. 2D segment trees extend the concept to handle queries on two-dimensional grids. On the right, you can see a Python implementation of a basic segment tree with build, query, and update operations. The time complexity for building the tree is O(n), while both query and update operations take O(log n) time. This makes segment trees significantly more efficient than naive approaches for problems requiring frequent range queries and updates. The advantages of segment trees include their efficiency for both queries and updates, versatility for various range operations, and relatively straightforward implementation.