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A Best Possible Bound for The Weighted Path Length of Binary Search Trees

1977
*
SIAM journal on computing (Print)
*

*The*

*weighted*

*path*

*length*

*of*optimum

*binary*

*search*

*trees*is

*bounded*above by Y'./3i + 2

*a*. + H where H is

*the*entropy

*of*

*the*frequency distribution, /3i is

*the*total

*weight*

*of*

*the*internal nodes, and aj ... This

*bound*is

*best*

*possible*.

*A*linear time algorithm

*for*constructing nearly optimal

*trees*is described. ... Knuth gives an algorithm

*for*constructing an optimum

*binary*

*search*

*tree*, i.e.,

*a*

*tree*with minimal

*weighted*

*path*

*length*. His algorithm operates in O(n 2) units

*of*time and O(n2) units

*of*space. ...

##
###
Best possible bounds on the weighted path length of optimum binary search trees
[chapter]

1975
*
Lecture Notes in Computer Science
*

We derive upper and lower

doi:10.1007/3-540-07407-4_4
fatcat:7dgegp7svzgkvhmytv3527jki4
*bounds**for**the**weighted**path**length*P opt*of*optimum*binary**search**trees*. In particular, where H is*the*entropy*of**the*frequency distribution. ... We also present an approximation algorithm which constructs nearly optimal*trees*. ... Prove good lower and upper*bounds**for**the**weighted*hath*length**of*optimum*binary**search**trees*, i.e.*the**trees*with minimal*weighted**path**length*. ...##
###
Nearly optimal binary search trees

1975
*
Acta Informatica
*

Therefore we define

doi:10.1007/bf00264563
fatcat:xzg7pz4babcorci36nheiombqi
*the**weighted**path**length**of**tree*T as: Knuth [4] gives an algorithm*for*constructing an optimum*binary**search**tree*, i.e.*a**tree*with minimal*weighted**path**length*. ... "One*of**the*popular methods*for*retrieving information by its 'name' is to store*the*names in*a**binary**tree*. ... Then (t/log 3)" H is*a*lower*bound*on*the**weighted**path**length*Popt*of*an optimal*binary**search**tree*. This*bound*is sharp*for*infinitely many distributions. Proo]. ...##
###
Optimal binary search trees

1997
*
Theoretical Computer Science
*

*For*an optimal

*binary*

*search*

*tree*T with

*a*subtree S(d) rooted at

*a*depth d from

*the*root

*of*T, we study

*the*ratio

*of*

*the*

*weight*

*of*S(d), to

*the*

*weight*

*of*T and correct

*a*result obtained by Hirschberg ... Knuth's algorithm is

*the*

*best*known so far

*for*

*the*general optimal

*binary*

*search*

*tree*problem. ... Acknowledgements

*The*author is thankful to all those who helped him in preparing this article, especially Dr Venkatesh Raman

*of*

*the*Institute

*of*Mathematical Sciences (email id: vraman@imsc.emet.in)

*for*...

##
###
Should Static Search Trees Ever Be Unbalanced?
[chapter]

2010
*
Lecture Notes in Computer Science
*

We present several methods to restructure an unbalanced k-ary

doi:10.1007/978-3-642-17517-6_12
fatcat:y66tuj7qyfa2zi62adyx76qkey
*search**tree*T into*a*new*tree*R that preserves many*of**the*properties*of*T while having*a*height*of*log k n + 1 which is one unit off*of**the*... In this paper we study*the*question*of*whether or not*a*static*search**tree*should ever be unbalanced. ...*The*first author wishes to especially thank Luc Devroye*for*his many key insights on this problem and*for*encouraging him to never give up. ...##
###
Should Static Search Trees Ever Be Unbalanced?
[article]

2010
*
arXiv
*
pre-print

In this paper we study

arXiv:1006.3715v1
fatcat:aqejvypclvfyfpsauo54roc33u
*the*question*of*whether or not*a*static*search**tree*should ever be unbalanced. ... We present several methods to restructure an unbalanced k-ary*search**tree*T into*a*new*tree*R that preserves many*of**the*properties*of*T while having*a*height*of*_k n +1 which is one unit off*of**the*optimal ...*The*first author wishes to especially thank Luc Devroye*for*his many key insights on this problem and*for*encouraging him to never give up. ...##
###
New binary codes

1994
*
IEEE Transactions on Information Theory
*

This code is called

doi:10.1109/18.312176
fatcat:xisgtqsygvghvaczxyp26gm5bm
*the*residual code with respect to tu (see [lo]). 2The notations are taken from [ I 11. 3Construction X is explained in [4, pp. 581-5821. ... In*a*similar way, it is*possible*to improve 'he puncturing*of**a*code by analyzing*the*dual code.*For*these ccnstructions it is necessary to*search**for*codewords*of*low*weight*in*the*dual code. ... CONCLUSION We can use*the**bounds*given here to set k ,*the*number*of**trees*to be*searched*in parallel, to guarantee average*search**lengths*shorter than some desired constant. ...##
###
Path map symbol release rules and the exponential metric tree (source coding)

1990
*
IEEE Transactions on Information Theory
*

*The*performance

*of*several

*path*map symbol release rules when used with exhaustive

*searching*

*of*

*the*exponential metric

*tree*is investigated. ... To design

*a*

*tree*coder

*for*source coding with

*a*fidelity criterion, one must choose

*a*suitable code generator, an efficient

*tree*

*search*algorithm, an appropriate distortion measure, and

*a*

*path*map symbol ... It

*searches*all

*possible*branches

*of*

*the*code

*tree*to depth L , and hence

*the*

*best*s^ is always found. ...

