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# Max Population Cities  **Wrong Needs Fixing**  ## Preprocessing  We store the tuple of coordinates in a range-tree. The auxiliary data structure at each node also keeps track of the maximum population cityamong all nodes  in their subtrees. the subtree rooted at this node.  We can achieve this while maintaining overall runtime if we build Building  our auxiliary data-structures from leaf-node up (**Figure 1**). Starting from the leafs, we perform data-structure takes \(O(n)\) work, where \(n\) is  the merging step number  of merge-sort on nodes stored in that sub-tree. We can store  the y-coordinates of our pointset. Every merge-step corresponds to maximum population for all nodes in  the auxiliary data-structure for a node graph  in our \(x\)-tree. a linear amount of time.  The only additional thing we have to We  do is keep track of this with a simple in-order traversal (recursively,  the maximum population city. In left and right subtrees are visited before  the first layer of merging root). When  we are comparing two leaves. We can reach an internal node, we just  compare the  population sizes of the two children  and store the maximum in larger of the two. Our traversal is \(O(n)\) and we do  \(O(1)\)extra  work per merge. For all sequential layers, we simply compare at each node so  the two population sizes and store total run-time is \(O(n)\). As such we have not altered  the largest value. big-oh complexity of building our auxiliary data-structure.  ## Queries  We query in mostly  the same way as general range trees. We first search our \(x\)-tree and then find \(O(\log n)\) auxiliary data-structures. In each of these, we find the \(O(\log n)\) sub-trees determining our valid \(y-\)-range. Instead of traversing this segment of the graph, we simply look at the population sizes stored in the root of the subtrees. We find the maximum such value for each auxiliary tree, and then compare these to find the overall maximum value.  - For each auxiliary tree i: \(O(\log n)\)  - Find the sub-trees determining the valid \(y\)-range: \(O(\log n)\)  - \(m_i\rightarrow\) the maximum root-value among all sub-trees: There are \(O(\log n)\) such sub-trees so this is \(O(\log n)\).  - Emit \(\max(m_j)\): There are \(O(\log n)\) auxiliary trees so this is \(O(\log n)\).  ## Complexity  **Preprocessing:** The process of building the auxiliary tree is a merge-step with \(O(1)\) additional work \(O(n)\)  per merge. layer with \(\log n\) layers.  As such, it runs in \(O(n\log n\) time. This is additive to the \(O(n\log n\) time required to build the \(x\)-tree, so we have overall \(O(n\log n\) pre-processing time. **Space:** We also maintain the \(O (n\log n)\) space bound of range-trees -- we are storing store  only one additional value per layer in the  auxiliary data-structure.   **Query Time:** To find the valid region we perform \(O(\log n\)\) look-ups in \(O(\log n\)\) auxiliary trees.