## On random Cartesian trees (1994)

Venue: | Random Structures Algorithms |

Citations: | 2 - 2 self |

### BibTeX

@ARTICLE{Devroye94onrandom,

author = {Luc Devroye},

title = {On random Cartesian trees},

journal = {Random Structures Algorithms},

year = {1994},

volume = {5},

pages = {305--327}

}

### OpenURL

### Abstract

Cartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. lithe search key and the priority key are independent, and the tree is built. based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees. In this article, we analyze the expected behavior when the keys are dependent: in most cases, the expected search, insertion, and deletion times are of). We indicate how these results can be used in the analysis of divide-and-conquer algorithms for maximal vectors and convex hulls. Finally, we look at distributions for which the expected time per operation grows like n a for a E [112, 1}.

### Citations

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Citation Context ...ion, so that events like Xi = X~ or Yk = Y, occur with zero probability . A few special data structures are worth mentioning : A . Ordinary Binary search Trees under the operations irrsExT and sEAxcx =-=[10]-=- can be considered as Cartesian trees in which the second key is the time of insertion : elements down any path in the tree have increasing time stamps . Of course, the second key is not usually store... |

395 | Application of random sampling in computational geometry - Clarkson, Shor - 1989 |

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Citation Context ... and that EI Xr E + El YI < co for some E >0 . This generalizes results in [5, 4, 11], and [13] . For other fast algorithms for maximal vectors, and some analysis, we refer to [1$, G, 21, 25, 3], and =-=[27]-=- . ∎ Remark 6 : Convex Hull Algorithms . Assume that we find the convex hull by first finding the outer layer, which is known to contain the convex hull, and then applying a convex hull algorithm . Co... |

142 | Randomized search trees
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Citation Context ... to have a battery of distributions readily available for drawing counterexamples . Let us describe one such family, in which (X, Y) is distributed 321s322 DEVRQYE on the top left triangle of [0,1] x =-=[0, 2]-=-. Assume that we have independent random variables W and U, where U is uniformly distributed on [0,1], and W has a decreasing density cp on [0,1] . Its distribution function is denoted by ~ . Next, we... |

100 |
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Citation Context ...] provided the first analyses of H,, . Surveys of known results can be found in [39, 22, and 31] . In [14, 16], the theory of branching processes is used in the analysis of H,, . Flajolet and Odlyzko =-=[19]-=- studied H„ under other models of randomization . This brings us finally to Hn . Using Lemma 2 and some large deviation inequalities, we have : Lemma 3 . Let H* be the worst-case insertion time of a r... |

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Citation Context ...described in [12] takes linear expected time . From Example 9, we recall that it suffices that the joint density f be bounded and that EI Xr E + El YI < co for some E >0 . This generalizes results in =-=[5, 4, 11]-=-, and [13] . For other fast algorithms for maximal vectors, and some analysis, we refer to [1$, G, 21, 25, 3], and [27] . ∎ Remark 6 : Convex Hull Algorithms . Assume that we find the convex hull by f... |

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(Show Context)
Citation Context ...described in [12] takes linear expected time . From Example 9, we recall that it suffices that the joint density f be bounded and that EI Xr E + El YI < co for some E >0 . This generalizes results in =-=[5, 4, 11]-=-, and [13] . For other fast algorithms for maximal vectors, and some analysis, we refer to [1$, G, 21, 25, 3], and [27] . ∎ Remark 6 : Convex Hull Algorithms . Assume that we find the convex hull by f... |

15 | On the most probable shape of a search tree grown from a random permutation - Mahmoud, Pittel - 1984 |

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Citation Context ... is a corollary of the independence of the X and Y coordinates) . It is known that the expected depth of the nth node in a random binary search tree on n nodes is asymptotic to 2 log n in many senses =-=[1, 26]-=- . The limit law of the depth of this node, and various other properties were obtained in [25, 26, 37, 35, 33, 24 ; 28, 17] . Various connections with the theory of random permutations [37] and the th... |

9 |
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Citation Context ...nd insertions take O(log n) time on the average, their implementations are of course different from those in binary search trees . C. Pagodas were first introduced by Francon, Viennot ; and Vuillemin =-=[20]-=- as an alternative for a priority queue . Barring certain technical modifications, the pagodas can be thought of as Cartesian trees in which the first key is a time stamp, i .e ., the time of insertio... |

