## Optimal Time-Space Trade-Offs for Non-Comparison-Based Sorting (2001)

Citations: | 3 - 0 self |

### BibTeX

@MISC{Pagh01optimaltime-space,

author = {Rasmus Pagh and Jakob Pagter},

title = {Optimal Time-Space Trade-Offs for Non-Comparison-Based Sorting },

year = {2001}

}

### OpenURL

### Abstract

### Citations

537 |
The input/output complexity of sorting and related problems
- Aggarwal, Vitter
- 1988
(Show Context)
Citation Context ...asoning that has lead to the study of so-called I/O-space trade-offs. Recently both upper and lower bounds on the I/O-space trade-off for sorting and element distinctness were proved in the I/O-model =-=[2]-=- by Arge and Pagter [7], building on the abovementioned upper and lower bound techniques. For completeness, we mention that Beame actually provides a tight timespace upper bound in the branching progr... |

84 | Sorting in linear time
- Andersson, Hagerup, et al.
- 1998
(Show Context)
Citation Context ...nsertion of elements less than the current minimum. 3 lg (k) n denotes the logarithm of n iterated ⌊k⌋ times, e.g. lg (2) n =lglgn.Fork>lg ∗ n we define lg (k) n =1. 3sreference T TS Andersson et al. =-=[6]-=- n lg lg n n 2+o(1) w w.h.p. Han [21] n lg lg n lg lg lg n n 2+o(1) w Pagter and Rauhe [24] n lg n to n 2 /w n 2 new n (lg lg n) 2 to n 2 /w n 2 new n lg lg n to n 2 /w n 2 w.h.p. Table 1: Time-space ... |

59 | A non-linear time lower bound for Boolean branching programs
- Ajtai
- 1999
(Show Context)
Citation Context ...on Turing machines. There are two main lines of motivation for such studies: one is the lower bound perspective, where restricting space allows you to prove general lower bounds for decision problems =-=[3, 4, 9, 10, 23]-=-; the other line is the upper bound perspective where one attempts to find time efficient algorithms that are also space efficient (or vice versa). Also, upper bounds are interesting for more “academi... |

55 | A time-space trade-off for sorting on a general sequential model of computation
- Borodin, Cook
- 1982
(Show Context)
Citation Context .... [13] founded the area of time-space trade-offs for sorting, by proving that any comparison-based sorting algorithm running in time T has TS =Ω(n 2 ). To permit more general bounds, Borodin and Cook =-=[11]-=- introduced the R-way branching program model which, for R =2 w , is at least as strong as a unit-cost RAM with word size w, read-only input, write-only output, and any instruction set. In this model ... |

44 | Time-space tradeoffs for branching programs
- Beame, Saks, et al.
- 1998
(Show Context)
Citation Context ...on Turing machines. There are two main lines of motivation for such studies: one is the lower bound perspective, where restricting space allows you to prove general lower bounds for decision problems =-=[3, 4, 9, 10, 23]-=-; the other line is the upper bound perspective where one attempts to find time efficient algorithms that are also space efficient (or vice versa). Also, upper bounds are interesting for more “academi... |

42 |
The recognition problem for the set of perfect squares
- Cobham
- 1966
(Show Context)
Citation Context ...rly tight. 2s1 Introduction 1.1 Motivation and results The study of time-space trade-offs, i.e., formulae that relate the most fundamental complexity measures, time and space, was initiated by Cobham =-=[15]-=-, who studied problems like recognizing the set of palindromes on Turing machines. There are two main lines of motivation for such studies: one is the lower bound perspective, where restricting space ... |

37 |
Universal classes of hash functions (extended abstract
- Carter, Wegman
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Citation Context ... fast sorting algorithms show how to decide element distinctness, but none of them in a very time-space efficient manner (more than a factor of w from the lower bound). Using universal hash functions =-=[14]-=- one can decide element distinctness in expected linear time, using O(n lg n + w) bits of space. By another approach based on hashing, Ajtai [4] provides a two-sided error randomized algorithm with ti... |

