## Self-improving algorithms

### Cached

### Download Links

- [www.cs.princeton.edu]
- [www.cs.princeton.edu]
- [www.cs.princeton.edu:80]
- [www.siam.org]
- [www.distcomp.ethz.ch]
- [disco.ethz.ch]
- [dcg.ethz.ch]
- [distcomp.ethz.ch]
- [www.dcg.ethz.ch]
- [siam.org]
- [www.cs.princeton.edu]
- [www.distcomp.ethz.ch]
- [distcomp.ethz.ch]
- [dcg.ethz.ch]
- [disco.ethz.ch]
- [disco.ethz.ch]
- [distcomp.ethz.ch]
- [dcg.ethz.ch]
- [www.distcomp.ethz.ch]
- [www.dcg.ethz.ch]
- [www.dcg.ethz.ch]
- [www.cs.princeton.edu]
- [www.cs.princeton.edu]
- [arxiv.org]
- [page.mi.fu-berlin.de]
- [www.cs.princeton.edu]
- [www.cs.princeton.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | in SODA ’06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm |

Citations: | 24 - 4 self |

### BibTeX

@INPROCEEDINGS{Ailon_self-improvingalgorithms,

author = {Nir Ailon and Bernard Chazelle and Kenneth L. Clarkson and Ding Liu and Wolfgang Mulzer and C. Seshadhri},

title = {Self-improving algorithms},

booktitle = {in SODA ’06: Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm},

year = {},

pages = {261--270}

}

### OpenURL

### Abstract

We investigate ways in which an algorithm can improve its expected performance by finetuning itself automatically with respect to an arbitrary, unknown input distribution. We give such self-improving algorithms for sorting and computing Delaunay triangulations. The highlights of this work: (i) an algorithm to sort a list of numbers with optimal expected limiting complexity; and (ii) an algorithm to compute the Delaunay triangulation of a set of points with optimal expected limiting complexity. In both cases, the algorithm begins with a training phase during which it adjusts itself to the input distribution, followed by a stationary regime in which the algorithm settles to its optimized incarnation. 1

### Citations

8595 |
Elements of Information Theory
- Cover, Thomas
- 1991
(Show Context)
Citation Context ...we have H(π(I)) = O(n). 2. Entropy and Comparison-based Algorithms. Before we consider sorting and Delaunay triangulations, let us first recall some useful properties of information theoretic entropy =-=[28]-=- and explain how it relates to our notion of comparison-based algorithms. Let X be a random variable with a finite range X . The entropy of X, H(X), is defined as H(X) := ∑ x∈X Pr[X = x] log(1/ Pr[X =... |

8543 |
Introduction to Algorithms
- Cormen, Leiserson, et al.
- 1990
(Show Context)
Citation Context ...omparison outcomes. Thus, any comparisonbased algorithm must perform at least H(X) comparisons in expectation. Note that our comparison-based algorithms include all the traditional sorting algorithms =-=[27]-=- (selection sort, insertion sort, quicksort, etc) as well as classic algorithms for Delaunay triangulations [12] (randomized incremental construction, divide and conquer, plane sweep). A notable excep... |

1926 |
Pattern Classification
- Duda, Hart, et al.
- 2001
(Show Context)
Citation Context ...n systems, etc). Input data is often lodged in a tiny slice of input space that cannot be captured by closed-form distributions. To make predictions about the slice is the essence of machine learning =-=[18,23,27]-=-. To take computational advantage of the slice is what self-improving algorithms are all about. Our Results The performance of a self-improving algorithm is measured with respect to an unknown memoryl... |

1870 | Randomized Algorithms - Motwani, Raghavan - 1995 |

1693 |
The Probabilistic Method
- Alon, Spencer
- 1992
(Show Context)
Citation Context ...he Hi ’s themselves are random variables depending on the choice of the V -list. Therefore, this is a conditional expectation. 5for the event that uℓ ∈ [ui, uj) = t. Let Y (t) = ∑ ℓ Chernoff’s bound =-=[4]-=-, for any β ∈ (0, 1], Y (t) ℓ Pr[Y (t) ≤ (1 − β)E[Y (t) ]] ≤ e −β2 E[Y (t) ]/2 . . Since all the Y (t) ℓ ’s are independent, by With probability at least 1 −n−4 , if E[Y (t)] > 4c log n, then Y (t) > ... |

