## Cell probe complexity - a survey

Venue: | In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop |

Citations: | 28 - 0 self |

### BibTeX

@INPROCEEDINGS{Miltersen_cellprobe,

author = {Peter Bro Miltersen},

title = {Cell probe complexity - a survey},

booktitle = {In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

The cell probe model is a general, combinatorial model of data structures. We give a survey of known results about the cell probe complexity of static and dynamic data structure problems, with an emphasis on techniques for proving lower bounds. 1

### Citations

759 | Approximate nearest neighbors: Towards removing the curse of dimensionality
- Indyk, Motwani
- 1998
(Show Context)
Citation Context ...nds are very good. The lack of good upper bounds for these problems is sometimes referred to as the curse of dimensionality. Recent, very interesting progress towards removing this curse were made in =-=[42, 43]-=-. There, very good worst case bounds are obtained for finding an approximate nearest neighbor if randomization is allowed by the query algorithm. Borodin, Ostrovsky and Rabani, show, using the greedy ... |

640 | N.: Communication complexity
- Kushilevitz, Nisan
- 1997
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Citation Context ...echnique to give a lower bound for a dynamic problem, without a constraint on the memory space. Actually, we will show it with the mild constraint s # 2 O(w) which can, in most cases, be removed (see =-=[41]-=-). The technique was first applied by Xiao [59]. Given a dynamic problem D, we simply define the following static data structure problem, for some parameter d: f : D Q # A, where Q is the set of query... |

267 | Probabilistic algorithms for Hamiltonian circuits and matchings - Angluin, Valiant - 1979 |

256 | Checking Computations in Polylogarithmic Time
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ... and so that any bit of x can be retrieved by looking only at a few bits of #(x). Thus, we are trying to construct a locally decodable source code, analogous to the locally decodable channel codes of =-=[9]-=-. Buhrman et al [16] show that the solution of Fredman, Komlos and Szemeredi is an optimal membership solution, also for w = 1 in the following sense: Any solution with w = 1, s = O(n log m) must have... |

217 |
Storing a sparse table with O(1) worst case access time
- Fredman, Komlós, et al.
- 1984
(Show Context)
Citation Context ...large compared to n i ). Still, even such a lower bound conveys useful information: It shows that a certain clean kind of upper bound does not exist. A seminal result of Fredman, Komlos and Szemeredi =-=[29]-=- gives the following optimal strongly transdichotomous upper bound on the complexity of membership and dictionary: Each have a solution with word size w = O(log m) using s = O(n) memory cells and with... |

201 |
Design and implementation of an efficient priority queue
- Boas, Kaas, et al.
- 1977
(Show Context)
Citation Context ...ons of the two problems behave very differently as we shall later see. A classical solution to predecessor/rank is balanced binary trees with w = O(log m), s = O(n), t = O(log n). Van Emde Boas et al =-=[57]-=- gave a solution to predecessor/rank with w = O(log m) with a time complexity of t = O(log log m). The space complexity is very high, but was later reduced to the strongly transdichotomously optimal v... |

196 | Efficient search for approximate nearest neighbor in high dimensional spaces
- Kushilevitz, Ostrovsky, et al.
(Show Context)
Citation Context ...nds are very good. The lack of good upper bounds for these problems is sometimes referred to as the curse of dimensionality. Recent, very interesting progress towards removing this curse were made in =-=[42, 43]-=-. There, very good worst case bounds are obtained for finding an approximate nearest neighbor if randomization is allowed by the query algorithm. Borodin, Ostrovsky and Rabani, show, using the greedy ... |

161 |
Trans-dichotomous algorithms for minimum spanning trees and shortest paths
- Fredman, Willard
(Show Context)
Citation Context ...the time complexity of this solution (i.e., the depth of its deepest tree) should be upper bounded by some function of n only. The term "strongly transdichotomous" was invented by Fredman an=-=d Willard [32, 33]-=-. We can motivate the strongly transdichotomous model as follows. We assume unit cost operations on the elements of the universe. In return, our algorithms should have a complexity where the number of... |

150 |
Surpassing the information theoretic bound with fusion trees
- Fredman, Willard
- 1993
(Show Context)
Citation Context ...the time complexity of this solution (i.e., the depth of its deepest tree) should be upper bounded by some function of n only. The term "strongly transdichotomous" was invented by Fredman an=-=d Willard [32, 33]-=-. We can motivate the strongly transdichotomous model as follows. We assume unit cost operations on the elements of the universe. In return, our algorithms should have a complexity where the number of... |

