## Data Structural Bootstrapping, Linear Path Compression, and Catenable Heap Ordered Double Ended Queues (1992)

Venue: | SIAM Journal on Computing |

Citations: | 16 - 7 self |

### BibTeX

@ARTICLE{Buchsbaum92datastructural,

author = {Adam L. Buchsbaum and Rajamani Sundar and Robert E. Tarjan},

title = {Data Structural Bootstrapping, Linear Path Compression, and Catenable Heap Ordered Double Ended Queues},

journal = {SIAM Journal on Computing},

year = {1992},

volume = {24},

pages = {1190--1206}

}

### Years of Citing Articles

### OpenURL

### Abstract

A deque with heap order is a linear list of elements with real-valued keys which allows insertions and deletions of elements at both ends of the list. It also allows the findmin (equivalently findmax) operation, which returns the element of least (greatest) key, but it does not allow a general deletemin (deletemax) operation. Such a data structure is also called a mindeque (maxdeque) . Whereas implementing mindeques in constant time per operation is a solved problem, catenating mindeques in sublogarithmic time has until now remained open. This paper provides an efficient implementation of catenable mindeques, yielding constant amortized time per operation. The important algorithmic technique employed is an idea which is best described as data structural bootstrapping: We abstract mindeques so that their elements represent other mindeques, effecting catenation while preserving heap order. The efficiency of the resulting data structure depends upon the complexity of a special case of pa...

### Citations

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Data Structures and Network Algorithms
- Tarjan
- 1996
(Show Context)
Citation Context ..., d is an output restricted deque. Such data structures can easily be implemented by doublylinked (in some cases singly-linked) lists yielding O(1) worst case times for each of the allowed operations =-=[Tar83]-=-. If each element in d has a real-valued key, we may also want to consider the following operation: findmin(d) Find and return an element of minimum key in d (or ; if d is empty). 1 Findmin does not m... |

143 |
Amortized computational complexity
- Tarjan
- 1985
(Show Context)
Citation Context ...path trees, and mininum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS84] independently showed how to implement minques in O(1) amortized time per operation =-=[Tar85]-=-. Gajewska and Tarjan [GT86] modified their techniques to produce mindeques with O(1) time per operation; they give both amortized and worst case solutions. Applications of mindeques include computing... |

142 | Randomized search trees
- Aragon, Seidel
- 1989
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Citation Context ...loiting the induced left-toright order of the leaves in a normal heap ordered tree arises in the pagodas of Francon et al [FVV78]. Similar data structures are the Cartesian tree [Vui80] and the treap =-=[AS89]-=-. These maintain one tree under both symmetric and heap orders (on two distinct keys per node). If the symmetrically ordered key represents the node's position in a linear list, the data structure sup... |

111 |
Leeuwen. Worst-case analysis of set union algorithms
- Tarjan, van
- 1984
(Show Context)
Citation Context ...erations, plus one final real pop or eject. If we consider the corresponding heap ordered tree, we see that the set of insertions, catenations, and deletions maps to an instance of disjoint set union =-=[TvL84]-=-. In particular, the insertions and catenations correspond to unions and the deletions to finds on the elements which are eventually deleted. That is, a sequence of pulls effects a path compression. F... |

107 | Nonlinearity of Davenport-Schinzel sequences and of a generalized path compression scheme
- Hart, Sharir
- 1984
(Show Context)
Citation Context ...e achieves constant amortized time per operation, we consider order preserving path compression. This is a generalization of special cases of path compression originally introduced by Hart and Sharir =-=[HS86]-=- and subsequently analyzed by Loebl and Nesetril [LN88a, LN88b, LN89] and Lucas [Luc90]. We prove a linear bound on deque ordered spine-only path compression, a case of order preserving path compressi... |

98 |
A Unifying Look at Data Structures
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Citation Context ...r list of items by exploiting the induced left-toright order of the leaves in a normal heap ordered tree arises in the pagodas of Francon et al [FVV78]. Similar data structures are the Cartesian tree =-=[Vui80]-=- and the treap [AS89]. These maintain one tree under both symmetric and heap orders (on two distinct keys per node). If the symmetrically ordered key represents the node's position in a linear list, t... |

35 | Planar graph decomposition and all pairs shortest paths - FREDERICKSON - 1991 |

27 |
River routing every which way but loose
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- 1984
(Show Context)
Citation Context ...tion which complicates the implementation of mindeques. 2.1 Related Work and Applications Queues with heap order (or minques) are useful in pagination [DF84, HL87, LH85, McC77] and VLSI river routing =-=[CS84]-=-. Booth and Westbrook [BW90] use catenable minques in the sensitivity analysis of minimum spanning trees, shortest path trees, and mininum cost network flow on planar graphs. Larmore and Hirschberg [L... |