972 | Computational Geometry, Algorithms and Applications - Berg, Kreveld, et al. - 1997 |

727 |
Amortized efficiency of list update and paging rules
- Sleator, Tarjan
- 1985
(Show Context)
Citation Context ... concepts have been studied before. List accessing algorithms and splay trees are textbook examples of how simple updating rules can speed up searching with respect to an adversarial request sequence =-=[2,9,22,29,30]-=-. It is interesting to note that self-organizing data structures were investigated over stochastic input models first [1,3,8,21,25,28]. It was the observation [7] that memoryless sources for list acce... |

674 | Learning quickly when irrelevant attributes abound: a new linear-threshold algorithm - Littlestone - 1988 |

669 | The Weighted Majority Algorithm - Littlestone, Warmuth - 1004 |

648 |
Online computation and competitive analysis
- Borodin, El-Yaniv
- 1998
(Show Context)
Citation Context ... concepts have been studied before. List accessing algorithms and splay trees are textbook examples of how simple updating rules can speed up searching with respect to an adversarial request sequence =-=[2,9,22,29,30]-=-. It is interesting to note that self-organizing data structures were investigated over stochastic input models first [1,3,8,21,25,28]. It was the observation [7] that memoryless sources for list acce... |

613 | Coordination of groups of mobile autonomous agents using nearest neighbor rules
- Jadbabaie, Lin, et al.
- 2003
(Show Context)
Citation Context ...umption stipulates that, over any time interval of a fixed length, every pair of birds should be able to communicate with each other, directly or indirectly via other birds. Jadbabaie, Lin, and Morse =-=[5]-=- proved the first of several convergence results under that assumption (eg, [11,12,17,20]). Several authors extended these results to variable-length intervals [4,8,10]. They established that the bird... |

594 |
An Introduction to Computational Learning Theory
- Kearns, Vazirani
- 1994
(Show Context)
Citation Context ...n systems, etc). Input data is often lodged in a tiny slice of input space that cannot be captured by closed-form distributions. To make predictions about the slice is the essence of machine learning =-=[18,23,27]-=-. To take computational advantage of the slice is what self-improving algorithms are all about. Our Results The performance of a self-improving algorithm is measured with respect to an unknown memoryl... |

389 | Applications of random sampling in computational geometry
- Clarkson, Shor
- 1987
(Show Context)
Citation Context ...To that end, we will use divide-and-conquer to compute the Voronoi diagram V(V ∪ I), using a scheme that has been used for nearest neighbor searching [24] and for randomized convex hull constructions =-=[20,25]-=-. It is well known that the Voronoi diagram of a point set is dual to the Delaunay triangulation, and that we can go from one to the other in linear time [12, Chapter 9]. Refer to Fig. 4.4(a). Conside... |

372 | Self-adjusting binary search trees - Sleator, Tarjan - 1985 |

313 | How to use expert advice - Cesa-Bianchi, Freund, et al. - 1997 |

277 | Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms
- Edelsbrunner, Mücke
- 1990
(Show Context)
Citation Context ...n a common circle). This is no loss of generality and does not restrict the distribution D, because the general position assumption can always be enforced by standard symbolic perturbation techniques =-=[29]-=-. Also we will assume that there is a bounding triangle that always contains all the points in I. Again, this does not restrict the distribution D in any way, because we can always simulate the boundi... |

274 |
Novel type of phase transition in a system of self-driven particles
- Vicsek, Czirk, et al.
- 1995
(Show Context)
Citation Context ...els as possible, we consider two specific examples that are highly representative of the many variants considered in the literature. Model K (for kinematic) is a variant of the classical Vicsek model =-=[22]-=-. The control does not allow for variations in speed—only headings can change—so the model is nonholonomic. Model D (for dynamic) averages over velocities: it includes inertia and is fully actuated, a... |

253 | Geometric range searching and its relatives, in
- Agarwal, Erickson
- 1999
(Show Context)
Citation Context ...linear. Geometric range searching seem to a good source of such problems. We are given some set of points and we want to build data structures that answer various geometric queries about these points =-=[2]-=-. Suppose the points came from some distribution. Can we speed up the construction of these structures? A different approach to self-improving algorithms would be to change the input model. We current... |

