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

2021
*
arXiv
*
pre-print

Balance in such a

arXiv:2108.11496v1
fatcat:lhix5vd5gfcl3ezhsg7ainbpha
*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. ...##
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Divider-based algorithms for hierarchical tree partitioning

2004
*
Discrete Applied Mathematics
*

The use of t-

doi:10.1016/s0166-218x(03)00443-8
fatcat:nyxggxstjjft7ldp4qqcgo5idy
*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 ...##
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Planar trees, slalom curves and hyperbolic knots
[article]

1999
*
arXiv
*
pre-print

We deduce from a rooted

arXiv:math/9906087v1
fatcat:sd2oasr5eff53flygydor4hega
*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 . ...##
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A Few Identities of the Takagi Function on Dyadic Rationals
[article]

2021
*
arXiv
*
pre-print

The number of unbalanced interior nodes of

arXiv:2111.05996v2
fatcat:a4btliu2tjhm5ngpdzlhl6sbaa
*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). ...##
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PK-Tree: A Spatial Index Structure for High Dimensional Point Data
[chapter]

2000
*
Information Organization and Databases
*

The proposed indexing structure can be viewed as combining aspects of the PR-quad or

doi:10.1007/978-1-4615-1379-7_20
fatcat:gnvxjzcyrzdhhodweujesi52ry
*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. ...##
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Essential Cycles in Graph Divides as a Link Representation

2006
*
Tokyo Journal of Mathematics
*

Graph

doi:10.3836/tjm/1170348182
fatcat:eucagorhjvfnvasd65qdvf6v6u
*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*. ...##
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A Kernel Support Vector Machine for Big Data Mining Based on Buffered k-D Tree

2016
*
ICIC Express Letters
*

The method adopts a buffered

doi:10.24507/icicel.10.12.2891
fatcat:bqflrujixnexzc4or4rmcrgvp4
*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. ...##
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Transitivity, lowness, and ranks in NSOP_1 theories
[article]

2020
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arXiv
*
pre-print

We show that, in such theories, Kim-independence is transitive and that ^

arXiv:2006.10486v1
fatcat:jzpsjerls5apfcwceyyd37jkgy
*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. ...##
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Computing the Girth of a Planar Graph in O(n logn) Time
[chapter]

2009
*
Lecture Notes in Computer Science
*

u,w) ∉ T
• For every edge (u,w) ∉ T check

doi:10.1007/978-3-642-02927-1_63
fatcat:pepv764bdnb2to6t5xrvbamwc4
*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 ...##
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MpFPC- a Parallelization Method for Fast Packet Classification

2022
*
IEEE Access
*

, so as to avoid the rule replication problem of the traditional decision-

doi:10.1109/access.2022.3165157
fatcat:ogp4p7sdp5fwnpwebcxxajuydi
*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, ...##
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Efficient Implementation of Tree Skeletons on Distributed-Memory Parallel Computers

2017
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Scalable Computing : Practice and Experience
*

In our implementation, we

doi:10.12694/scpe.v18i1.1231
fatcat:f55zmmhuirbljldevhvuqcdary
*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. ...##
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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. ...##
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A 2/sup d/-tree-based blocking method for microaggregating very large data sets

2006
*
First International Conference on Availability, Reliability and Security (ARES'06)
*

This paper presents a new blocking method based on 2

doi:10.1109/ares.2006.1
dblp:conf/IEEEares/SolanasMDM06
fatcat:nn7lelvu5ja2fe77nz5nmuh74i
*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. ...##
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Mining Frequent Itemsets from Secondary Memory
[article]

2004
*
arXiv
*
pre-print

Several

arXiv:cs/0405069v1
fatcat:afckrtlmd5gspdogep52da6b3e
*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*...##
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Divide-and-conquer algorithms on two-dimensional meshes
[chapter]

1998
*
Lecture Notes in Computer Science
*

The Reflecting and Growing mappings have been proposed to map parallel

doi:10.1007/bfb0057966
fatcat:uklxtpcsyre2jfagkfo3kngemm
*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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