## New Lower Bound Techniques For Dynamic Partial Sums and Related Problems (2003)

Venue: | SIAM Journal on Computing |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Husfeldt03newlower,

author = {Thore Husfeldt and Theis Rauhe},

title = {New Lower Bound Techniques For Dynamic Partial Sums and Related Problems},

journal = {SIAM Journal on Computing},

year = {2003},

volume = {32},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the complexity of the dynamic partial sum problem in the cell-probe model. We give the model access to nondeterministic queries and prove that the problem remains hard. We give the model access to the right answer as an oracle and prove that the problem remains hard. This suggests which kind of information is hard to maintain. From these results, we derive a number of lower bounds for dynamic algorithms and data structures: We prove lower bounds for dynamic algorithms for existential range queries, reachability in directed graphs, planarity testing, planar point location, incremental parsing, and fundamental data structure problems like maintaining the majority of the prefixes of a string of bits. We prove a lower bound for reachability in grid graphs in terms of the graph's width. We characterize the complexity of maintaining the value of any symmetric function on the prefixes of a bit string. Keywords. cell-probe model, partial sum, dynamic algorithm, data structure AMS subject classifications. 68Q17, 68Q10, 68Q05, 68P05

### Citations

235 | Almost optimal lower bounds for small depth circuits - H˚astad - 1989 |

138 | Should tables be sorted
- Yao
- 1981
(Show Context)
Citation Context ...risingly interesting” [13], and it has been the focal point of many investigations of dynamic complexity in a variety of models [15, 31]. We reason within the cell-probe model of Fredman [12] and Ya=-=o [30]-=- with some extensions to cope with our stronger modes of computation. The model can be viewed as a nonuniform version of the random access computer with arbitrary register instructions. Lower bounds a... |

128 |
The cell probe complexity of dynamic data structures
- Fredman, Saks
- 1989
(Show Context)
Citation Context ...each update). However, in many partial sum problems— and in many dynamic problems in general—we cannot have both. This trade-off between update time and query time was established by Fredman and S=-=aks [15], who-=- showed that, with the parity query parity(i): return x1 + ···+ xi mod 2, the partial sum problem requires time Ω(log n/ log log n) per operation on the unitcost RAM with logarithmic cell size. I... |

84 | On data structures and asymmetric communication complexity
- Miltersen, Nisan, et al.
- 1998
(Show Context)
Citation Context ... by Fredman and Saks [15] and got its name in [7]. The prefix parity problem was solved in [15], but no nontrivial lower bounds for the majority or equality problems followfrom that. The results from =-=[6, 21, 22, 29] can be -=-seen to imply Ω(log log n/ log log log n) lower bounds using an entirely different technique based on Ajtai’s result [2]; and [19] reports Ω((log n/ log log n) 1/2 ) for equality and Ω(log n/(... |

57 | Optimal bounds for the predecessor problem and related problems
- Beame, Fich
(Show Context)
Citation Context ... by Fredman and Saks [15] and got its name in [7]. The prefix parity problem was solved in [15], but no nontrivial lower bounds for the majority or equality problems followfrom that. The results from =-=[6, 21, 22, 29] can be -=-seen to imply Ω(log log n/ log log log n) lower bounds using an entirely different technique based on Ajtai’s result [2]; and [19] reports Ω((log n/ log log n) 1/2 ) for equality and Ω(log n/(... |

49 |
A lower bound for finding predecessors in Yao’s cell probe model
- Ajtai
- 1988
(Show Context)
Citation Context ...rity or equality problems followfrom that. The results from [6, 21, 22, 29] can be seen to imply Ω(log log n/ log log log n) lower bounds using an entirely different technique based on Ajtai’s res=-=ult [2]; an-=-d [19] reports Ω((log n/ log log n) 1/2 ) for equality and Ω(log n/(log log n) 2 ) for the majority. 2. Nondeterminism in dynamic algorithms. 2.1. Example: Range queries. We can illustrate our con... |

49 | Marked ancestor problems
- Alstrup, Husfeldt, et al.
- 1998
(Show Context)
Citation Context ...r operation, bridging the gap between the two results. Apart from the bound for the existential range query problem, for which the authors recently proved a stronger bound using a different technique =-=[3], al-=-l these bounds are newand the best known. Related work. Fredman introduced the partial sum problem as a “toy problem which is both tractable and surprisingly interesting” [13], and it has been the... |