221 |
Stability of multiagent systems with time-dependent communication links
- Moreau
- 2005
(Show Context)
Citation Context ...her birds. Jadbabaie, Lin, and Morse [5] proved the first of several convergence results under that assumption (eg, [11,12,17,20]). Several authors extended these results to variable-length intervals =-=[4,8,10]-=-. They established that the bird group always ends up as a collection of separate flocks (perhaps only one), each one converging toward its own speed and heading. Some authors have shown how to do awa... |

193 | Toward efficient agnostic learning - Kearns, Schapire, et al. - 1994 |

189 | Efficient search for approximate nearest neighbor in highdimensional spaces - Kushilevitz, Ostrovsky, et al. - 1998 |

151 |
Applications of random sampling
- Clarkson, Shor
- 1989
(Show Context)
Citation Context ...To that end, we will use divide-and-conquer to compute the Voronoi diagram V(V ∪ I), using a scheme that has been used for nearest neighbor searching [13] and for randomized convex hull constructions =-=[11, 14]-=-. It is well known that the Voronoi diagram of a point set is dual to the Delaunay triangulation, and that we can go from one to the other in linear time. Consider the Voronoi diagram of V , V(V ). By... |

151 | Flocking for multi-agent dynamic systems: algorithms and theory
- Olfati-Saber
(Show Context)
Citation Context ...f birds should be able to communicate with each other, directly or indirectly via other birds. Jadbabaie, Lin, and Morse [5] proved the first of several convergence results under that assumption (eg, =-=[11,12,17,20]-=-). Several authors extended these results to variable-length intervals [4,8,10]. They established that the bird group always ends up as a collection of separate flocks (perhaps only one), each one con... |

149 | Computational Complexity: A Modern Approach - ARORA, BARAK - 2009 |

145 |
Data Structures and Algorithms 1: Sorting and Searching. EATCS,Monographs on Theoretical Computer Science
- Mehlhorn
- 1984
(Show Context)
Citation Context ...rch structures. • The Di-trees: For any i > 0, let B V i be the predecessor2 of a random y from Di in the V -list, and let H V i be the entropy of B V i . The Di-tree is an optimum binary search tree =-=[26]-=- over the keys of the V -list, where the access probability of vk is ∑ j { pi,j | vk ≤ j < vk+1 } 3 , for any 0 ≤ k ≤ n: the same distribution used to define HV i . This allows us to compute BV i usin... |

140 | Fundamental Problems of Algorithmic Algebra - Yap - 2000 |

134 | Efficient algorithms for online decision problems - Kalai, Vempala - 2003 |

131 | Adaptive game playing using multiplicative weights - Freund, Schapire - 1999 |

113 |
A randomized algorithm for closest-point queries
- CLARKSON
- 1988
(Show Context)
Citation Context ...V ∪ I) in linear time using the conflict sets Zt. To that end, we will use divide-and-conquer to compute the Voronoi diagram V(V ∪ I), using a scheme that has been used for nearest neighbor searching =-=[13]-=- and for randomized convex hull constructions [11, 14]. It is well known that the Voronoi diagram of a point set is dual to the Delaunay triangulation, and that we can go from one to the other in line... |

113 |
Detecting strange attractors in turbulence, in Dynamical systems and turbulence: lecture notes in mathematics
- Takens
- 1981
(Show Context)
Citation Context ...dding theorem asserts that univariate time series obtained from deterministic dynamical systems can be geometrized canonically as a (usually) low-dimensional attractor set in finite-dimensional space =-=[31]-=-. Hidden Markov models for speech recognition can be remarkably effective with only a few thousand states. Anecdotal evidence can also be gleaned from the current trend toward personalization in the d... |

111 | INDYK: Approximate clustering via core-sets - BĂDOIU, HAR-PELED, et al. |

105 | Algorithmic Geometry - Boissonnat, Yvinec - 1998 |

82 | The Discrepancy Method - Chazelle - 2000 |

80 | Stable flocking of mobile agents, Part I: Fixed topology - Tanner, Jadbabaie, et al. - 2003 |

79 | Alinear-time algorithm for computing the Voronoi diagram of a convex polygon - Aggarwal, Guibas, et al. - 1989 |

