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### A Class of Trees Having Near-Best Balance [article]

Laura Monroe
2021 arXiv   pre-print
Balance in such a tree is highly desirable for efficiency in calculation. The best balance is attained with a divide-and-conquer approach.  ...  These trees are easily constructed from the binary decomposition of the number of terms in the problem. They also permit much more flexibility than the optimally balanced divide-and-conquer trees.  ...  The Colless index on a divide-and-conquer tree is the number of D-nodes in that tree. Proof.  ...

### Divider-based algorithms for hierarchical tree partitioning

Irene Finocchi, Rossella Petreschi
2004 Discrete Applied Mathematics
The use of t-dividers, combined with a reduction to a classical scheduling problem, yields an algorithm that, given a n-vertex tree T , builds in O(n log n) worst-case time a hierarchical decomposition  ...  Furthermore, every high-level representation of the tree obtained from such decompositions is guaranteed to be a tree.  ...  After a t-divider of T c has been found, consider the k subtrees T 1 · · · T k obtained from T c by removing the t-divider d and create k children c 1 · · · c k of node c in the hierarchy tree: ∀i ∈ [1  ...

### Planar trees, slalom curves and hyperbolic knots [article]

Norbert A'Campo
1999 arXiv   pre-print
We deduce from a rooted tree in the disk a slalom divide and a slalom knot.  ...  Divide and knot of a planar rooted tree. Let B be a rooted planar tree in the unit disk D ⊂ R 2 and let P B be its divide. The knot K B of the tree B is the knot of its divide P B (see ), i.e.  ...  Conway spheres and Bonahon-Siebenmann decomposition for slalom knots Let B be a rooted tree with slalom divide P B ⊂ D and slalom knot K B .  ...

### A Few Identities of the Takagi Function on Dyadic Rationals [article]

Laura Monroe
2021 arXiv   pre-print
The number of unbalanced interior nodes of divide-and-conquer trees on n leaves is known to form a sequence of dilations of the Takagi function on dyadic rationals.  ...  D-nodes in divide-and-conquer trees In this section, we develop several identities on δ(n), the number of D-nodes in a divide-andconquer tree.  ...  There is only one divide-and-conquer tree on n leaves, up to tree isomorphism. Notation 3.0.5. The number of D-nodes in a divide-and-conquer tree is denoted by δ(n).  ...

### PK-Tree: A Spatial Index Structure for High Dimensional Point Data [chapter]

Wei Wang, Jiong Yang, Richard Muntz
2000 Information Organization and Databases
The proposed indexing structure can be viewed as combining aspects of the PR-quad or K-D tree but where unnecessary nodes are eliminated.  ...  Empirical evidence both on real data sets and generated data sets shows that the PK-tree outperforms the recently proposed spatial indexes based on the R-tree such as the SR-tree and X-tree by a w i d  ...  De nition 1.4 Given a nite set of points D over index space C 0 and dividing ratio tuple R, a PK-tree o f rank K (K > 1) is de ned a s follows. 1.  ...

### Essential Cycles in Graph Divides as a Link Representation

Tomomi KAWAMURA
2006 Tokyo Journal of Mathematics
Graph divide links were introduced as an extension of the class of A'Campo's divide links.  ...  We show that some of such links cannot be represented as the links of any graph divides including less circles or less cycles.  ...  Let K be a non-trivial knot. We suppose that K is not a graph divide link. By Theorem 2.1, there exists an oriented divide Q K with L ori (Q K ) = K.  ...

### A Kernel Support Vector Machine for Big Data Mining Based on Buffered k-D Tree

Xia Gao, Ruijun Li
2016 ICIC Express Letters
The method adopts a buffered k-D tree to divide data samples in a training set into different leaf structures together with buffer cells.  ...  The KSVM classifiers are updated according to the data samples buffered in k-D tree.  ...  Generally speaking, a buffered k-D tree is a k-D tree associated with a set of buffers.  ...

### Transitivity, lowness, and ranks in NSOP_1 theories [article]

Artem Chernikov, Byunghan Kim, Nicholas Ramsey
2020 arXiv   pre-print
We show that, in such theories, Kim-independence is transitive and that ^K-Morley sequences witness Kim-dividing.  ...  Therefore, Theorem 4.7(1) implies that, if for all ϕ(x; y) ∈ L(B), q(y) ∈ S(B), and k < ω, D 1 (π ′ , ϕ, k, q) = D 1 (π, ϕ, k, q), then π ′ does not Kim-divide over B. Question 4.9.  ...  (d) The tree (c η ) η∈ω <ω \{∅} is s-indiscernible over B.  ...

