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Your question is not the right one. An AVL tree is a binary tree that has additional properties. First it is a search tree, which means we can easily find each number in the tree. Second it is balanced, meaning that there are no leafs very far form the root. (Formal definitions on request.) Assume you have a set of $n$ numbers in advance, and an arbitrary &...


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AVL trees are a kind of binary search trees. As such, they implement the following operations, among else: Initialize an empty tree. Add a value to the tree. Remove a value from the tree. Search a value in the tree. Try to use some of these operations to answer your question.


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Augment the tree to store, for each subtree, the minimum of the prefix sums of the sequence of elements in that subtree. Then you can check the supergood condition in $O(\log n)$ time. It's your exercise, so I'll let you work out the details. Hint #1: every prefix can be decomposed into a union of $O(\log n)$ subtrees. Hint #2: the minimum of the prefix-...


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Maintain an AVL tree $T$ containing the indices of the unoccupied rooms. For each vertex $v$ of the tree additionally maintain the number $\eta(v)$ of vertices in the subtree of $T$ rooted at $v$. 1) Init: create an AVL tree with keys $\{1, \dots, n\}$. This requires $O(n)$ time. 2) GetUnOcc($l$, $r$): Find the vertex $u$ associated with the successor of $...


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No, it is not an AVL tree. A binary tree is defined to be an AVL tree if $${\text{BalanceFactor}}(N)\in {\{-1,0,1\}} $$ holds for every node $N$ in the tree, where $$\text{BalanceFactor}(N):=\text{Height(RightSubtree(N))} - \text{Height(LeftSubtree(N))}$$ Consider Node 2. Its left subtree is a degenerated tree of height 2. Its right subtree is empty. ...


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Both of them will work, and both of them support insertions${}^1$ and deletion${}^2$ of the minimum element in $O(\log n)$ worst-case time (where $n$ is the number of elements currently in the data structure). It really depends what you mean by "better". An AVL tree will have the additional binary-search-tree property that the heaps do not have, ...


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Apparently, you want to rebalance the AVL tree. You can either do a simple left rotation or a right-left rotation. I prefer not to mention right-right rotation since that term is misleading and ambiguous. You can visualize what is happening. Go to AVL tree visualization, a page created by David Galles. Insert 2, 1, 4, 3, 5 in that order. Now look ...


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We want to show that the number of nodes $n$ in a height-balanced binary tree with height $h$ grows exponentially with $h$ and at least as fast as the Fibonacci sequence. Let $N_h$ denote the minimum number of nodes in a height-balanced binary tree having height $h$. Recall that in a height-balanced binary tree of height $h$, the subtree rooted at one of ...


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It is on the right direction to try some kind of tree augmented with a global counter that stores the number of all unordered pairs of nodes $\{u,v\}$ such that $v.value+u.value \le d$. The kind of tree you are looking for is a balanced (binary) search tree. Being balanced such as an AVL tree, it supports insertion, deletion and lookup of a number with $O(\...


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This can be solved with split and join operations, both achievable in $\mathcal{O}(\log n)$ for Red-Black and AVL trees. For Red-Black Trees this is doable either leveraging finger trees or via extending a regular Red-Black Tree as in Ron Wein's "Efficient Implementation of Red-Black Trees with Split and Catenate Operations" implemented in CGAL ...


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Please skip that article at tutorialspoint. Instead, read this article at GeekforGeeks, which is much better. You may want to check my answer to another question. The common abbreviations for Left rotation, Right rotation, Left-Right rotation, Right-Left rotation are L, R, LR and RL respectively, as expected. These rotations can be applied to all kinds of ...


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Let $T$ be an AVL tree of $n > 0$ nodes having height $h \ge 0$. You can prove that $n > F(h)$ by induction on $h$. For $h=0$ the claim is true since $n=1$ and $1 > 0 = F(0) = F(h)$. For $h=1$ the claim is true since $n \ge 2$ and $2 > 1 = F(1) = F(h)$. For $h \ge 2$, let $r$ be the root of the AVL tree. The subtree $T_\ell$ rooted in the left ...


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Searching in an AVL tree is anyways only at most O(log n) (see on @Wikipedia: https://en.wikipedia.org/wiki/AVL_tree?wprov=sfta1). If you do not care about the higher cost of rebalancing AVL tree instead of BSTree, this tree will have even the advantage of always maintaining the AVL-min-cost path on the left edge. Search the left edge has also O(log n) so no ...


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There are some implementations of Balanced Binary Search Trees, also known as Self-balancing Binary Search Trees. One of the aspects that can be considered when balancing a tree is it's subtrees height difference. This height aspect aims to keep the tree as flat as possible. The more flat the tree is, the less levels it will have and the search will be ...


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Adding one element to a set, versus deleting one element from a set, are indeed rather symmetric operations. You are right. However, in the case where the sets are represented by a tree, the two operations turn out to be asymmetric. When adding an item one branch may be too long, and one well-chosen rotation will rebalance. When deleting an item one branch ...


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No it is not. They both have the same running time but the heap is way lighter for a couple of reasons. For the asymptotic running time, note that a heap can be built in linear time meanwhile applying $n$ operations of extract min takes a total of $O(n \log n)$. On the other hand, constructing an AVL tree requires $O(n\log n)$ operations meanwhile traversing ...


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Denote by $d(h)$ the minimum height of a leaf in an AVL tree of height $h$. One subtree of the root necessarily has height $h-1$, and the other one has height either $h-2$ or $h-1$ by the defining property of AVL trees. Therefore $$ d(h) = \min(d(h-1),d(h-2)) + 1. $$ Also, one checks that $d(0) = d(1) = 0$ (if one measures height as the maximum number of ...


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AVL trees, like all other binary search trees, have the following guarantee: Suppose that $v$ is a node with left child $l$ and right child $r$. If $x$ is a node in the subtree rooted at $l$ and $y$ is a node in the subtree rooted at $r$ then $key(x) \leq key(v) \leq key(y)$. Using this, you can easily prove by induction that the inorder traversal of a ...


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What you are asking is simply how to balance a binary search tree (BST) $T$. By the way, the phrase "un-balanced AVL tree" is somewhat confusing since AVL tree is height balanced by definition. If there is enough space for a whole new tree, then we can simply construct a new AVL tree from scratch, by removing one element from the $T$ and inserting it to the ...


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Try and create the "worst" AVL tree, that is, the AVL tree in which the height is as large as possible, with as least nodes as possible. Once you get to the Fibonacci sequence (the answer to the question above), you can use the fact the the $n$-th Fibonacci number is $O(\phi ^n)$ (where $\phi = 1.618\ldots$), it should be easy to conclude that the worst ...


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I think, I should get 37 as the root node 25 as it's left child and 52 as it's right child but I dunno. You plan is correct. What you need to do is a left-right rotation as shown in the third column of the table below. That is, a left rotation at node 25 followed by a right rotation at 52. It is symmetric to the right-left rotation as shown in the fourth ...


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