75 | Flocking in fixed and switching networks - Tanner, Jadbabaie, et al. - 2007 |

71 | Online convex optimization in the bandit setting: Gradient descent without a gradient - Flaxman, Kalai, et al. - 2005 |

70 |
On self-organizing sequential search heuristics
- Rivest
- 1976
(Show Context)
Citation Context ... speed up searching with respect to an adversarial request sequence [2,9,22,29,30]. It is interesting to note that self-organizing data structures were investigated over stochastic input models first =-=[1,3,8,21,25,28]-=-. It was the observation [7] that memoryless sources for list accessing are not terribly realistic that partly motivated work on the adversarial models. It is highly plausible that both approaches are... |

69 | Emergent behavior in flocks
- Cucker, Smale
(Show Context)
Citation Context ...do away with the recurrent connectivity assumption by changing the model suitably. Tahbaz-Salehi and Jadbabaie [18], for example, assume that the birds fly on the surface of a torus. Cucker and Smale =-=[3]-=- use a broadcast model that extends a bird’s influence to the entire group while scaling it down as a function of distance. In a similar vein, Ji and Egerstedt [6] introduce a hysteresis rule to ensur... |

66 | The k-server problem - Koutsoupias - 2009 |

65 | A survey of adaptive sorting algorithms
- Estivill-Castro, Wood
- 1992
(Show Context)
Citation Context ...1H(π(I))): this tradeoff is optimal for distributions of high enough entropy. The algorithm reaches its steady state within O(nε log n) rounds. Remark: Much research has been done on adaptive sorting =-=[19]-=-, especially on algorithms that exploit near-sortedness. Our approach is conceptually different. As we mentioned in the previous section, we seek to exploit properties, not of individual inputs, but o... |

64 | Applications of weighted Voronoi diagrams and randomisation to variance-based ko clustering. }~oc. 10th Annu. ACId S~n~s - Inaba, Katoh, et al. - 1994 |

62 | Online geometric optimization in the bandit setting against an adaptive adversary - McMahan, Blum - 2004 |

61 | An optimal algorithm for intersecting three-dimensional convex polyhedra
- Chazelle
- 1992
(Show Context)
Citation Context ...ion within V(Zs) ∩ s (Claim 3.8) Sorted list of V ∪ I T(V ∪ I) Build sorted V from sorted V ∪ I (trivial) Build T(I) from T(V ∪ I) [12] (analysis) merge sorted V and I (analysis) merge T(V ) and T(I) =-=[10]-=- (analysis) recover indices B V i (trivial) from sorted I (analysis) recover triangles B V i T(I) (Lemma 3.11) in T(V ) from Figure 1: Delaunay triangulation algorithm as a generalization of the sorti... |

60 | On k-Nearest Neighbor Voronoi Diagrams in the Plane - LEE - 1982 |

58 | Testing of clustering - Alon, Dar, et al. - 2003 |

55 | RABANI: Approximation schemes for clustering problems - VEGA, KARPINSKI, et al. - 2003 |

55 | Improved approximation algorithms for geometric set cover, Discrete Comput
- Clarkson, Varadarajan
(Show Context)
Citation Context ...ing instance points with the same BV value Expect O(1) values of I within each bucket (of the same BV index) Optimal weighted binary trees Ti Delaunay Triangulation Delaunay disks Range space ε-net V =-=[15, 24]-=-, ranges are disks, ε = 1/n log n training instance points in each Delaunay disk Expect O(1) points of I in each Delaunay disk of V Entropy-optimal planar point location data structures Ti [6] Sorting... |

53 |
Efficient computation of continuous skeletons
- Kirkpatrick
- 1979
(Show Context)
Citation Context .... Given I and T (I), for every xi in I we can compute the triangle BV i in T (V ) that contains xi in total expected time O(n). Proof. First, we compute T (V ∪ I) from T (V ) and T (I) in linear time =-=[19, 36]-=-. Thus, we now know T (V ∪ I) and T (V ), and we want to find for every point xi ∈ I the triangle BV i of T (V ) that contains it. For the moment, let us be a little less ambitious and try to determin... |

51 | Tsitsiklis, “Convergence speed in distributed consensus and averaging - Olshevsky, N - 2009 |