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Register a new userOptimal Alphabetic Ternary Trees
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Added by
David Morgenthaler
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We give a new algorithm to construct optimal alphabetic ternary trees, where every internal node has at most three children. This algorithm generalizes the classic Hu-Tucker algorithm, though the overall computational complexity has yet to be determined.
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We show the $O(log n)$ time extract minimum function of efficient priority queues can be generalized to the extraction of the $k$ smallest elements in $O(k log(n/k))$ time, where we define $log(x)$ as $max(log_2(x), 1)$. We first show heap-ordered tree selection (Kaplan et al., SOSA 19) can be applied on the heap-ordered trees of the classic Fibonacci heap to support the extraction in $O(k log(n/k))$ amortized time. We then show selection is possible in a priority queue with optimal worst-case guarantees by applying heap-ordered tree selection on Brodal queues (SODA 96), supporting the operation in $O(k log(n/k))$ worst-case time. Via a reduction from the multiple selection problem, $Omega(k log(n/k))$ time is necessary if insertion is supported in $o(log n)$ time. We then apply the result to lazy search trees (Sandlund & Wild, FOCS 20), creating a new interval data structure based on selectable heaps. This gives optimal $O(B+n)$ time lazy search tree performance, lowering insertion complexity into a gap $Delta_i$ to $O(log(n/|Delta_i|))$ time. An $O(1)$ time merge operation is also made possible when used as a priority queue, among other situations. If Brodal queues are used, runtimes of the lazy search tree can be made worst-case in the general case of two-sided gaps. The presented data structure makes fundamental use of soft heaps (Chazelle, J. ACM 00), biased search trees, and efficient priority queues, approaching the theoretically-best data structure for ordered data.
In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.
We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online $O(log log n)$-competitive search tree data structure in this model, matching the best known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one which we call Steiner-closed search trees, which may be of independent interest. Moreover our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees, and secondly from these trees into paths.
PQ-trees and PC-trees are data structures that represent sets of linear and circular orders, respectively, subject to constraints that specific subsets of elements have to be consecutive. While equivalent to each other, PC-trees are conceptually much simpler than PQ-trees; updating a PC-trees so that a set of elements becomes consecutive requires only a single operation, whereas PQ-trees use an update procedure that is described in terms of nine transformation templates that have to be recursively matched and applied. Despite these theoretical advantages, to date no practical PC-tree implementation is available. This might be due to the original description by Hsu and McConnell in some places only sketching the details of the implementation. In this paper, we describe two alternative implementations of PC-trees. For the first one, we follow the approach by Hsu and McConnell, filling in the necessary details and also proposing improvements on the original algorithm. For the second one, we use a different technique for efficiently representing the tree using a Union-Find data structure. In an extensive experimental evaluation we compare our implementations to a variety of other implementations of PQ-trees that are available on the web as part of academic and other software libraries. Our results show that both PC-tree implementations beat their closest fully correct competitor, the PQ-tree implementation from the OGDF library, by a factor of 2 to 4, showing that PC-trees are not only conceptually simpler but also fast in practice. Moreover, we find the Union-Find-based implementation, while having a slightly worse asymptotic runtime, to be twice as fast as the one based on the description by Hsu and McConnell.
We consider a natural variant of the well-known Feedback Vertex Set problem, namely the problem of deleting a small subset of vertices or edges to a full binary tree. This version of the problem is motivated by real-world scenarios that are best modeled by full binary trees. We establish that bo
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