36 | Super-linear time-space tradeoff lower bounds for randomized computation
- Beame, Saks, et al.
(Show Context)
Citation Context ...on Turing machines. There are two main lines of motivation for such studies: one is the lower bound perspective, where restricting space allows you to prove general lower bounds for decision problems =-=[3, 4, 9, 10, 23]-=-; the other line is the upper bound perspective where one attempts to find time efficient algorithms that are also space efficient (or vice versa). Also, upper bounds are interesting for more “academi... |

27 | A general sequential time-space tradeoff for finding unique elements
- Beame
- 1991
(Show Context)
Citation Context ... any w ≥ lg n. Thiswaslater 4 The high probability bound for this algorithm is not stated in the paper, but can be achieved by increasing the range of the hash functions employed. 4simproved by Beame =-=[8]-=-, who showed that any sorting algorithm running in time T has TS =Ω(n 2 ). The proof counts only accesses to the input, and hence holds for any kind of instruction set as long as we can only read O(1)... |

27 |
Polynomial hash functions are reliable (extended abstract
- Dietzfelbinger, Gil, et al.
(Show Context)
Citation Context ...(n lg n) words of space with constant factor overhead, w.h.p. (the perhaps simplest implementation of this being a hash table with chaining, using the reliable hash functions of Dietzfelbinger et al. =-=[16]-=-). Theorem 4 (Thorup [25]) Let s, t : N → R+ be non-decreasing. Suppose there is an algorithm sorting n words in time O(nt(n)), using s(n)w bits of space. Then there exists a monotone priority queue s... |

12 |
Time-space tradeoffs for algebraic problems on general sequential machines
- ABRAHAMSON
- 1991
(Show Context)
Citation Context ...nd hence holds for any kind of instruction set as long as we can only read O(1) input words at a time. As noted by Beame, his result also holds for average time and space by using ideas of Abrahamson =-=[1]-=-. Yao’s minimax principle [28] then provides the same lower bound for the expected time and space usage of any randomized Las Vegas algorithm. Element distinctness For the related problem of element d... |

7 |
Friedhelm Meyer auf der Heide, Eli Upfal, Avi Wigderson. “A Tradeoff Between Search and Update Timefor the Implicit Dictionary Problem
- Borodin, Fich
- 1988
(Show Context)
Citation Context ... expected time and space usage of any randomized Las Vegas algorithm. Element distinctness For the related problem of element distinctness, time-space trade-offs are also well-studied. Borodin et al. =-=[12]-=- proved a lower bound of TS =Ω(n 3/2 ) for comparison-based algorithms, which was later improved by Yao [29] to TS =Ω(n 2−ɛ(n) ), where ɛ(n) =5/ √ lg n. In the general setting of R-way branching progr... |

3 |
A Time-Space Tradeoff for Sorting on
- Borodin, Fischer, et al.
- 1981
(Show Context)
Citation Context ...tically, employing a deterministic linear space priority queue using a non-standard AC0 instruction set to provide constant time operations [20]. 1.2 History 1.2.1 Lower bounds Sorting Borodin et al. =-=[13]-=- founded the area of time-space trade-offs for sorting, by proving that any comparison-based sorting algorithm running in time T has TS =Ω(n 2 ). To permit more general bounds, Borodin and Cook [11] i... |

2 |
Sorting and Searching Revisited,” in Algorithm Theory
- Andersson
- 1996
(Show Context)
Citation Context ... algorithm into an efficient monotone priority queue. This is described in more detail in section 2.2.1. For further information on sorting on the RAM, we refer the reader to the surveys of Andersson =-=[5]-=- and Hagerup [20]. Sorting in other models Time-space upper bounds become relevant from a practical point of view when the number of input elements is huge, a line of reasoning that has lead to the st... |

1 |
versus Non-Determinism for Linear Time RAMs with Memory Restrictions
- Determinism
- 1999
(Show Context)
Citation Context |

1 |
I/O-Space Trade-Offs, 7th Scandinavian Workshop on Algorithm Theory
- Arge, Pagter
- 2000
(Show Context)
Citation Context ...o the study of so-called I/O-space trade-offs. Recently both upper and lower bounds on the I/O-space trade-off for sorting and element distinctness were proved in the I/O-model [2] by Arge and Pagter =-=[7]-=-, building on the abovementioned upper and lower bound techniques. For completeness, we mention that Beame actually provides a tight timespace upper bound in the branching program model for elements i... |