48 | Lower bounds for union-split-find related problems on random access machines
- Miltersen
- 1994
(Show Context)
Citation Context ... by Fredman and Saks [15] and got its name in [7]. The prefix parity problem was solved in [15], but no nontrivial lower bounds for the majority or equality problems followfrom that. The results from =-=[6, 21, 22, 29] can be -=-seen to imply Ω(log log n/ log log log n) lower bounds using an entirely different technique based on Ajtai’s result [2]; and [19] reports Ω((log n/ log log n) 1/2 ) for equality and Ω(log n/(... |

42 |
On the complexity of maintaining partial sums
- Yao
- 1985
(Show Context)
Citation Context ...e partial sum problem as a “toy problem which is both tractable and surprisingly interesting” [13], and it has been the focal point of many investigations of dynamic complexity in a variety of mod=-=els [15, 31]-=-. We reason within the cell-probe model of Fredman [12] and Yao [30] with some extensions to cope with our stronger modes of computation. The model can be viewed as a nonuniform version of the random ... |

41 |
The complexity of maintaining an array and computing its partial sums
- Fredman
- 1982
(Show Context)
Citation Context ...a different technique [3], all these bounds are newand the best known. Related work. Fredman introduced the partial sum problem as a “toy problem which is both tractable and surprisingly interesting=-=” [13]-=-, and it has been the focal point of many investigations of dynamic complexity in a variety of models [15, 31]. We reason within the cell-probe model of Fredman [12] and Yao [30] with some extensions ... |

36 |
Optimal algorithms for list indexing and subset rank
- Dietz
- 1989
(Show Context)
Citation Context ...nistic queries (defined and discussed in section 2), the partial sum problem requires time Ω(log n/ log log n) per operation. It is known that this is also the deterministic complexity of the proble=-=m [9, 15]-=-, so nondeterminism does not help. Theorem 3 studies the same problem in a promise setting, where the (deterministic) query algorithm receives an almost correct answer for free. The updates are as bef... |

32 |
Dynamic point location in general subdivisions
- Baumgarten, Jung, et al.
- 1994
(Show Context)
Citation Context ...t the existential problem for orthogonal range queries in the plane requires time Ω(log 1/2 n) per operation (Proposition 2). We also present bounds for planar point location in monotone subdivision=-=s [5, 26], re-=-achability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is... |

32 | Lower bounds for fully dynamic connectivity problems in graphs
- Fredman, Henzinger
- 1998
(Show Context)
Citation Context ...ability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is kn=-=own [10, 14, 17, 23]-=- that this is also a lower bound for reachability in grid graphs. However, grid graphs of constant width allow a reachability algorithm in time O(log log n) per operation [4], an exponential improveme... |

26 | Searching constant width mazes captures the AC0 hierarchy
- Barrington, Lu, et al.
- 1998
(Show Context)
Citation Context ... It is known [10, 14, 17, 23] that this is also a lower bound for reachability in grid graphs. However, grid graphs of constant width allow a reachability algorithm in time O(log log n) per operation =-=[4]-=-, an exponential improvement. We prove a lower bound that is parameterized by the width w of the graph: Proposition 10 states that dynamic reachability for grid graphs of width w = O(log n/ log log n)... |

23 | Fully dynamic point location in a monotone subdivision
- Preparata, Tamassia
- 1989
(Show Context)
Citation Context ...t the existential problem for orthogonal range queries in the plane requires time Ω(log 1/2 n) per operation (Proposition 2). We also present bounds for planar point location in monotone subdivision=-=s [5, 26], re-=-achability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is... |

19 |
Dynamic maintenance of planar digraphs, with applications. Algorithmica
- Tamassia, Preparata
- 1990
(Show Context)
Citation Context ...ueries in the plane requires time Ω(log 1/2 n) per operation (Proposition 2). We also present bounds for planar point location in monotone subdivisions [5, 26], reachability in upward planar digraph=-=s [28], an-=-d incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is known [10, 14, 17, 23] that this is also a l... |

13 |
Observations on the complexity of generating quasi-gray codes
- Fredman
- 1978
(Show Context)
Citation Context ...able and surprisingly interesting” [13], and it has been the focal point of many investigations of dynamic complexity in a variety of models [15, 31]. We reason within the cell-probe model of Fredma=-=n [12]-=- and Yao [30] with some extensions to cope with our stronger modes of computation. The model can be viewed as a nonuniform version of the random access computer with arbitrary register instructions. L... |

