The cell probe complexity of dynamic range counting

Larsen, Kasper Green

doi:10.1145/2213977.2213987

Cited by 48 publications

(28 citation statements)

References 15 publications

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“…Very recently, Larsen [11] showed how to combine the cell sampling approach of Panigrahy et al [16] with the chronogram technique of Fredman and Saks [5]. This combination essentially allows one to argue that when answering a query, one has to probe Ω(lg n/ lg(wt u )) cells from each epoch instead of Ω(1), yielding lower bounds of t q = Ω((lg n/ lg(wt u ))…”

Section: B Previous Resultsmentioning

confidence: 99%

“…Secondly, we apply the recent technique of Larsen [11] to obtain a lower bound of t q = Ω(lg |F| lg n/ lg(wt u / lg |F|) lg(wt u )) for the dynamic polynomial evaluation problem over a field of size at least Ω(n 2 ). Here t u is the worst case update time and t q is the query cost of any randomized data structure with a constant error probability δ < 1/2.…”

Section: Our Resultsmentioning

confidence: 99%

“…Here d denotes the maximum of the number of different queries and different updates to the data structure problem. Furthermore, Larsen's lower bound [11] for weighted range counting peaks at max{t u , t q } = Ω((lg n/ lg lg n)…”

Section: Our Resultsmentioning

confidence: 99%

“…However, in the dynamic setting, we need several additional properties to obtain a contradiction. As demonstrated by Larsen [11], we can reach a contradiction by encoding a subset of queries for which to simulate the query algorithm and obtain the corresponding answers. Since the answer to an evaluation query reveals at most lg |F| bits, encoding these queries must take less than lg |F| bits per query to obtain a contradiction.…”

Section: A Bounding the Probes To Epoch I (Proof Of Lemma 2)mentioning

confidence: 99%

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Higher Cell Probe Lower Bounds for Evaluating Polynomials

Larsen

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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Abstract-In this paper, we study the cell probe complexity of evaluating an n-degree polynomial P over a finite field F of size at least n 1+Ω(1) . More specifically, we show that any static data structure for evaluating P (x), where x ∈ F, must use Ω(lg |F|/ lg(Sw/n lg |F|)) cell probes to answer a query, where S denotes the space of the data structure in number of cells and w the cell size in bits. This bound holds in expectation for randomized data structures with any constant error probability δ < 1/2. Our lower bound not only improves over the Ω(lg |F|/ lg S) lower bound of Miltersen [TCS'95], but is in fact the highest static cell probe lower bound to date: For linear space (i.e. S = O(n lg |F|/w)), our query time lower bound simplifies to Ω(lg |F|), whereas the highest previous lower bound for any static data structure problem having d different queries is Ω(lg d/ lg lg d), which was first achieved by Pǎtraşcu and Thorup [SICOMP'10].We also use the recent technique of Larsen [STOC'12] to show a lower bound of tq = Ω(lg |F| lg n/ lg(wtu/ lg |F|) lg(wtu)) for dynamic data structures for polynomial evaluation over a finite field F of size Ω(n 2 ). Here tq denotes the expected query time and tu the worst case update time. This lower bound holds for randomized data structures with any constant error probability δ < 1/2. This is only the second time a lower bound beyond max{tu, tq} = Ω(max{lg n, lg d/ lg lg d}) has been achieved for dynamic data structures, where d denotes the number of different queries and updates to the problem. Furthermore, it is the first such lower bound that holds for randomized data structures with a constant probability of error.

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Section: B Previous Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: A Bounding the Probes To Epoch I (Proof Of Lemma 2)mentioning

confidence: 99%

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Higher Cell Probe Lower Bounds for Evaluating Polynomials

Larsen

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…In data structures, the largest proven lower bounds are polylogarithmic [8], and a major challenge is to prove polynomial lower bounds like n Ω(1) . Mihai proposed a very interesting line of attack via his so-called multiphase problem in [13].…”

Section: Problem 2: Multiphase Problemmentioning

confidence: 99%

Mihai Pǎtraşcu

Thorup

2013

SIGACT News

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Mihai's talent showed early. In high school he received numerous medals at national (Romanian) and international olympiads including prizes in informatics, physics and applied math. He received gold medals at the International Olympiad in Informatics (IOI) in both 2000 and 2001. He remained involved with olympiads and was elected member of the International Scientific Committee for the International Olympiad of Informatics since 2010.Mihai's main research area was data structure lower bounds. In data structures we try to understand how we can efficiently represent, access, and update information. Mihai revolutionized and revitalized the lower bound side, in many cases matching known upper bounds. The lower bounds were proved in the powerful cell-probe model that only charges for memory access, hence which captures both RAM and external memory. Already in 2004 [17], as a second year undergraduate student, with his supervisor Erik Demaine as non-alphabetic second author, he broke the Ω(log n/ log log n) lower bound barrier that had impeded dynamic lower bounds since 1989 [6], and showed the first logarithmic lower bound by an elegant short proof, a true combinatorial gem. The important conclusion was that binary search trees are optimal algorithms for the textbook problem of maintaining prefix sums in a dynamic array. They also proved an Ω(log n) lower bound for dynamic trees, matching Sleator and Tarjan's upper bound from 1983 [20]. In 2005 he received from the Computing Research Association (CRA) the Outstanding Undergraduate Award for best undergraduate research in the US and Canada.I was myself lucky enough to meet Mihai in 2004, starting one of most intense collaborations I have experienced in my career. It took us almost two years to find the first separation between near-linear and polynomial space in data structures [19]. What kept us going on this hard problem was that we always had lots of fun on the side: playing squash, going on long hikes, and having beers celebrating every potentially useful idea we found on the way. A strong friendship was formed.Mihai published more than 10 papers while pursuing his undergraduate studies at MIT from 2002 to 2006. Nevertheless he finished with a perfect 5.0/5.0 GPA. Over the next 2 years, he did his PhD at MIT. His thesis "Lower Bound Techniques for Data Structures" [11] is a must-read for researchers who want to get into data structure lower bounds.During Mihai's PhD, I got to be his mentor at AT&T, and in 2009, after a year as Raviv Postdoctoral Fellow at IBM Almaden, he joined me at AT&T. We continued our work on lower bounds, but I also managed to get him interested in hashing which is of immense importance to real computing. We sought schemes that were both truly practical and theoretically powerful [15].

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Lower Bounds for Differentially Private RAMs

Persiano

Yeo

2019

Lecture Notes in Computer Science

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The cell probe complexity of dynamic range counting

Cited by 48 publications

References 15 publications

Higher Cell Probe Lower Bounds for Evaluating Polynomials

Higher Cell Probe Lower Bounds for Evaluating Polynomials

Mihai Pǎtraşcu

Lower Bounds for Differentially Private RAMs

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