## New Bounds on Optimal Binary Search Trees (2006)

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### BibTeX

@MISC{Harmon06newbounds,

author = {Dion Harmon and Erik Demaine and Rodolfo Ruben Rosales and Pavel I. Etingof and Dion Harmon},

title = {New Bounds on Optimal Binary Search Trees},

year = {2006}

}

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### Abstract

### Citations

8985 |
Introduction to algorithms
- Cormen, Leiserson, et al.
- 2001
(Show Context)
Citation Context ...he upper bound results come in a variety of flavors. Some BST algorithms have upper bounds on search and rearrangement cost per access, such as RB trees, AVL trees, or even static balanced BSTs. (See =-=[9]-=- Chapter 14, [1], and [21] respectively.) These BSTs are all online and usually guarantee search cost O(logn) for each access and also for other modifications such as adding and splitting the trees. H... |

259 |
Sorting and searching. In Volume 3 of The Art of Computer Programming ( printing
- Knuth
- 1973
(Show Context)
Citation Context ...gn) for each access and also for other modifications such as adding and splitting the trees. However these 3 BSTs were discovered by a number of independent authors in the 1950’s ([9] in reference to =-=[22]-=-) 24strees cannot achieve better costs per access even for serial access of the keys (which has cost at most O(n)). Other online BST algorithms give upper bounds for the total cost over the whole sequ... |

184 |
An algorithm for the organization of information
- Adel’son-Vel’skii, Landis
- 1962
(Show Context)
Citation Context ...esults come in a variety of flavors. Some BST algorithms have upper bounds on search and rearrangement cost per access, such as RB trees, AVL trees, or even static balanced BSTs. (See [9] Chapter 14, =-=[1]-=-, and [21] respectively.) These BSTs are all online and usually guarantee search cost O(logn) for each access and also for other modifications such as adding and splitting the trees. However these 3 B... |

181 |
Scaling and related techniques for geometry problems
- Gabow, Bentley, et al.
- 1984
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Citation Context ...K be a set of keys and L be an ordering of the keys. The Cartesian tree on the keys is a binary tree, but not a binary search tree. The construction is discussed in detail by Gabow Bentley and Tarjan =-=[14]-=-. A Cartesian tree is constructed by placing the lowest key r as the root, then constructing the left and right subtrees of the root recursively using the keys less than r in L and greater than r in L... |

117 | Optimum binary search trees - Knuth - 1971 |

50 |
Sleator and Robert Endre Tarjan. Self-adjusting binary search trees
- Dominic
- 1985
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Citation Context ... thesis presents new bounds on the behavior of binary search trees. This work was initially undertaken in an attempt to prove or disprove the dynamic optimality conjecture posed by Sleator and Tarjan =-=[30]-=- in 1985. Though we neither proved nor disproved the conjecture, the tools and methods of attack we discovered may prove useful for latter attempts at the problem. There are two major results presente... |

47 | On the dynamic finger conjecture for splay trees. Part II: The proof
- Cole
(Show Context)
Citation Context ...si−1 (including the end points). The dynamic finger property implies that the cost of the splay tree algorithm on S is O(m + n + log d(i, S)). This is described and proved by Richard Cole et al. [8], =-=[7]-=-) in a highly non-trivial fashion. (It is tough.) • Sequential access. If we access each key in the tree in order, the splay algorithm has cost O(n). This was proved in the classic paper by Sleator an... |

42 |
Alternatives to splay trees with O(log n) worst-case access times
- Iacono
(Show Context)
Citation Context ...performance of (predominantly) splay trees are not tight; there are some sequences on which splay trees perform in linear time for the sequence while the upper bounds give cost logarithmic per access =-=[18]-=-. Iacono hypothesized a tighter upper bound. Let S = (s1, . . .,sm) be a sequence of keys, let ti(y) be the distinct number of times accessed after the last access to y and at or before si. Let di(y) ... |

32 |
Lower bounds for accessing binary search trees with rotations
- Wilber
- 1989
(Show Context)
Citation Context ...constant factor. This implies that it is the best known lower bound for the behavior of offline binary search trees, although we cannot show that it is strictly better than the second bound of Wilber =-=[33]-=-. The tango algorithm is an online O(log log n)-competitive binary search tree algorithm. It is the first online o(log n)-competitive BST algorithm. 15sCollaboration. The work in this thesis is the re... |

30 |
Sequential access in splay trees takes linear time
- Tarjan
- 1985
(Show Context)
Citation Context ...ghly non-trivial fashion. (It is tough.) • Sequential access. If we access each key in the tree in order, the splay algorithm has cost O(n). This was proved in the classic paper by Sleator and Tarjan =-=[32]-=-. • Preorder access. If we start with an arbitrary initial tree T and use splay to access the nodes in preorder for T, the splay algorithm runs with cost O(|T |) [4]. This is a slight generalization o... |