22 |
Fully persistent lists with catenation
- Driscoll, Sleator, et al.
- 1994
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Citation Context ... leave open the problem of how to catenate mindeques. This paper provides an efficient implementation of catenable mindeques. The important algorithmic technique employed is an idea of Driscoll et al =-=[DST91]-=-, which is best described as data structural bootstrapping: The mindeques of Gajewska and Tarjan are abstracted so that their elements represent other mindeques, effecting catenation while preserving ... |

22 |
Deques with heap order
- Gajewska, Tarjan
- 1986
(Show Context)
Citation Context ...tructure is also called a mindeque (maxdeque). The restricted access and lack of deletemin distinguish mindeques from general heaps and allow faster operation times than do heaps. Gajewska and Tarjan =-=[GT86]-=- show how to implement mindeques with constant time (amortized or worst-case) per operation; they leave open the problem of how to catenate mindeques. This paper provides an efficient implementation o... |

16 | A linear algorithm for analysis of minimum spanning and shortest-path trees of planar graphs, Algorithmica 11 (4
- Booth, Westbrook
- 1994
(Show Context)
Citation Context ...mplementation of mindeques. 2.1 Related Work and Applications Queues with heap order (or minques) are useful in pagination [DF84, HL87, LH85, McC77] and VLSI river routing [CS84]. Booth and Westbrook =-=[BW90]-=- use catenable minques in the sensitivity analysis of minimum spanning trees, shortest path trees, and mininum cost network flow on planar graphs. Larmore and Hirschberg [LH85] and Cole and Siegal [CS... |

10 | Linearity and unprovability of set union problem strategies I. Linearity of strong postorder - Loebl, Neˇsetˇril - 1997 |

10 |
Postorder disjoint set union is linear
- Lucas
- 1990
(Show Context)
Citation Context ...mpression. This is a generalization of special cases of path compression originally introduced by Hart and Sharir [HS86] and subsequently analyzed by Loebl and Nesetril [LN88a, LN88b, LN89] and Lucas =-=[Luc90]-=-. We prove a linear bound on deque ordered spine-only path compression, a case of order preserving path compression employed by our data structure. Our result is important in the following respects. I... |

9 |
Description and analysis of an Efficient Priority Queue Representation
- FRANÇON, VIENNOT, et al.
- 1978
(Show Context)
Citation Context ...s of D. The concept of representing a heap ordered linear list of items by exploiting the induced left-toright order of the leaves in a normal heap ordered tree arises in the pagodas of Francon et al =-=[FVV78]-=-. Similar data structures are the Cartesian tree [Vui80] and the treap [AS89]. These maintain one tree under both symmetric and heap orders (on two distinct keys per node). If the symmetrically ordere... |

8 | Optimal pagination of B-trees with variablelength items - Diehr, Faaland - 1984 |

6 |
Real-time simulation of concatenable double-ended queues by double-ended queues
- Kosaraju
- 1979
(Show Context)
Citation Context ... data structure supports heap ordered list access operations. The idea of bootstrapping mindeques to implement catenable mindeques in the above recursive fashion generalizes the technique of Kosaraju =-=[Kos79]-=-, by which he designs catenable deques (not heap ordered) by decomposing the deques into contiguous pieces and storing 3 (a) 1 1 5 4 1 2 0 0 8 20 8 10 17 10 15 11 (b) 1 1 5 4 1 2 0 0 8 20 8 10 17 10 1... |

5 | Efficient optimal pagination of scrolls
- Larmore, Hirschberg
- 1985
(Show Context)
Citation Context ...4]. Booth and Westbrook [BW90] use catenable minques in the sensitivity analysis of minimum spanning trees, shortest path trees, and mininum cost network flow on planar graphs. Larmore and Hirschberg =-=[LH85]-=- and Cole and Siegal [CS84] independently showed how to implement minques in O(1) amortized time per operation [Tar85]. Gajewska and Tarjan [GT86] modified their techniques to produce mindeques with O... |

4 | Computing external farthest neighbors for a simple polygon - Agarwal, Aggarwal, et al. - 1991 |

4 | New applications of failure functions - Hirschberg, Larmore - 1987 |

4 |
Worst-Case Data Structures for the Priority Queue with Attrition
- Sundar
- 1989
(Show Context)
Citation Context ...heap ordered deque or mindeque. Analogous data structures are obtained by adding findmin to stacks, queues, and output-restricted queues. A related data structure is the priority queue with attrition =-=[Sun89]-=-. We can also consider the findmax operation but will restrict ourselves without loss of generality to findmin for the remainder of this paper. Finally, if d 1 and d 2 are lists of the same type and o... |

1 | Postorder hierarchy for path compressions and set union - Loebl, Nesetril - 1988 |

1 | Pagination of B 3 -trees with variable-length records - McCreight - 1977 |