11 | Lower bounds for dynamic transitive closure, planar point location, and parentheses matching
- Husfeldt, Rauhe, et al.
- 1996
(Show Context)
Citation Context ...quality problems followfrom that. The results from [6, 21, 22, 29] can be seen to imply Ω(log log n/ log log log n) lower bounds using an entirely different technique based on Ajtai’s result [2]; =-=and [19] rep-=-orts Ω((log n/ log log n) 1/2 ) for equality and Ω(log n/(log log n) 2 ) for the majority. 2. Nondeterminism in dynamic algorithms. 2.1. Example: Range queries. We can illustrate our concept of no... |

11 |
Complexity models for incremental computation, Theoretical Computer Science 130:203-236
- Miltersen, Subramanian, et al.
- 1994
(Show Context)
Citation Context ...ability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is kn=-=own [10, 14, 17, 23]-=- that this is also a lower bound for reachability in grid graphs. However, grid graphs of constant width allow a reachability algorithm in time O(log log n) per operation [4], an exponential improveme... |

10 | Dynamic algorithms for the Dyck languages
- Frandsen, Husfeldt, et al.
- 1995
(Show Context)
Citation Context ...ration (Proposition 2). We also present bounds for planar point location in monotone subdivisions [5, 26], reachability in upward planar digraphs [28], and incremental parsing of balanced parentheses =-=[11]. We-=- showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is known [10, 14, 17, 23] that this is also a lower bound for reachability in grid graphs. However, g... |

9 |
Range searching. In Handbook of Discrete and Computational Geometry
- Agarwal
- 1997
(Show Context)
Citation Context ...og(btu)), which is optimal. However, before the present paper, no lower bound better than Ω(log log n/ log log log n) was known for this problem (which is rather central—see the discussion by Agar=-=wal [1]-=-), so the result provides an exponential yet, by now, outdated improvement. Following [3], we start with the existential marked ancestor problem. Consider a full rooted tree with nodes V ,numberofleav... |

8 | Hardness result for dynamic problems by extensions of Fredman and Saks chronogram method - Husfeldt, Rauhe |

7 |
Lower bounds for data structure problems on RAMs
- Ben-Amram, Galil
- 1991
(Show Context)
Citation Context ... become the model of choice for lower bounds for dynamic computation. Theorems 1 and 3 are proved by extending the chronogram method, which was introduced by Fredman and Saks [15] and got its name in =-=[7]. -=-The prefix parity problem was solved in [15], but no nontrivial lower bounds for the majority or equality problems followfrom that. The results from [6, 21, 22, 29] can be seen to imply Ω(log log n/... |

7 | The complexity of symmetric functions in boundeddepth circuits - Brustmann, Wegener - 1987 |

7 | Fully dynamic planarity testing in planar embedded graphs - Italiano, Poutré, et al. - 1993 |

7 | On-line planar graph embedding
- Tamassia
- 1996
(Show Context)
Citation Context ...ates insert and delete edges as above, and the query is as follows: planar(u, v): return “yes” iff the graph remains upward planar after insertion of edge (u, v). This problem was studied by Tamas=-=sia [27], -=-who found an O(log n) upper bound. Proposition 7. Upward planarity testing requires time Ω(log n/ log log n) per operation. A classical problem in computational geometry is planar point location: gi... |

5 | Dynamic connectivity in digital images
- Eppstein
- 1996
(Show Context)
Citation Context ...ability in upward planar digraphs [28], and incremental parsing of balanced parentheses [11]. We showthat these problems require time Ω(log n/ log log n) per operation (Propositions 5–8). It is kn=-=own [10, 14, 17, 23]-=- that this is also a lower bound for reachability in grid graphs. However, grid graphs of constant width allow a reachability algorithm in time O(log log n) per operation [4], an exponential improveme... |

4 | Generalized lower bounds derived from Hastad's main lemma - Moran - 1987 |

3 | Fully dynamic transitive closure in plane dags with one source and one sink
- Husfeldt
- 1995
(Show Context)
Citation Context |

2 |
New bounds in cell probe model. Doctoral dissertation
- Xiao
- 1992
(Show Context)
Citation Context |