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

Kurt Mehlhorn
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.  ... 
doi:10.1137/0206017 fatcat:hmgrqvvn4jcddpe4ttejbyfsny

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

Kurt Mehlhorn
1975 Lecture Notes in Computer Science  
We derive upper and lower 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.  ... 
doi:10.1007/3-540-07407-4_4 fatcat:7dgegp7svzgkvhmytv3527jki4

Nearly optimal binary search trees

Kurt Mehlhorn
1975 Acta Informatica  
Therefore we define 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].  ... 
doi:10.1007/bf00264563 fatcat:xzg7pz4babcorci36nheiombqi

Optimal binary search trees

S.V. Nagaraj
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  ... 
doi:10.1016/s0304-3975(96)00320-9 fatcat:c6qiteegkrflfiktenkbffaa7e

Should Static Search Trees Ever Be Unbalanced? [chapter]

Prosenjit Bose, Karim Douïeb
2010 Lecture Notes in Computer Science  
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 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.  ... 
doi:10.1007/978-3-642-17517-6_12 fatcat:y66tuj7qyfa2zi62adyx76qkey

Should Static Search Trees Ever Be Unbalanced? [article]

Prosenjit Bose, Karim Douïeb
2010 arXiv   pre-print
In this paper we study 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.  ... 
arXiv:1006.3715v1 fatcat:aqejvypclvfyfpsauo54roc33u

New binary codes

B. Groneick, S. Grosse
1994 IEEE Transactions on Information Theory  
This code is called 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.  ... 
doi:10.1109/18.312176 fatcat:xisgtqsygvghvaczxyp26gm5bm

Path map symbol release rules and the exponential metric tree (source coding)

Wen-Whei Chang, J.D. Gibson
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.  ... 
doi:10.1109/18.57212 fatcat:ed7hugw4kfd63kpw2i3qxbgq3e

Optimal Binary Search Trees with Near Minimal Height [article]

Peter Becker
2010 arXiv   pre-print
Suppose we have n keys, n access probabilities 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  ... 
arXiv:1011.1337v1 fatcat:jhywdwx4qbchvaabb45fxrvs6q

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

David Eppstein, Denis Kurz, Marc Herbstritt
2018 International Symposium on Parameterized and Exact Computation  
to minimize 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.  ... 
doi:10.4230/lipics.ipec.2017.16 dblp:conf/iwpec/EppsteinK17 fatcat:t2lvfofzdzdzbpqz2o2lbikf3e

k-best enumeration [article]

David Eppstein
2014 arXiv   pre-print
We survey k-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.  ... 
arXiv:1412.5075v1 fatcat:tc6frzf42bcedejhlcdjfoe4rm

K-Best Solutions of MSO Problems on Tree-Decomposable Graphs [article]

David Eppstein, Denis Kurz
2017 arXiv   pre-print
to minimize 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.  ... 
arXiv:1703.02784v1 fatcat:rdphndgxnnbq5dpx7o7fhluxji

k-Best Enumeration [chapter]

David Eppstein
2016 Encyclopedia of Algorithms  
We survey k-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.  ... 
doi:10.1007/978-1-4939-2864-4_733 fatcat:beise4wvfjdwbgprvxt7zkwm4y

Binary search algorithm

Anthony Lin
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.  ... 
doi:10.15347/wjs/2019.005 fatcat:tzr6dqdkrzhtbfl36tcpbi5rau
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