##
###
Optimal Binary Search Trees with Near Minimal Height
[article]

2010
*
arXiv
*
pre-print

Suppose we have n keys, n access probabilities

arXiv:1011.1337v1
fatcat:jhywdwx4qbchvaabb45fxrvs6q
*for**the*keys, and n+1 access probabilities*for**the*gaps between*the*keys. Let h_min(n) be*the*minimal height*of**a**binary**search**tree**for*n keys. ... So far,*the**best*known algorithms*for**the*construction*of*height-restricted optimal*binary**search**trees*have running time O(L n^2), whereby L is*the*maximal permitted height. ...*For**a**binary**search**tree*T we define*the**weighted**path**length*wpl(T ) by wpl(T ) := n i=1 β i (b i + 1) + n j=0 α j*a*j*The**weighted**path**length*is*the*expected number*of*node visits resp. comparisons ...##
###
Page 2802 of Mathematical Reviews Vol. , Issue 81G
[page]

1981
*
Mathematical Reviews
*

Author’s summary: “

*A*simple proof is given*of**a**best**possible*upper*bound*on*the*cost*of**the**binary**search**tree*.” 81g:68087 Vaishnavi, V. K.; Kriegel, H. P.; Wood, D. Optimum multiway*search**trees*. ... 81¢:68087 Horibe, Yasuichi*A*note on cost*bound**of**binary**search**tree*. J. Combin. Inform. System Sci. § (1980), no. 1, 36-37. ...##
###
K-Best Solutions of MSO Problems on Tree-Decomposable Graphs

2018
*
International Symposium on Parameterized and Exact Computation
*

to minimize

doi:10.4230/lipics.ipec.2017.16
dblp:conf/iwpec/EppsteinK17
fatcat:t2lvfofzdzdzbpqz2o2lbikf3e
*the*sum*of**the**weights*in*the*selected sets, we can find*the*k*best*solution values*for*n-vertex graphs*of**bounded*treewidth in time O(n + k log n). ... In particular, this applies to finding*the*k shortest simple*paths*between given vertices in directed graphs*of**bounded*treewidth, giving an exponential speedup in*the*per-*path*cost over previous algorithms ...*Path*copying is*a*technique introduced by Sarnak et al.*for*making any*tree*-based ephemeral data structure (such as*a**binary*heap or*binary**search**tree*) fully persistent. ...##
###
k-best enumeration
[article]

2014
*
arXiv
*
pre-print

We survey k-

arXiv:1412.5075v1
fatcat:tc6frzf42bcedejhlcdjfoe4rm
*best*enumeration problems and*the*algorithms*for*solving them, including in particular*the*problems*of*finding*the*k shortest*paths*, k smallest spanning*trees*, and k*best*matchings in*weighted*...*the*removal),*a*property that simplifies*the**search**for**a*second-*best**tree*as needed*for*Gabow's partitioning technique. ...*The**length**of**the**path*is then*the*shortest*path*distance from s to t, plus*the*sum*of**the**lengths*added by each detour. ...##
###
K-Best Solutions of MSO Problems on Tree-Decomposable Graphs
[article]

2017
*
arXiv
*
pre-print

to minimize

arXiv:1703.02784v1
fatcat:rdphndgxnnbq5dpx7o7fhluxji
*the*sum*of**the**weights*in*the*selected sets, we can find*the*k*best*solutions*for*n-vertex graphs*of**bounded*treewidth in time O(n+k n). ... In particular, this applies to*the*problem*of*finding*the*k shortest simple*paths*between given vertices in directed graphs*of**bounded*treewidth, giving an exponential speedup in*the*per-*path*cost over ...*The*research*of*David Eppstein was supported in part by NSF grants CCF-1228639, CCF-1618301, and CCF-1616248. ...##
###
k-Best Enumeration
[chapter]

2016
*
Encyclopedia of Algorithms
*

We survey k-

doi:10.1007/978-1-4939-2864-4_733
fatcat:beise4wvfjdwbgprvxt7zkwm4y
*best*enumeration problems and*the*algorithms*for*solving them, including in particular*the*problems*of*finding*the*k shortest*paths*, k smallest spanning*trees*, and k*best*matchings in*weighted*...*A*more sophisticated algorithm*of*Frederickson*for**searching*an implicit heap improves this time*bound*to O(k) [74] . ...*the*removal),*a*property that simplifies*the**search**for**a*second-*best**tree*as needed*for*Gabow's partitioning technique. ...##
###
Binary search algorithm

2019
*
WikiJournal of Science
*

*The*

*binary*

*search*

*tree*and B-

*tree*data structures are based on

*binary*

*search*. ... In In computer science,

*binary*

*search*, also known as half-interval

*search*, [1] logarithmic

*search*, [2] or

*binary*chop, [3] is

*a*

*search*algorithm that finds

*a*position

*of*

*a*target value within

*a*sorted ... review

*of*

*the*article. ...

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