Questions tagged [splay-trees]

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9
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1answer
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Splay tree with odd number of rotations

When inserting an item into a splay tree, rotations are performed in pairs based on either a zig-zag or zig-zig pattern. When there is an odd number of rotations to be performed, one could either do ...
7
votes
4answers
1k views

Is there a binary tree structure with fast access to recently accessed elements and worst $O \left( \log n \right )$ complexity?

The idea of splay trees is very nice as they move frequently accessed elements to the top, which can gain a considerable speed up in many applications. The drawback is that in the worst case an ...
4
votes
1answer
72 views

Make appends not degrade a splay tree

Splay trees offer armotized O(log n) access to tree elements. However, if you keep repeatedly appending elements to a splay tree, without splaying any other elements, it degrades into a linked list ...
2
votes
1answer
28 views

Doubt on Performance Comparison of Splay Trees

I'm trying to understand splay tree performance compared to a standard balanced tree like AVL/red-black. In practice, I am pretty skeptical of the amortized bounds Tarjan and Sleator proves in their ...
2
votes
0answers
27 views

Can every node of a link/cut tree be accessed in $O(n)$ time?

Per the Sequential Access Theorem we can access every node of a splay tree in $O(n)$ time, when accessing the nodes in a specific order. Given a link/cut tree, is it possible to access all of its ...
2
votes
0answers
93 views

Splay trees: why are depths of nodes on the access path halved?

The original paper describing splay trees Self-Adjusting Binary Search Trees by Sleator and Tarjan claims that: Splaying not only moves x to the root, but roughly halves the depth of every node ...
1
vote
1answer
69 views

Prove that a sequence of increasing find operations on a splay tree takes $\mathcal{O}(n)$ time

When studying about splay trees, I found the following statement: Suppose we have a splay tree and a sequence of Find operations, where the elements we are searching for are in increasing order. ...
1
vote
1answer
43 views

Example of tree with > 6 vertices, tree would have depth = n after splay() deepest vertex

How to build tree with more than 6 vertices, that after operation splay() would have depth = number of vertices? Is it possible? UPD: Example for n = 4: insert 60 insert 10 insert 20 insert 50 ...
1
vote
0answers
13 views

Splay tree amortized analysis cost using Access Lemma

Currently studying for an algorithms exam and I came across this question and solution, but I can't understand the solution where it references nodes of depth less than $4\log n$ and not restructuring....
1
vote
0answers
69 views

Splay Tree - Insert Permutation

Let $T$ be a Splay Tree. For a given permutation $\sigma$ on a set $S = \{1,2,3,...n \}$ we defined the following function: ...
1
vote
0answers
24 views

Accounting value of Splay trees?

In Splay trees, by definition - the required element x - rises to the root of the tree, using the operations: zig, zig-zig, zig-zag. And the formula zig of the step is this: ...
1
vote
0answers
45 views

Why is maximum size of root is 2n + 1 in Splay trees?

In the amortized analysis of Splaying in Dynamic trees, let us consider a splay tree $T$ with $n$ keys and $v$ be a node of $T$. We define $size(v)$ as the number of nodes in the subtree rooted at $...
0
votes
1answer
93 views

Merging two splay trees whose ranges may overlap in $O(\log N)$

I have two splay trees, $A$ and $B$. When every element in $A$ is smaller than every element in $B$, we can merge them in $O(\log N)$. My question is; when all elements of $A$ are not necessarily ...
0
votes
1answer
117 views

Rotations in splay trees

I am having some difficulties splaying the element 4 to the root. Considering the following splay tree. 0 \ 1 \ 2 \ 3 \ 4 ...
0
votes
0answers
28 views

(Searching + Splaying) in a Splay Tree

Given a tree To search for value 1, I do the following splay operations: i) Zag on 7 ii) Zag on 1 And hence obtain 1 in the root. But why would this be incorrect? i) Zag on 1 ii) Zag on 1 The ...
0
votes
0answers
20 views

How to reconstruct an existing splay tree by insertion?

I'm trying to figure out the same problem as stated in this question. In brief, I want to reconstruct an existing splay tree (printed on paper) on Splay Tree Visualization by inserting the values in ...
0
votes
1answer
81 views

The validity of the potential function for splay tree

The paper "Self-Adjusting Binary Search Trees" defines (Page 658) the potential function for analyzing the amortized cost of a sequence of $m$ splay operations as the sum of the ranks of all nodes in ...
0
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0answers
28 views

Proof of Zig-Zig step

There was a question connected with one of the video lecture lessons that I'm currently watching. Let two trees be given - the original and the tree after the zig-zig step: Calculate the cost of ...
0
votes
0answers
319 views

Proof by induction for a splay tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay ...