waterloo using kd-trees, we can report all points in a two-dimensional range in time o(n 1/2 k), where n is the number of points stored in the data structure and k is the number of reported points. in this problem, you will describe a data structure that needs o(n) space and answers two-dimensional range reporting queries in o(n 1/3 log(n) k) time. we assume for simplicity that all points have different x-coordinates.
To solve this problem, we can use the Divide and Conquer approach with a modified version of kd-trees.
1. Divide the points into groups of size n^(1/3). We will have a total of n^(2/3) groups.
2. For each group, sort the points based on their y-coordinates.
3. Build a binary search tree on each sorted group based on the y-coordinate. Each node in the binary search tree will store the range of y-coordinates covered by its left and right subtrees.
4. Build a kd-tree using the x-coordinate of the points. Each node in the kd-tree will store the range of x-coordinates covered by its left and right subtrees.
5. Store the root of each binary search tree corresponding to each group in the kd-tree nodes.
6. For a range query (x1, x2, y1, y2), traverse the kd-tree to find the nodes whose x-coordinate range intersects with the range (x1, x2).
7. For each of these nodes, traverse the corresponding binary search tree to find the points whose y-coordinate range intersects with the range (y1, y2).
The overall space complexity of this approach is O(n) because we only need to store the points in the kd-tree and the binary search trees.
The time complexity of a range query is O(n^(1/3)log(n)k). To understand why, consider the following:
- Traversing the kd-tree to find the nodes whose x-coordinate range intersects with the query range takes O(n^(1/3)) time in the worst case.
- For each of these nodes, traversing the corresponding binary search tree takes O(log(n)) time in the worst case.
- Finally, we retrieve the k points whose y-coordinate range intersects with the query range, which takes O(k) time.
Therefore, the total time complexity of a range query is O(n^(1/3)log(n)k), as required.
This data structure and algorithm optimize the time complexity of range reporting queries while maintaining a sublinear space complexity.
To answer two-dimensional range reporting queries in o(n^(1/3) log(n) k) time using o(n) space, we can use a data structure called an R-tree.
The R-tree is a balanced tree structure that is specifically designed for efficient two-dimensional range queries. It works by dividing the space into rectangular regions and storing the points in these regions. Each node of the R-tree contains a bounding box that encloses all the points in its subtree.
To construct the R-tree, we can follow these steps:
1. Divide the space into a grid of small cells. Each cell can hold a maximum number of points.
2. Insert all the points into the R-tree recursively. Start at the root and split nodes as necessary to maintain the maximum number of points in each cell.
3. Update the bounding boxes in each node to enclose all the points in its subtree.
Once the R-tree is constructed, we can perform two-dimensional range reporting queries efficiently. Here are the steps:
1. Start at the root of the R-tree.
2. Check if the bounding box of the current node overlaps with the query range. If not, stop searching in this subtree.
3. If the current node is a leaf node, report all the points in the leaf node that fall within the query range.
4. If the current node is an internal node, recursively apply the same steps to its child nodes.
The time complexity of this approach is o(n^(1/3) log(n) k) because we need to visit a portion of the R-tree that is proportional to n^(1/3) to find the k points within the query range. The space complexity is O(n) as we need to store all the points in the R-tree.
Note that this approach assumes that all points have different x-coordinates, which simplifies the implementation of the R-tree.
To solve this problem, we can use a data structure called BSP (Binary Space Partitioning) tree. A BSP tree is a binary tree that recursively divides the two-dimensional space into two halves at each level based on a splitting line.
Here's how we can construct the BSP tree:
1. Sort the points based on their x-coordinates in non-decreasing order.
2. Choose the point with the median x-coordinate as the root of the BSP tree.
3. Recursively construct the left and right subtrees of the root by splitting the points into two halves based on a vertical line passing through the median point.
- The left subtree contains all points with x-coordinates less than the median.
- The right subtree contains all points with x-coordinates greater than the median.
4. Repeat steps 2 and 3 for each subtree until all points are inserted into the BSP tree.
Now, let's explain how we can answer two-dimensional range reporting queries efficiently using this BSP tree data structure:
1. Traverse the BSP tree starting from the root and visit each node that intersects with the query range.
2. At each node, do the following:
- If the query range completely covers the node's range, report all the points in that subtree and stop traversing further down.
- If the query range does not intersect with the node's range, stop traversing further down.
- If the query range partially overlaps with the node's range, recursively traverse both the left and right subtrees of that node.
The time complexity of this approach is determined by the number of nodes visited during the traversal, which is proportional to the height of the tree and the number of reported points. Since the tree is balanced, its height is approximately log(n), where n is the number of points. Therefore, the time complexity for each range reporting query is O(log(n) * k).
The space complexity of the BSP tree is O(n) since we need to store all the points in the tree.
In summary, by using the BSP tree data structure, we can answer two-dimensional range reporting queries in O(n^(1/3) * log(n) * k) time and O(n) space. This is an improvement over the O(n^(1/2) * k) time complexity achieved by using KD-trees.