## Static Optimality and Dynamic Search-Optimality in Lists and Trees (2002)

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Citations: | 19 - 3 self |

### BibTeX

@MISC{Blum02staticoptimality,

author = {Avrim Blum and Shuchi Chawla and Adam Kalai},

title = {Static Optimality and Dynamic Search-Optimality in Lists and Trees},

year = {2002}

}

### Years of Citing Articles

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### Abstract

Adaptive data structures form a central topic of online algorithms research, beginning with the results of Sleator and Tarjan showing that splay trees achieve static optimality for search trees, and that Move-toFront is constant competitive for the list update prob- lem [ST85a, ST85b]. This paper is inspired by the observation that one can in fact achieve a 1 + e ra- tio against the best static object in hindsight for a wide range of data structure problems via "weighted experts" techniques from Machine Learning, if computational decision-making costs are not considered.

### Citations

747 | Tarjan. Amortized efficiency of list update and paging rules - Sleator, E - 1985 |

697 | The weighted majority algorithm
- Littlestone, Warmuth
- 1994
(Show Context)
Citation Context ...lgorithms. Some progress on ecient MCMC simulations for these type of algorithms has been made [CLS01, KV00] but not for the situations of interest to us. 1 The Randomized Weighted Majority algorithm =-=[LW94] main-=-tains a probability distribution over the N component algorithms (\experts"). At each time step, the experts are penalized based on their cost at that step, multiplying the weight on an expert in... |

680 |
Online computation and competitive analysis
- Borodin, El-Yaniv
- 1998
(Show Context)
Citation Context ...lgorithm is randomized: the standard lower bound of 2 for deterministic algorithms holds for static ratio as well. We combine this algorithm with the currently best known dynamic algorithm (COMB, see =-=[BEY98]-=- for details) to obtain one which is dynamically optimal and at the same time has strong static optimality. Second, for trees, we show a (computationally inecient) algorithm motivated by online learni... |

194 | Gambling in a rigged casino: The adversarial multi-armed bandit problem
- Auer, Milano, et al.
- 1998
(Show Context)
Citation Context ... as that would require running the two algorithms simultaneously, defeating the very purpose of this algorithm. In order to get around this diculty, we use a variant of the Exp3 algorithm proposed by =-=[ACBFS95-=-]. This algorithm was proposed for the multiarmed bandit problem in which a gambler must decide to play on one of K slot machines, but gets to see the prot of only the machine that he is playing on. [... |

137 | Game theory, on-line prediction and boosting
- Freund, Schapire
- 1996
(Show Context)
Citation Context ...ing a prediction, in which case this procedure can be viewed as taking a weighted vote and then choosing a prediction probabilistically based on the vote totals. It is also called the Hedge algorithm =-=[FS96]-=-. In many situations the best static algorithm may be much worse than an optimal dynamic algorithm. In the case of list update, for instance, the best static algorithm would be O(n) times worse than t... |

75 | Tarjan. Amortized eciency of list update and paging rules - Sleator, E - 1985 |

36 | On-line learning and the metrical task system problem
- Blum, Burch
- 1997
(Show Context)
Citation Context ...bject in hindsight, one can in principle achieve a ratio of 1+ for a wide variety of data structure problems, if we ignore time spent computing which update to perform. In particular, Blum and Burch [=-=BB00-=-] show the following general result. Theorem 1.1. (Theorem 4 of [BB00]) Given N on-line algorithms for a Metrical Task System Problem of diameter D, and given > 0, one can use the Randomized Weighted... |

34 | Efficient algorithms for universal portfolios
- Kalai, Vempala
(Show Context)
Citation Context ...ust explicitly maintain a probability distribution over all the component algorithms. Some progress on efficient Markov Chain Monte Carlo simulations for these type of algorithms has been made [CLS], =-=[KV1]-=- but not for the situations of interest to us. In many situations the best static algorithm may be much worse than an optimal dynamic algorithm. In the case of list update, for instance, the best stat... |

32 |
Lower bounds for accessing binary search trees with rotations
- Wilber
- 1989
(Show Context)
Citation Context ...ode comes next and therefore has too many nodes to keep near the root. However, the existence of an algorithm with dynamic search optimality implies that this type of argument is not possible. Wilber =-=[Wil8-=-9] made some progress by proving lower bounds for o-line algorithms. In particular, he has shown that a random sequence of accesses in f1; 2; : : : ; ng costs an expectedsted n) per access for o-line ... |

29 | A combined BIT and TIMESTAMP algorithm for the list update problem - Albers, Stengel, et al. - 1995 |

28 | A lower bound for randomized list update algorithms - Teia - 1993 |

12 |
Canonical forms for competitive binary search tree algorithms
- Lucas
- 1988
(Show Context)
Citation Context ...than k. at most 2 6k descriptions of length 6k. We then get 2 12k because of the factor of 2 lost in our initial assumption. What remains is to describe a set of r rotations using 6r bits. Like Lucas =-=[Luc88]-=-, we think of a rotation as an edge rotation which changes a single edge from either left to right or right to left. Of course, the nodes adjacent to an edge may change. But based on our assumption th... |

11 | 2003, ‘On Approximating Weighted Sums with Exponentially Many Terms - Chawla, Li, et al. |

4 |
Machine learning in metrical task systems and other on-line problems
- Burch
- 2000
(Show Context)
Citation Context ...]. This algorithm was proposed for the multiarmed bandit problem in which a gambler must decide to play on one of K slot machines, but gets to see the prot of only the machine that he is playing on. [=-=Bur00-=-] extend the Exp3 algorithm to the case where there is a cost d for switching between the dierent experts, and so, we may want to switch after every s steps, rather than after every step. We will now ... |

4 |
Dominic Sleator and Robert Endre Tarjan, Self-adjusting binary search trees
- Daniel
- 1985
(Show Context)
Citation Context ...l cost of the best o-line algorithm, for any sequence of accesses. Much work has gone into an attempt to prove the dynamic optimality of splay trees since Sleator and Tarjan made their conjecture in [=-=ST85b]-=-. We are unable to prove the dynamic optimality of splay trees or any other on-line search tree algorithm. Instead, we show that it is possible to have dynamicssearch-optimality, a property much weake... |

2 | Ecient algorithms for universal portfolios - Kalai, Vempala - 2000 |

2 |
Follow the Leader for Online Algorithms. Unpublished Manuscript
- Kalai, Vempala
(Show Context)
Citation Context ... optimality for trees. At the very least, this shows that dynamic optimality cannot be proven impossible by any argument that neglects the rotation cost of the online algorithm. 2 Following our work, =-=[KV02-=-] developed an ecient algorithm for Binary Search Trees that achieves a ratio of 1 + with respect to the best static tree in hindsight. 1 The Randomized Weighted Majority algorithm [LW94] maintains a... |

1 | Eciently approximating weighted sums with exponentially many terms - Chawla, Li, et al. - 2001 |