138 | Should tables be sorted
- Yao
- 1981
(Show Context)
Citation Context ...1.5 Bibliographical remarks The cell probe model originates in the 1968 book Perceptrons by Minsky and Papert [50]. In more modern times it was taken up by Fredman [27] (for dynamic problems) and Yao =-=[61]-=- (for static problems). The late 1990's have seen a revitalisation of the area with several FOCS and STOC papers dealing with the subject. The Were-you-last? game is from Frandsen, Miltersen and Skyum... |

130 |
The Design of Dynamic Data Structures
- Overmars
- 1983
(Show Context)
Citation Context ...e take the parameter n to be an upper bound that the size of S must always satisfy. Alternatively, we could require space and time to depend on the current size of S at any time. As shown by Overmars =-=[51]-=-, there is not much di#erence between these two conventions, and since the former is simpler, this is the one we adopt. All the results reported in this section is for word size w = O(log m). For dyna... |

129 | Polylogarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity
- Holm, Lichtenberg, et al.
(Show Context)
Citation Context ...time per operation of O(n 1/3 ), due to Henzinger and King [37]. If we allow the time bound for the operations to be amortized, there is an O(log 2 n) solution due to Holm, de Lichtenberg, and Thorup =-=[38]-=-, improved to O(log n log log n) in [56]. Fredman and Henzinger [30] and Miltersen et al [49] show a lower bound of#43 n/ log log n) by a reduction from a dynamic prefix problem (a class of problems d... |

128 |
The cell probe complexity of dynamic data structures
- Fredman, Saks
- 1989
(Show Context)
Citation Context ... will be explained in Section 4. Dynamic rank seems much harder than dynamic predecessor. Dietz [18] gives an upper bound of time O(log m/ log log m) per operation, using space O(m). Fredman and Saks =-=[31]-=-, show a matching lower bound: Any m-strongly transdichotomous solution, no matter what space is used, must use time #me m/ log log m) (and thus#11 n/ log log n). Thus, n-strongly transdichotomously, ... |

84 | On data structures and asymmetric communication complexity
- Miltersen, Nisan, et al.
- 1998
(Show Context)
Citation Context ... is via a communication complexity lower bound proved using Ajtai's technique of probability amplification in product spaces. In Section 4, we present a somewhat simpler proof, due to Miltersen et al =-=[48]-=-, of a somewhat weaker lower bound using this technique. Natural generalization of the search problems above to higher dimensions include problems usually studied in computational geometry, such as ra... |

69 | Membership in constant time and almost minimum space
- Brodnik, Munro
- 1999
(Show Context)
Citation Context ...nsdichotomous bounds, the space can be improved somewhat for membership. Clearly, the best we can hope for is a space use of log # m n # bits, i.e., b = log # m n # /w memory cells. Brodnik and Munro =-=[15]-=- and Pagh [52] show how to achieve s = b + o(b) and t = O(1). Word size w = O(log m) is not the only interesting value to study. Indeed, solutions to the membership problem for word size w = 1 was the... |

66 | Corrigendum (Maintenance of a minimum spanning forest in a dynamic plane graph
- Eppstein, Italiano, et al.
- 1993
(Show Context)
Citation Context ... case of dynamic undirected connectivity is the plane case where V has some fixed embedding in R 2 and we must insert edges under the constraint that no two edges cross. For this case, Eppstein et al =-=[23]-=- have a solution with a worst case time complexity of O(log n) per operation. The lower bound of#11 n/ log log n) is still valid, even for the case where V forms a grid and all edges must be grid edge... |

63 | Fully dynamic algorithms for maintaining all-pairs shortest paths and transitive closure in digraphs
- King
- 1999
(Show Context)
Citation Context ...ns a Boolean indicating whether there is a path form u to v in the graph. The upper bounds known for directed graph connectivity are much worse than the bounds known for undirected graph connectivity =-=[44]-=-. Interestingly, no better lower bounds are known for the directed case. An interesting special case of directed dynamic graph reachability is the case of upward planar source-sink graphs. There, the ... |

60 | Optimal bounds for the predecessor problem
- Beame, Fich
- 1999
(Show Context)
Citation Context ...function of m. Building on the work of Van Emde Boas and Willard, Andersson obtained a pure transdichotomous solution with s = O(n) and t = O( # log n). In a recent breakthrough paper, Beame and Fich =-=[12]-=- improved these bounds to the bound s = O(n), t = O(min(log log m/ log log log m, # log n/ log log n)) and proved the new bound transdichotomously optimal in the following sense: If s = n O(1) then an... |