### Computing the Girth of a Planar Graph in O(n logn) Time [chapter]

Oren Weimann, Raphael Yuster
2009 Lecture Notes in Computer Science
u,w) ∉ T • For every edge (u,w) ∉ T check d(u)+d(w)+weight(u,w) Shortest Cycle in Planar Graphs v Sunday, July 12, 2009 k-outerplanar graphs: O(knlogn) • Divide & Conquer by k-sized separator  ...  • Shortest Cycle in Planar Graphs v Sunday, July 12, 2009 k-outerplanar graphs: O(knlogn) • Divide & Conquer by k-sized separator • From every build shortest paths tree T in O(n) time [Henzinger  ...

### MpFPC- a Parallelization Method for Fast Packet Classification

Yuzhu Cheng, Qiuying Shi
2022 IEEE Access
, so as to avoid the rule replication problem of the traditional decision-tree-based method.  ...  , but compared with the traditional algorithm, a rule mapping preprocessing process is added before constructing the classification decision tree, which effectively removes the rule redundancy and conflict  ...  Definition 1: Let cell spaces u = (l (u) 1 , . . . , l (u) k )(d (u) 1 , . . . , d (u) k ), v = (l (v) 1 , . . . , l (v) k ) (d (v) 1 , . . . , d (v) k ), if (1) R(u, v, F i ) = 'crossed,' then P(u, v,  ...

### Efficient Implementation of Tree Skeletons on Distributed-Memory Parallel Computers

Kiminori Matsuzaki
2017 Scalable Computing : Practice and Experience
In our implementation, we divide a binary tree to segments based on the idea of m-bridges with high locality, and represent local segments as serialized arrays for high sequential performance.  ...  Parallel tree skeletons are basic computational patterns that can be used to develop parallel programs for manipulating trees.  ...  The tree skeleton map takes two functions k l and k n and a binary tree, and applies k l to each leaf and k n to each internal node.  ...

### Page 5123 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews
Summary: “A variant of k-d trees, the divided k-d tree, is de- scribed that has some important advantages over ordinary k-d trees.  ...  (NL-UTRE-C) Divided k-d trees. Algorithmica 6 (1991), no. 6, 840-858.  ...

### A 2/sup d/-tree-based blocking method for microaggregating very large data sets

A. Solanas, A. Martmez-Balleste, J. Domingo-Ferrer, J.M. Mateo-Sanz
2006 First International Conference on Availability, Reliability and Security (ARES'06)
This paper presents a new blocking method based on 2 d -trees for intelligently partitioning very large data sets for microaggregation.  ...  Initially, the 2 d -tree has a single leaf and it is divided into 2 d leaves if the number of records inside its bounds is bigger than L.  ...  Thus, we propose to use a very well-known data structure such as a 2 d -tree for dividing the data taking into account a userdefinable number d of dimensions.  ...

### Mining Frequent Itemsets from Secondary Memory [article]

Gösta Grahne, Jianfei Zhu
2004 arXiv   pre-print
Several divide-and-conquer algorithms are given for mining from disks. Many novel techniques are introduced.  ...  If k ≪ n, we can expect that the aggressive divide and conquer algorithm will significantly outperform the naive one. D T .ab T .cd T .b T .c T .d T .  ...  In the figure, since the partial FP-tree for t(T α ) of t(D α ) transactions can be accommodate in main memory, we can expect that the FP-tree containing i k , . . . , i 1 , where k = ⌊n · t(T α )/t(D  ...

### Divide-and-conquer algorithms on two-dimensional meshes [chapter]

Miguel Valero-García, Antonio González, Luis Díaz de Cerio, Dolors Royo
1998 Lecture Notes in Computer Science
The Reflecting and Growing mappings have been proposed to map parallel divide-and-conquer algorithms onto two-dimensional meshes.  ...  In this paper we consider a more general scenario in which the parallel divide-and-conquer algorithm can be started from an arbitrary node of the mesh.  ...  0 2 k 1 - 1 - , ∈ D a b , i ( ) D a 2 k 1 - b - , i ( ) if a 0 2 k 1 - 1 - , ∈ b 2 k 1 -2 k 1 - , ∈ ,     D 2 k 1 - a -b , i ( ) if a 2 k 1 -2 k 1 - , ∈ b 0 2 k 1 - 1 - , ∈ ,     D 2 k 1 - a -2  ...
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