9 |
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(Show Context)
Citation Context ...e time taken by these operations is reduced in a simple manner to the quantities introduced thus far . Without enlarging our model it is not possible to meaningfully discuss the operation DECREASEKEY =-=[38]-=- . We retain from this brief introduction that the quantities of interest to us are D,,, D, 1 , D n,i and Hn . We also introduce the worst-case insertion or deletion time Observe that D n ~ i D,, and ... |

7 | On the joint distribution of the insertion path length and the number of comparisons in search trees - Mahmoud, Pittel - 1988 |

5 |
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Citation Context ...}=nP Z.>t where Z2 , . . . , Zn are independent Bernoulli random variables taking the value one with probability 2/2, . . . , 2/n, respectively [17] . We use Chernoff's exponential bounding technique =-=[7]-=- : let S = > (11j) and let A > o be a constant to be picked further on . Then -z P{Hn > t} nE exp n - A t + A Zi -z n 2 2e'' 1 i i - ne -'" f 1 - i=z <- n exp(-J1t + 2S(e A - 1)) . Take A = log(t/(2S)... |

5 |
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(Show Context)
Citation Context ...d where c = 4 .31107 . . . is the solution of -*1 in probability c log(n) E{H„} -~- c log(n) clog( 2e -) =1 ; c~2 211 ll 311s3 1 2 Remark 1 . For the random binary search tree, Robson [36] and Pittel =-=[35]-=- provided the first analyses of H,, . Surveys of known results can be found in [39, 22, and 31] . In [14, 16], the theory of branching processes is used in the analysis of H,, . Flajolet and Odlyzko [... |

4 |
A note on the height of binary search trees
- unknown authors
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(Show Context)
Citation Context ...th a little extra work, one can also show that LN - ELN - \/Var{LN} n the normal distribution . Next, observe that H,~ is distributed as the height of a random binary search tree . Thus, from Devroye =-=[14, 16]-=- : Lemma 2 . Let Hn be the height of a random Cartesian tree under the independent model . Then and where c = 4 .31107 . . . is the solution of -*1 in probability c log(n) E{H„} -~- c log(n) clog( 2e ... |

3 |
Devroye, Applications of the theory of records in the study of random trees
- unknown authors
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(Show Context)
Citation Context ... of the depth of this node, and various other properties were obtained in [25, 26, 37, 35, 33, 24 ; 28, 17] . Various connections with the theory of random permutations [37] and the theory of records =-=[17]-=- were pointed out over the years . Thus, from [17] : Lemma 1 . Every D, 1 in a random Cartesian tree under the independent model is distributed as Ln , the depth of the last node inserted in arandom b... |

3 |
More combinatorial problems on certain trees
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(Show Context)
Citation Context ...} Ln - E {Ln } \/Var(L n ) DEVROYEsON RANDOM CARTESIAN TREES where-! denotes convergence in distribution and N is a standard normal random variable . The exact distribution of Ln was derived by Lynch =-=[29]-=- and I nuth [26] (see also [37, p . 144]) . We next turn to D nT1 . Clearly, D, s Dn,1 , so that Lemma 1 describes already part of the story . In fact, D 1 is very close to D . We have the following d... |

2 |
Branching processes in the analysis of the heights of trees
- unknown authors
- 1987
(Show Context)
Citation Context ...th a little extra work, one can also show that LN - ELN - \/Var{LN} n the normal distribution . Next, observe that H,~ is distributed as the height of a random binary search tree . Thus, from Devroye =-=[14, 16]-=- : Lemma 2 . Let Hn be the height of a random Cartesian tree under the independent model . Then and where c = 4 .31107 . . . is the solution of -*1 in probability c log(n) E{H„} -~- c log(n) clog( 2e ... |

2 | Fundamentals of the Average Case Analysis of Particular Algorithms - unknown authors - 1984 |

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2 |
Priority search trees
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(Show Context)
Citation Context ...n which the first key is drawn independently from a fixed distribution. This insures the random binary search tree distribution regardless of how the second keys are picked . E. Priority search Trees =-=[34]-=- are not Cartesian trees although they too are designed to store information for double use as a dictionary and a priority queue . This is achieved by creating a binary search tree with respect to the... |

2 |
Mathematical analysis of combinatorial algorithms
- unknown authors
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(Show Context)
Citation Context ... senses [1, 26] . The limit law of the depth of this node, and various other properties were obtained in [25, 26, 37, 35, 33, 24 ; 28, 17] . Various connections with the theory of random permutations =-=[37]-=- and the theory of records [17] were pointed out over the years . Thus, from [17] : Lemma 1 . Every D, 1 in a random Cartesian tree under the independent model is distributed as Ln , the depth of the ... |