29 | Rectilinear and polygonal p-piercing and p-center problems
- Sharir, Welzl
- 1996
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Citation Context ...he plane, the rectilinear p-piercing problem is to find p points such that each region intersects at least one of the points, or determine that this is not possible. See the paper by Sharir and Welzl =-=[29]-=- for a more detailed description and references to the literature. 41sThe complexity results for rectilinear p-piercing problems are perhaps not surprising: they are NP-complete if we are required to ... |

21 | The p-center problem: heuristic and optimal algorithms - Drezner - 1984 |

21 | Tight bounds for the partial-sums problem - Patrascu, Demaine - 2004 |

19 | Static optimality and dynamic search-optimality in lists and trees
- Blum, Chawla, et al.
(Show Context)
Citation Context ...). 1.3.3.3 Search optimality Upon cursory examination of the BST problem, it is not clear that online BSTs can get even their search costs (see Section 1.2.3) to be O(OPT(S)). Blum, Chowla, and Kalai =-=[2]-=- used a learning algorithm to show that online BSTs can (in principle) achieve this. Their result is an optimality result under a different cost model. Blum, Chowla, and Kalai describe an online BST a... |

16 |
o(loglogn)-competitive dynamic binary search trees
- Wang, Derryberry, et al.
- 2006
(Show Context)
Citation Context ...his set. Most of the results presented in this thesis can be extended to the case where we can add and remove keys, and search for keys not present in the tree. See papers by Sleator and others [12], =-=[5]-=- for some ideas on how to perform these extensions for the tango algorithm, for example. 18 5 6 7 8s1.2.2.1 Searching for nodes The order invariant allows us to search for keys in the tree. In the mod... |

16 | Approximating a minimum Manhattan network
- Gudmundsson, Levcopoulos, et al.
- 1999
(Show Context)
Citation Context ...tan network on a set of points is a Manhattan network with minimum length. Another definition is one with a minimum number of Steiner points. See the paper by Gudmundsson, Levcopoulos, and Narasimhan =-=[16]-=- for a more extensive discussion and pointers to the literature. This paper provides a polynomial time algorithm for computing an O(1)-competitive Manhattan network of minimum length. The complexity o... |

16 | Optimal bi-weighted binary trees and the complexity of maintaining partial sums
- Hampapuram, Fredman
- 1993
(Show Context)
Citation Context ... upper bounds on the behavior of specific BST algorithms. Many, perhaps most, are related to splay trees [30]. • Lower bounds. These are primarily due to Wilber [33], but are proved elsewhere as well =-=[17]-=-, [28]). See Sections 3.3 and 3.4 for detailed descriptions and a generalization of these bounds. Note that this bound generalization was discovered independently [12], [11]), though without the box m... |

13 |
Data Structures and Algorithms I: Sorting and Searching
- Mehlhorn
- 1984
(Show Context)
Citation Context ...s We strongly suspect an online linearly splittable BST algorithm may be constructed using red-black trees, and it is likely that splay trees are linearly splittable. Mehlhorn shows a related result (=-=[26]-=-) in Exercise 33 on page 311. A BST joining algorithm is similar to a BST splitting algorithm, but the joining algorithm begins with a forest of isolated nodes, and puts them into a single BST through... |

12 |
Canonical forms for competitive binary search tree algorithms
- Lucas
- 1988
(Show Context)
Citation Context ...provided the algorithm does not wander about the tree too much without making structural changes, this cost is within a constant factor of the cost as defined in Section 1.2.3. An argument from Lucas =-=[24]-=- implies that we do not have to wander about too much, so any BST algorithm that does is inefficient, and may be replaced by an algorithm that achieves equivalent structural changes with lower cost. T... |

9 | A rounding algorithm for approximating minimum Manhattan networks
- Chepoi, Nouioua, et al.
(Show Context)
Citation Context ... and pointers to the literature. This paper provides a polynomial time algorithm for computing an O(1)-competitive Manhattan network of minimum length. The complexity of this problem is still unknown =-=[6]-=-, although approximation algorithms for the minimum length Manhattan network are improving. These results do not provide us with much useful information for the analysis of box stabbing problems, unfo... |

9 |
On the competitiveness of linear search
- Munro
- 2000
(Show Context)
Citation Context ...provides a new geometric method of analyzing the behavior of binary search trees. The box model provides interesting results bearing on Munro’s paper on linear search and optimal BST search from 2000 =-=[27]-=-. In particular, we provide evidence that suggests that the offline BST algorithm Munro proposed is dynamically optimal and may be made online. We also use the box model to describe a new class of low... |

