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Optimal parallel selection has complexity O(Log Log N)

Miklós Ajtai, János Komlós, W.L. Steiger, Endre Szemerédi
1989 Journal of computer and system sciences (Print)  
We show that in the deterministic comparison model for parallel computation, p=n processors can select the kth smallest item from a set of n numbers in O(log log n) parallel time.  ...  This optimal time bound holds even if p = o(n).  ...  ACKNOWLEDGMENT We thank Jeff Salowe for help in preparing the paper.  ... 
doi:10.1016/0022-0000(89)90035-4 fatcat:z7kk67veffhm7fttm2gxwuwwvq

MST construction in O(log log n) communication rounds

Zvi Lotker, Elan Pavlov, Boaz Patt-Shamir, David Peleg
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03  
This result is the first to break the Ω(log n) parallel time complexity barrier with small message sizes.  ...  For this model, we present a distributed algorithm that constructs a minimumweight spanning tree in O(log log n) communication rounds, where in each round any process can send a message to each other process  ...  This can be done in O(log n) time. To reduce the time complexity to O(log log n), it is necessary to speed up the process by making the cluster sizes grow quadratically in each phase.  ... 
doi:10.1145/777412.777428 dblp:conf/spaa/LotkerPPP03 fatcat:j3krvcrcb5ca3glgpmwadwlk7q

MST construction in O(log log n) communication rounds

Zvi Lotker, Elan Pavlov, Boaz Patt-Shamir, David Peleg
2003 Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures - SPAA '03  
This result is the first to break the Ω(log n) parallel time complexity barrier with small message sizes.  ...  For this model, we present a distributed algorithm that constructs a minimumweight spanning tree in O(log log n) communication rounds, where in each round any process can send a message to each other process  ...  This can be done in O(log n) time. To reduce the time complexity to O(log log n), it is necessary to speed up the process by making the cluster sizes grow quadratically in each phase.  ... 
doi:10.1145/777426.777428 fatcat:4cdmg4apbzghxn7niolz5ovhxe

Mutual Exclusion with O(log^2 Log n) Amortized Work

Michael A. Bender, Seth Gilbert
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
This paper presents a new algorithm for mutual exclusion in which each passage through the critical section costs amortized O(log 2 log n) RMRs with high probability.  ...  The algorithm operates in a standard asynchronous, local spinning, sharedmemory model with an oblivious adversary. It guarantees that every process enters the critical section with high probability.  ...  This research was supported in part by NSF Grants CCF 0937822, CCF 1114809, CCF 0634793, and CCF 0540897, DOE Grant DE-FG02-08ER25853, and NUS FRC R-252-000-443-133.  ... 
doi:10.1109/focs.2011.84 dblp:conf/focs/BenderG11 fatcat:loqdendnlzcktc3nbbnadixyhi

A deterministic poly(log log N)-time N-processor algorithm for linear programming in fixed dimension

Miklos Ajtai, Nimrod Megiddo
1992 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing - STOC '92  
The parallel time bound (counting only the arithmetic operations) is O((log log n) d ) where d is the number of variables. In the one-dimensional case this bound is optimal.  ...  It is shown that for any x e d n umb e r o f v ariables, the linear programming problems with n linear inequalities can be solved deterministically by n parallel processors in sub-logarithmic time.  ...  Deng 5] gave a parallel algorithm which runs in O(log n) time, using O(n= log n) processors.  ... 
doi:10.1145/129712.129744 dblp:conf/stoc/AjtaiM92 fatcat:spcyu4clq5gd3fpf4obmbhc6py

Minimum-Weight Spanning Tree Construction in O(log log n) Communication Rounds

Zvi Lotker, Boaz Patt-Shamir, Elan Pavlov, David Peleg
2005 SIAM journal on computing (Print)  
For this model, we present a distributed algorithm which constructs a minimum-weight spanning tree in O(log log n) communication rounds, where in each round any process can send a message to every other  ...  If message size is Θ(n ) for some > 0, then the number of communication rounds is O(log 1 ).  ...  This can be done in O(log n) time. To reduce the time complexity to O(log log n), it is necessary to speed up the process by making the cluster sizes grow quadratically in each phase.  ... 
doi:10.1137/s0097539704441848 fatcat:e5nknu7onnhqlenfmyqz33mvmm

An n log n Algorithm for Online BDD Refinement

Nils Klarlund
1995 BRICS Report Series  
<br />We apply our algorithm to show that automata with exponentially<br />large, but implicitly represented alphabets, can be minimized in time<br />O(n log n), where n is the total number of BDD nodes  ...  <br />In this paper, we consider a natural online BDD refinement problem<br />and show that it can be solved in O(n log n) if n bounds the size of the<br />BDD and the total size of update operations.  ...  Thus the total time is O(n log n+k).  ... 
doi:10.7146/brics.v2i29.19931 fatcat:vuy4lqvhjjak5avelvpms4jaau

Breaking the log n barrier on rumor spreading [article]

