## Overview (2010)

### BibTeX

@MISC{Demaine10overview,

author = {Prof Erik Demaine},

title = {Overview},

year = {2010}

}

### OpenURL

### Abstract

In the last lecture we discussed Binary Search Trees(BST) and introduced them as a model of computation. A quick recap: A search is conducted with a pointer starting at the root, which is free to move about the tree and perform rotations; however, the pointer must at some point in the operation visit the item being searched. The cost of the search is simply the total number of distinct nodes in the trees that have been visited by the pointer during the operation. We measure the total cost of executing a sequence of searches S = �s1, s2, s3...�, where each search si is chosen from among the fixed set of n keys in the BST. We have witnessed that there are access sequences which require o(log(n)) time per operation. There are also some deterministic sequences on n queries (for example, the bit reversal permutation) which require a total running time of Ω(n log(n)) for any BST algorithm. This disparity however does not rule out the possibility of having an instance optimal BST. By this we mean: Let OPT(S) denote the minimal cost for executing the access sequence S in the BST model, or the cost of the best BST algorithm which has access to the sequence apriori. It is believed that splay trees are the “best BST”. However, they are not known to have o(log(n)) competitive ratio. Also, notice that we are only concerned with the cost of the specified operations on the BST and we are not

### Citations

32 |
Lower bounds for accessing binary search trees with rotations
- Wilber
- 1989
(Show Context)
Citation Context ...s (more tightly fitting xj), or nothing happens (the access is uninteresting for xj). The Wilber number is the number of alternations between ai increasing and bi decreasing. Wilber’s 2nd lower bound =-=[Wil89]-=- states that the sum of the Wilber numbers of all xj’s is a lower bound on the total cost of the access sequence, for any BST. It should be noted that this only holds in an amortized sense (for the to... |

8 | Key independent optimality
- Iacono
(Show Context)
Citation Context ... the expectation is taken over a random permutation π of the set of keys. Key-independent optimality is defined as usual with respect to the optimal offline BST. Theorem 6 (key independent optimality =-=[Iac02]-=-). A BST has the key-independent optimality property iff it has the working-set property. In particular, splay trees are key-independently optimal. 3Proof. (sketch) We must show that: � � � m Eπ[dyna... |

6 |
Mihai Pǎtra¸scu. Dynamic optimality — almost
- Demaine, Harmon, et al.
(Show Context)
Citation Context ...RB(X) + |X| 2 minASS⊠(X) minASS�(X) + minASS�(X) |add�(X)| + |add�(X) + 2|X| = |IRB�(X)| + |IRB�(X) + 2|X| ≤ ≤ ≤ ≤ 2maxIRB(X) + 2|X| 2maxIRB(X) + 4|X| 4minASS⊠(X) 4minASS(X) 5 Tango Trees Tango trees =-=[DHIP04]-=- are an O(lg lg n)-competitive BST. They represent an important step forward from the previous competitive ratio of O(lg n), which is achieved by standard balanced trees. The running time of Tango tre... |

1 |
The geometry of binary search trees. SODA ’09
- Demaine, Harmon, et al.
(Show Context)
Citation Context ...so far is the O(log log(n)) competitive ratio achieved by the Tango Trees - we shall see them in the later part of the lecture. Another perspective, is the recently proposed geometric view of the BST =-=[DHIKP09]-=-. In this approach, an correspondence between the BST model of computation and points in R2 is given. Informally, call a set P of points arborally satisfied if, for any two points a, b ∈ P not on a co... |