52 | A non-linear time lower bound for Boolean branching programs
- Ajtai
- 1999
(Show Context)
Citation Context ...implies that there is a problem in P which cannot be solved by read-O(1)-times, polynomial size branching programs. Until recently, proving this was a well known open problem. However, recently Ajtai =-=[6]-=-, in a breakthrough paper, proved that there is indeed such a problem in P so the evidence is not quite as discouraging anymore (and also reminding us that the lower bound barriers of complexity theor... |

49 |
A lower bound for finding predecessors in Yao’s cell probe model
- Ajtai
- 1988
(Show Context)
Citation Context ...transdichotomous bound on t must have t =#156 log m/ log log log m) and any n-strongly transdichotomous bound on t must have t = # # log n/ log log n). The lower bounds improve a lower bound by Ajtai =-=[5]-=- and was obtained independently by Xiao [59]. The lower bound proof technique is via a communication complexity lower bound proved using Ajtai's technique of probability amplification in product space... |

49 | Marked ancestor problems
- Alstrup, Husfeldt, et al.
- 1998
(Show Context)
Citation Context ...thod, to be described in Section 4. Lower bounds for computational geometry search problem, such as dynamic range query and point location problems where shown by Husfeldt, Rauhe and collaborators in =-=[40, 39, 7]-=-, using very interesting refinements and extensions of the chronogram method which are not covered in this survey. In particular, a lower bound of#14 n/ log log n) for dynamic point location and the d... |

49 | Are bitvectors optimal
- Buhrman, Miltersen, et al.
(Show Context)
Citation Context ...eresting value to study. Indeed, solutions to the membership problem for word size w = 1 was the subject of the original work of Minsky and Papert and has recently been studied again by Buhrman et al =-=[16]-=-. The membership problem in the cell probe model with word size w = 1 has an interesting coding theoretic interpretation: We are looking for a representation #(S) of any n-subset S of {1, . . . , m} u... |

49 |
Near-optimal fully-dynamic graph connectivity
- Thorup
- 2000
(Show Context)
Citation Context ...Henzinger and King [37]. If we allow the time bound for the operations to be amortized, there is an O(log 2 n) solution due to Holm, de Lichtenberg, and Thorup [38], improved to O(log n log log n) in =-=[56]-=-. Fredman and Henzinger [30] and Miltersen et al [49] show a lower bound of#43 n/ log log n) by a reduction from a dynamic prefix problem (a class of problems described below). An interesting special ... |

48 | Lower bounds for union-split-find related problems on random access machines
- Miltersen
- 1994
(Show Context)
Citation Context ...heir chronogram method which we describe in Section 4. A classification of the complexity of dynamic word and prefix problems based on the algebraic properties of M was begun in [26] and continued in =-=[46, 40, 39, 12]-=-. Note that the Were-you-last? game, described in the introduction, is a restricted version of the dynamic word problem for the monoid ({0, 1}, #), where # is Boolean OR. 2.2.5 Dynamic algebraic probl... |

47 | Lower bounds for high dimensional nearest neighbor search and related problems - Borodin, Ostrovsky, et al. - 1999 |

46 |
Sorting and searching on the word RAM
- Hagerup
- 1998
(Show Context)
Citation Context ...lls and with each operation using a decision assignment tree of depth t = O(log n). For an excellent account of the transdichotomous model from an algorithmic point of view, see the survey of Hagerup =-=[35]-=-. A strongly transdichotomous lower bound of # f(n)) on the time for a search problem under some space constraint means that there is no strongly transdichotomous upper bound of o(f(n)). Thus, we have... |

41 |
The complexity of maintaining an array and computing its partial sums
- Fredman
- 1982
(Show Context)
Citation Context ...log n/ log log n) for w = O(log n). The dynamic prefix problem with M = Z/2Z was studied first, by Fredman in his seminal paper "The complexity of maintaining an array and computing its partial s=-=ums" [28]. He showe-=-d a lower bound of t =#867 n/ log log n) for w = 1 for this problem, using the "which-side" technique we describe in Section 3. The same lower bound for this problem was shown to be valid ev... |

39 | Faster deterministic sorting and searching in linear space - Andersson - 1996 |

36 |
Optimal algorithms for list indexing and subset rank
- Dietz
- 1989
(Show Context)
Citation Context ...dynamic case without the space constraint. This is a general reduction first observed by Xiao [59] which will be explained in Section 4. Dynamic rank seems much harder than dynamic predecessor. Dietz =-=[18]-=- gives an upper bound of time O(log m/ log log m) per operation, using space O(m). Fredman and Saks [31], show a matching lower bound: Any m-strongly transdichotomous solution, no matter what space is... |