Chen Avin, Robert Elsässer
2015 arXiv   pre-print
This algorithm can also cope with F= O(n/2^√( n)) node failures, in which case all but O(F) nodes become informed within O(√( n)) rounds, w.h.p.  ...  O( n) rounds has been a well known upper bound for rumor spreading using push&pull in the random phone call model (i.e., uniform gossip in the complete graph).  ...  in a spreading time of O( log n log log n ).  ... 
arXiv:1512.03022v1 fatcat:7qrg3fahz5avvam7i42s2wruo4

Population protocols for leader election and exact majority with O(log^2 n) states and O(log^2 n) convergence time [article]

Andreas Bilke, Colin Cooper, Robert Elsaesser, Tomasz Radzik
2017 arXiv   pre-print
[SODA 2017] showed O(^2 n)-state protocols for both problems, with the exact majority protocol converging in time O(^3 n), and the leader election protocol converging in time O(^6.3 n) w.h.p. and O(^5.3  ...  We present a protocol which elects the leader in O(^2 n) time w.h.p. and in expectation and uses Θ(^2 n) states per agent.  ...  W.h.p. in O(log n) (parallel) time all nodes have the message.  ... 
arXiv:1705.01146v1 fatcat:ynf7cxqjxbchdhu53bjp73dnb4

Sorting Short Keys in Circuits of Size o(n log n) [article]

Gilad Asharov, Wei-Kai Lin, Elaine Shi
2020 arXiv   pre-print
in such cases.  ...  Therefore, if the keys to be sorted are short, say, k < o(log n), our result is asymptotically better than the classical AKS sorting network (ignoring log^* terms); and we also overcome the n log n barrier  ...  This work is in part supported by an NSF CAREER Award under the award number CNS-1601879, a Packard Fellowship, an ONR YIP award, and a DARPA Brandeis award.  ... 
arXiv:2010.09884v2 fatcat:ji2762hgevbpxijglqczbp5sxu

Representing hard lattices with O(n log n) bits

Miklós Ajtai
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
We present a variant of the Ajtai-Dwork public-key cryptosystem where the size of the public-key is only O(n log n) bits and the encrypted text/clear text ratio is also O(n log n).  ...  This is true with the assumption that all of the participants in the cryptosystem share O(n 2 log n) random bits which have to be picked only once and the users of the cryptosystem get them e.g. together  ...  Moreover in the new system the public key consists of only O(n log n) bits and we encrypt a single bit by O(n log n) bits.  ... 
doi:10.1145/1060590.1060604 dblp:conf/stoc/Ajtai05 fatcat:dbp4q3d7u5gh5niiihzvmxhm4a

Efficient Range ORAM with 핆(log 2 N) Locality [article]

Anrin Chakraborti, Adam J. Aviv, Seung Geol Choi, Travis Mayberry, Daniel S. Roche, Radu Sion
2018 IACR Cryptology ePrint Archive  
log N ).  ...  can also speed up standard ORAM constructions, e.g., resulting in a 2x faster Path ORAM variant.  ...  number of parallel seeks per access is O log N • 1 + log N k .  ... 
dblp:journals/iacr/ChakrabortiACMR18 fatcat:es7anaoqxzhnbfhg4hiqwyffjy

Parallel Minimum Cuts in O(m log^2(n)) Work and Low Depth [article]

Daniel Anderson, Guy E. Blelloch
2021 arXiv   pre-print
We present an O(m log^2(n)) work, O(polylog(n)) depth parallel algorithm for minimum cut.  ...  and Gianinazzi [SPAA'18, (2018), pp. 1-11] which performs O(m log^4(n)) work in O(polylog(n)) depth.  ...  Acknowledgments This work was supported in part by NSF grants CCF-1408940 and CCF-1629444.  ... 
arXiv:2102.05301v1 fatcat:5wbcjmxcbfgy5hn6lnah6fxylm

Perfect Information Leader Election in log*n+O(1) Rounds

Alexander Russell, David Zuckerman
2001 Journal of computer and system sciences (Print)  
Our protocol takes log n + O(1) rounds, each player sending at most log n bits per round.  ...  k times.  ...  The time to construct such a function deterministically is n O(n) . Proof.  ... 
doi:10.1006/jcss.2001.1776 fatcat:p6a6mqhivrcxbimyz25eeje6c4

NanoGRAM: Garbled RAM with $\widetilde{O}(\log N)$ Overhead [article]

Andrew Park, Wei-Kai Lin, Elaine Shi
2022 IACR Cryptology ePrint Archive  
For sufficiently large blocks where W = Ω(log 2 N ), our scheme achieves O(λ • W log N ) cost per memory access, where the dependence on N is optimal (barring poly log log factors), in terms of the evaluator's  ...  N is the total number of blocks, and O(•) hides poly log log factors.  ...  Acknowledgments This work is in part supported by a DARPA SIEVE grant, a Packard Fellowship, NSF awards under the grant numbers 2128519 and 2044679, and a grant from ONR.  ... 
dblp:journals/iacr/ParkLS22 fatcat:nm7oxobdjzhepjp3bnifv34lv4
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