8 | A simplified and dynamic unified structure
- Bădoiu, Demaine
(Show Context)
Citation Context ...h we take S) between y and si, including endpoints. Iacono hypothesizes the amortized time to access si using splay trees is O(log min y∈S (ti(y) + di(y) + 1)) Iacono [18] and then Demaine and Bădoiu =-=[3]-=- used a data structure loosely based on BSTs, to achieve this upper bound with O(log n) worst-case run time for any access. 1.3.5 Conjectured performance and dynamic optimality If the cost of a BST al... |

8 | Key independent optimality
- Iacono
(Show Context)
Citation Context ...age 37) on the keys when they sorted by last access time (the most recent access is ordered first). This was only been stated in Wilber’s paper [33], though it has been summarized elsewhere elsewhere =-=[20]-=-. Although the second Wilber bound is Ω(n log n) on some sequences 4 , it is not known if this bound is optimal, or even as large as the cut set lower bounds described in Chapter 3. 1.3.3 Assorted opt... |

7 | A lower bound framework for binary search trees with rotations
- Derryberry, Sleator, et al.
- 2005
(Show Context)
Citation Context ...t in this set. Most of the results presented in this thesis can be extended to the case where we can add and remove keys, and search for keys not present in the tree. See papers by Sleator and others =-=[12]-=-, [5] for some ideas on how to perform these extensions for the tango algorithm, for example. 18 5 6 7 8s1.2.2.1 Searching for nodes The order invariant allows us to search for keys in the tree. In th... |

6 |
Splaying a search tree in preorder takes linear time
- Chaudhuri, Hoft
- 1993
(Show Context)
Citation Context ...assic paper by Sleator and Tarjan [32]. • Preorder access. If we start with an arbitrary initial tree T and use splay to access the nodes in preorder for T, the splay algorithm runs with cost O(|T |) =-=[4]-=-. This is a slight generalization of sequential access. Note that neither of these two arguments is a potential argument; they use detailed arguments about the operation of splay trees to achieve thei... |

6 | An explanation of splaying - Subramanian - 1993 |

5 |
Mihai Pătra¸scu, Dynamic Optimality—Almost
- Demaine, Harmon, et al.
(Show Context)
Citation Context ...n. The work in this thesis is the result of a close collaboration with following people: Erik D. Demaine, John Iacono, and Mihai Pǎtra¸scu. In particular,the results from Chapter 4 appeared in FOCS05 =-=[10]-=-. 1.1 Outline of Thesis Chapter 1 introduces some notation and conventions for BSTs and reviews the literature. We then introduce the box model in Chapter 2, and results related to Munro’s 2000 paper ... |

2 |
Rectilinear mcenter problem
- Ko, Lee, et al.
- 1987
(Show Context)
Citation Context ...he smallest p such that the regions can be pierced by p points [25], but for any fixed p there is an algorithm solving the p-piercing problem that is polynomial in the number of rectangles ([29],[13],=-=[23]-=-). Interesting though these results may be, it is not immediately clear how to apply them to an analysis of box stabbing problems. Taking advantage of the superficial resemblance does not provide imme... |

1 |
General splay: a basic theory and calculus
- McClurkin
- 1999
(Show Context)
Citation Context ...e latter is the splay tree algorithm [30]. Many upper bounds have been proved on the behavior of splay trees, and many of these may be extended to more general (online) rearrangement heuristics [31], =-=[15]-=-. Let S = (s1, s2, . . ., sm) be a sequence of keys of length m with n distinct keys. The run time of a splay tree on S has certain upper bounds. Many of these were proved in the classic paper by Slea... |

1 |
An attempt at showing the box model is NP-complete using reduction to planar 3NAE (and more or less to planar 3SAT). We cannot come up with a convincing splitter
- Email, Sept
(Show Context)
Citation Context ...e lattice of the problem of appropriate size if it exists, but we have not yet been able to prove it is NP-hard. Some reduction approaches using planar 3SAT look promising, but are not yet conclusive =-=[19]-=-. Other questions. With the introduction of a new model, one might ask if there are other models. Is there another (substantially different) model of the BST access 72sproblem that makes computing an ... |

1 |
On the complexity of some common geometric locaiton problems
- Megiddo, Supowit
- 1984
(Show Context)
Citation Context ...sThe complexity results for rectilinear p-piercing problems are perhaps not surprising: they are NP-complete if we are required to find the smallest p such that the regions can be pierced by p points =-=[25]-=-, but for any fixed p there is an algorithm solving the p-piercing problem that is polynomial in the number of rectangles ([29],[13],[23]). Interesting though these results may be, it is not immediate... |