35 | Sublogarithmic searching without multiplications - Andersson - 1995 |

32 | Lower bounds for fully dynamic connectivity problems in graphs
- Fredman, Henzinger
- 1998
(Show Context)
Citation Context ...we allow the time bound for the operations to be amortized, there is an O(log 2 n) solution due to Holm, de Lichtenberg, and Thorup [38], improved to O(log n log log n) in [56]. Fredman and Henzinger =-=[30]-=- and Miltersen et al [49] show a lower bound of#43 n/ log log n) by a reduction from a dynamic prefix problem (a class of problems described below). An interesting special case of dynamic undirected c... |

30 | Data Structures for Travelling Salesmen
- Fredman, Johnson, et al.
- 1995
(Show Context)
Citation Context ...nds an ancestor of v with a switch that is on (if one exists). Tight upper and lower bounds on the marked ancestor problem was found by [7], the lower bound using the chronogram method. Fredman et al =-=[34]-=- show lower bounds using the chronogram method for certain dynamic graph problems encountered in the implementation of heuristics for the travelling salesman problem. 2.2.3 Union-Find Union-Find is th... |

27 | Maintaining minimum spanning trees in dynamic graphs
- Henzinger, King
- 1997
(Show Context)
Citation Context ...ally use and allow in dynamic graph algorithms. The best worst case solution to the dynamic undirected connectivity problem has a worst case time per operation of O(n 1/3 ), due to Henzinger and King =-=[37]-=-. If we allow the time bound for the operations to be amortized, there is an O(log 2 n) solution due to Holm, de Lichtenberg, and Thorup [38], improved to O(log n log log n) in [56]. Fredman and Henzi... |

23 | Low redundancy in static dictionaries with O(1) lookup time
- Pagh
- 1999
(Show Context)
Citation Context ...bounds, the space can be improved somewhat for membership. Clearly, the best we can hope for is a space use of log # m n # bits, i.e., b = log # m n # /w memory cells. Brodnik and Munro [15] and Pagh =-=[52]-=- show how to achieve s = b + o(b) and t = O(1). Word size w = O(log m) is not the only interesting value to study. Indeed, solutions to the membership problem for word size w = 1 was the subject of th... |

19 | A lower bound on the complexity of approximate nearest-neighbor searching on the hamming cube
- Chakrabarti, Chazelle, et al.
- 1999
(Show Context)
Citation Context ...essive result was obtained recently by Barkol and Rabani [10]: For the nearest neighbor problem, if s = n O(1) , then any n-strongly transdichotomous bound on t must have t = n#94 . Chakrabarti et al =-=[17]-=- show a lower bound for approximating the nearest neighbor, using the amplification version of the communication complexity technique. Interestingly, this lower bound matches the upper bound shown for... |

19 |
Dynamic maintenance of planar digraphs, with applications. Algorithmica
- Tamassia, Preparata
- 1990
(Show Context)
Citation Context ... case. An interesting special case of directed dynamic graph reachability is the case of upward planar source-sink graphs. There, the dynamic reachability can be solved in time O(log n) per operation =-=[55]-=-. Husfeldt, Rauhe and Skyum [40, 39] show a lower bound of#25 n/ log log n) per operation, using the chronogram method. Dynamic planarity testing is the problem of maintaining a plane graph with a fix... |

18 |
Logarithmic worst case range queries are possible
- Willard
- 1983
(Show Context)
Citation Context ...cessor/rank with w = O(log m) with a time complexity of t = O(log log m). The space complexity is very high, but was later reduced to the strongly transdichotomously optimal value s = O(n) by Willard =-=[58]. Note tha-=-t we here have a "mixed transdichotomous " solution with s being a function of n and t being a function of m. Building on the work of Van Emde Boas and Willard, Andersson obtained a pure tra... |

17 | Dynamic word problems
- Frandsen, Miltersen, et al.
- 1997
(Show Context)
Citation Context ...(for static problems). The late 1990's have seen a revitalisation of the area with several FOCS and STOC papers dealing with the subject. The Were-you-last? game is from Frandsen, Miltersen and Skyum =-=[26]-=- where a tighter analysis of the game can be found, improving the constant factors somewhat. 2 Problems, problems, problems... In this section we give a taxonomic overview of natural problems appropri... |

