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41
Optimal Bounds for the Predecessor Problem and Related Problems
 Journal of Computer and System Sciences
, 2001
"... We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved ..."
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Cited by 55 (0 self)
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We obtain matching upper and lower bounds for the amount of time to find the predecessor of a given element among the elements of a fixed compactly stored set. Our algorithms are for the unitcost word RAM with multiplication and are extended to give dynamic algorithms. The lower bounds are proved for a large class of problems, including both static and dynamic predecessor problems, in a much stronger communication game model, but they apply to the cell probe and RAM models.
Logarithmic lower bounds in the cellprobe model
 SIAM Journal on Computing
"... Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the ..."
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Cited by 34 (4 self)
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Abstract. We develop a new technique for proving cellprobe lower bounds on dynamic data structures. This enables us to prove Ω(lg n) bounds, breaking a longstanding barrier of Ω(lg n/lg lg n). We can also prove the first Ω(lgB n) lower bound in the external memory model, without assumptions on the data structure. We use our technique to prove better bounds for the partialsums problem, dynamic connectivity and (by reductions) other dynamic graph problems. Our proofs are surprisingly simple and clean. The bounds we obtain are often optimal, and lead to a nearly complete understanding of the problems. We also present new matching upper bounds for the partialsums problem. Key words. cellprobe complexity, lower bounds, data structures, dynamic graph problems, partialsums problem AMS subject classification. 68Q17
Cell probe complexity  a survey
 In 19th Conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 1999. Advances in Data Structures Workshop
"... 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 ..."
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Cited by 29 (0 self)
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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
An optimal dynamic spanner for doubling metric spaces
 In [KL04] [KR07] [Laa00] [Laa02] [Lee10] [LLKS85] [LP01] [Mit99] Proceedings of the 16th annual European symposium on Algorithms, ESA ’08
, 2008
"... For a set S of points in a metric space, a tspanner is a graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the actual distance between the points. In this paper, we consider points residing in a metric space of doublin ..."
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Cited by 16 (3 self)
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For a set S of points in a metric space, a tspanner is a graph on the points of S such that between any pair of points there is a path in the spanner whose total length is at most t times the actual distance between the points. In this paper, we consider points residing in a metric space of doubling dimension λ, and show how to construct a dynamic (1+ε)spanner with constant degree and O(log n) update time (when λ and ε are taken as constants). Our update time is optimal up to a constant.
UNIFYING THE LANDSCAPE OF CELLPROBE LOWER BOUNDS
, 2008
"... We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • co ..."
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Cited by 11 (0 self)
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We show that a large fraction of the datastructure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: • highdimensional problems, where the goal is to show large space lower bounds. • constantdimensional geometric problems, where the goal is to bound the query time for space O(n·polylogn). • dynamic problems, where we are looking for a tradeoff between query and update time. (In this case, our bounds are slightly weaker than the originals, losing a lglgn factor.) Our reductions also imply the following new results: • an Ω(lgn/lglgn) bound for 4dimensional range reporting, given space O(n · polylogn). This is quite timely, since a recent result [39] solved 3D reporting in O(lg 2 lgn) time, raising the prospect that higher dimensions could also be easy. • a tight space lower bound for the partial match problem, for constant query time. • the first lower bound for reachability oracles. In the process, we prove optimal randomized lower bounds for lopsided set disjointness.
Random Access to GrammarCompressed Strings
, 2011
"... Let S be a string of length N compressed into a contextfree grammar S of size n. We present two representations of S achieving O(log N) random access time, and either O(n · αk(n)) construction time and space on the pointer machine model, or O(n) construction time and space on the RAM. Here, αk(n) is ..."
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Cited by 9 (0 self)
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Let S be a string of length N compressed into a contextfree grammar S of size n. We present two representations of S achieving O(log N) random access time, and either O(n · αk(n)) construction time and space on the pointer machine model, or O(n) construction time and space on the RAM. Here, αk(n) is the inverse of the k th row of Ackermann’s function. Our representations also efficiently support decompression of any substring in S: we can decompress any substring of length m in the same complexity as a single random access query and additional O(m) time. Combining these results with fast algorithms for uncompressed approximate string matching leads to several efficient algorithms for approximate string matching on grammarcompressed strings without decompression. For instance, we can find all approximate occurrences of a pattern P with at most k errors in time O(n(min{P k, k 4 + P } + log N) + occ), where occ is the number of occurrences of P in S. Finally, we are able to generalize our results to navigation and other operations on grammarcompressed trees. All of the above bounds significantly improve the currently best known results. To achieve these bounds, we introduce several new techniques and data structures of independent interest, including a predecessor data structure, two ”biased” weighted ancestor data structures, and a compact representation of heavypaths in grammars.
New Lower Bound Techniques For Dynamic Partial Sums and Related Problems
 SIAM Journal on Computing
, 2003
"... We study the complexity of the dynamic partial sum problem in the cellprobe 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 kin ..."
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Cited by 8 (1 self)
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We study the complexity of the dynamic partial sum problem in the cellprobe 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. cellprobe model, partial sum, dynamic algorithm, data structure AMS subject classifications. 68Q17, 68Q10, 68Q05, 68P05
Fullydynamic orthogonal range reporting on RAM
, 2003
"... In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of p ..."
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Cited by 7 (2 self)
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In a natural variant of the comparison model, we show that there exists a constant ! < 1 such that the fullydynamic ddimensional orthogonal range reporting problem for d 2 can be solved in time O(log n) for updates and time O((log n= log log n) + r) for queries. Here n is the number of points stored and r is the number of points reported. The space usage is n). In the standard comparison model the result holds for d 3.
Stream sampling for varianceoptimal estimation of subset sums
 In ACMSIAM Symposium on Discrete Algorithms
, 2009
"... From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size k that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of online reservoir sampling, thinking of the generic sample as a reservoir. We present ..."
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Cited by 6 (5 self)
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From a high volume stream of weighted items, we want to maintain a generic sample of a certain limited size k that we can later use to estimate the total weight of arbitrary subsets. This is the classic context of online reservoir sampling, thinking of the generic sample as a reservoir. We present an efficient reservoir sampling scheme, VarOptk, that dominates all previous schemes in terms of estimation quality. VarOptk provides variance optimal unbiased estimation of subset sums. More precisely, if we have seen n items of the stream, then for any subset size m, our scheme based on k samples minimizes the average variance over all subsets of size m. In fact, the optimality is against any offline scheme with k samples tailored for the concrete set of items seen. In addition to optimal average variance, our scheme provides tighter worstcase bounds on the variance of particular subsets than previously possible. It is efficient, handling each new item of the stream in O(log k) time, which is optimal even on the word RAM. Finally, it is particularly well suited for combination of samples from different streams in a distributed setting. 1
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
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Cited by 6 (3 self)
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We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.