2 |
A data structure for manipulating priority queues
- unknown authors
- 1978
(Show Context)
Citation Context ...ly, we look at distributions for which the expected time per operation grows like n a for a E [112, 1} . © 1994 John Wiley & Sons, Inc . 1 . INTRODUCTION Cartesian trees were introduced, by Vuillemin =-=[40,41]-=- as a data structure for storing data according to two keys : they are binary search trees with respect to the first key, and the nodes have the heap property with respect to the second key . The mini... |

1 | Fast linear expected-time algorithms for computing maxima and convex hulls - unknown authors - 1994 |

1 | On the average number of maxima in a set of vectors - unknown authors - 1989 |

1 |
Devroye, A note on finding convex hulls via maximal vectors
- unknown authors
(Show Context)
Citation Context ...described in [12] takes linear expected time . From Example 9, we recall that it suffices that the joint density f be bounded and that EI Xr E + El YI < co for some E >0 . This generalizes results in =-=[5, 4, 11]-=-, and [13] . For other fast algorithms for maximal vectors, and some analysis, we refer to [1$, G, 21, 25, 3], and [27] . ∎ Remark 6 : Convex Hull Algorithms . Assume that we find the convex hull by f... |

1 |
Moment inequalities for random variables in computational geometry
- unknown authors
- 1983
(Show Context)
Citation Context ...M = 0(nn/2)for any r>0 . lim P{Mn > (C + €)\/d}=0, . E{M} lim sup Remark 4. If d Coo, we have EMn = O(n2) by Theorem 3 . The same result could have been obtained by the moment inequalities of Devroye =-=[12]-=- . ∎ Remark S : Finding the Maximal Vectors by Divide-and-Conquer . Two collections of maximal vectors can be merged in linear time if both are sorted according to the same coordinate . By the expecte... |

1 |
On the expected time required to construct the outer layer
- unknown authors
- 1985
(Show Context)
Citation Context ...ree . If M,~ is the number of maximal vectors, we have M Hn . Thus, all the bounds of Theorems 1 and 2 apply as well to Mn . Let us collect these in Theorem 3, which generalizes results by the author =-=[13,16]-=- : Theorem 3. Assume that da C Then, for every e >0, where Also, 1im P{Mn > (C + E)(dan)a1~2a--1)} = s C = 2e(4 log 4)(121 E{M} < 1 lim sup n--~~ C(da n )a/(2a --1) 323s3 2 4 and EMS = D(nr°`'(2c _'})... |

1 | gentler average-case analysis for convex hulls and maximal vectors - unknown authors - 1990 |

1 | Scaling and related techniques for geometry problems - unknown authors - 1984 |

1 | A Handbook of Algorithms and Data Structures - unknown authors - 1984 |

1 |
An efficient algorithm for determining the convex hull of a finite planar set
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- 1972
(Show Context)
Citation Context ... by first finding the outer layer, which is known to contain the convex hull, and then applying a convex hull algorithm . Consider the time required for the second step only . With Graham's algorithm =-=[23]-=-, we obtain an expected complexity equal to o(EMn log Mn) = O(EM) log n = o(n) when da < for some a > 0 (see Example 9) . Thus, on the average, finding convex hulls for these distributions is (complex... |

1 |
The height of binary search trees, A ust
- unknown authors
- 1979
(Show Context)
Citation Context ... model . Then and where c = 4 .31107 . . . is the solution of -*1 in probability c log(n) E{H„} -~- c log(n) clog( 2e -) =1 ; c~2 211 ll 311s3 1 2 Remark 1 . For the random binary search tree, Robson =-=[36]-=- and Pittel [35] provided the first analyses of H,, . Surveys of known results can be found in [39, 22, and 31] . In [14, 16], the theory of branching processes is used in the analysis of H,, . Flajol... |

1 |
A unifying look at data structures
- unknown authors
- 1980
(Show Context)
Citation Context ...ly, we look at distributions for which the expected time per operation grows like n a for a E [112, 1} . © 1994 John Wiley & Sons, Inc . 1 . INTRODUCTION Cartesian trees were introduced, by Vuillemin =-=[40,41]-=- as a data structure for storing data according to two keys : they are binary search trees with respect to the first key, and the nodes have the heap property with respect to the second key . The mini... |

1 | Wheeden and A . Zygmund, Measure and Integral - unknown authors - 1977 |