17 |
Ecient search for approximate nearest neighbor in high dimensional spaces
- Kushilevitz, Ostrovsky, et al.
- 2000
(Show Context)
Citation Context ...nds are very good. The lack of good upper bounds for these problems is sometimes referred to as the curse of dimensionality. Recent, very interesting progress towards removing this curse were made in =-=[42, 43]-=-. There, very good worst case bounds are obtained for finding an approximate nearest neighbor if randomization is allowed by the query algorithm. Borodin, Ostrovsky and Rabani, show, using the greedy ... |

17 |
Design and implementation of an ecient priority queue
- Boas, Kaas, et al.
- 1977
(Show Context)
Citation Context ...ions of the two problems behave very di#erently as we shall later see. A classical solution to predecessor/rank is balanced binary trees with w = O(log m), s = O(n), t = O(log n). Van Emde Boas et al =-=[57]-=- gave a solution to predecessor/rank with w = O(log m) with a time complexity of t = O(log log m). The space complexity is very high, but was later reduced to the strongly transdichotomously optimal v... |

17 |
New Bounds in Cell Probe Model
- Xiao
- 1992
(Show Context)
Citation Context ...56 log m/ log log log m) and any n-strongly transdichotomous bound on t must have t = # # log n/ log log n). The lower bounds improve a lower bound by Ajtai [5] and was obtained independently by Xiao =-=[59]-=-. The lower bound proof technique is via a communication complexity lower bound proved using Ajtai's technique of probability amplification in product spaces. In Section 4, we present a somewhat simpl... |

16 | Optimal bi-weighted binary trees and the complexity of maintaining partial sums - Hampapuram, Fredman - 1993 |

16 | On the cell probe complexity of polynomial evaluation
- Miltersen
- 1995
(Show Context)
Citation Context ...e or less inexhaustible supply of other natural dynamic algebraic problems to look at, and for most of them, not nearly as tight bounds are known. We refer the reader to Reif and Tate [53], Miltersen =-=[47]-=- and Frandsen, Hansen, and Miltersen [24] for more examples. For this survey, we just note that one additional example of a dynamic algebraic problem was already discussed under the heading dynamic wo... |

15 | Dictionary look-up with one error - Yao, Yao - 1997 |

13 |
Tighter bounds for nearest neighbor search and related problems in the cell probe model
- BARKOL, RABANI
(Show Context)
Citation Context ...trongly transdichotomous bound on t must have t =#264 n) for both the nearest neighbor problem and the partial match problem. An even more impressive result was obtained recently by Barkol and Rabani =-=[10]-=-: For the nearest neighbor problem, if s = n O(1) , then any n-strongly transdichotomous bound on t must have t = n#94 . Chakrabarti et al [17] show a lower bound for approximating the nearest neighbo... |

13 |
Observations on the complexity of generating quasi-gray codes
- Fredman
- 1978
(Show Context)
Citation Context ... research in cell probe complexity. 1.5 Bibliographical remarks The cell probe model originates in the 1968 book Perceptrons by Minsky and Papert [50]. In more modern times it was taken up by Fredman =-=[27]-=- (for dynamic problems) and Yao [61] (for static problems). The late 1990's have seen a revitalisation of the area with several FOCS and STOC papers dealing with the subject. The Were-you-last? game i... |

12 |
The complexity of some simple retrieval problems
- Elias, Flower
- 1975
(Show Context)
Citation Context ... of the Long Problem. Thus, upper bounds obtained for the Long Problem are valid for any other problem as well. The cell probe complexity of the Long Problem for w = 1 was studied by Elias and Flower =-=[21]-=- and Miltersen [45] and the following results were obtained. Theorem 2 Let s # n = #log D#. There is a solution to the Long Problem on domain D using s bits and with time complexity t # n - #log log(s... |

11 | Lower bounds for dynamic transitive closure, planar point location, and parentheses matching
- Husfeldt, Rauhe, et al.
- 1996
(Show Context)
Citation Context ... lower bound technique, due to Fredman and Saks [31]. The technique has been refined considerably beyond what we show here, most notably in the work of Rauhe and collaborators. We refer the reader to =-=[31, 13, 40, 39, 7, 8]-=- for details. Recall that the dynamic rank problem is the problem of maintaining a subset S of U = {1, . . . , m} under Insert, Delete, and Rank, where Rank(x) returns the number of elements of S smal... |

11 | Complexity models for incremental computation, Theoretical Computer Science 130:203-236 - Miltersen, Subramanian, et al. - 1994 |

10 | Worst-case and amortised optimality in union-find - Alstrup, Ben-Amram, et al. - 1999 |