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Complexity of Boolean functions on PRAMs - Lower bound techniques
[chapter]

1992
*
Lecture Notes in Computer Science
*

Determining time necessary for computing important

doi:10.1007/3-540-55488-2_34
fatcat:xryrodty2bar3faapp6jfribry
*functions**on*parallel machines is*one**of*the most important problems in*complexity*theory for parallel algorithms. ... In this survey paper, we discuss the results that have been obtained for three types*of*parallel random access machines (*PRAMs*): CREW, ROBUST and EREW. † throughout the paper, log stands for log to the ... The overall strategy*of*determining*lower**bounds*for*Boolean**functions**on**PRAMs*is the following: First a*complexity*measure m for*Boolean**functions*is defined. ...##
###
Exact lower time bounds for computing Boolean functions on CREW PRAMs

1994
*
Journal of computer and system sciences (Print)
*

The time

doi:10.1016/s0022-0000(05)80003-0
fatcat:5yp6wsy5uvfqdppw4iwwrkt26u
*complexity**of**Boolean**functions**on*abstract concurrent-read exclusive-write parallel random access machines (CREW*PRAMs*) is considered. ...*lower**bound*valid for all*Boolean**functions*in the*bounded*error model. ... ACKNOWLEDGMENTS The first author thanks Friedhelm Meyer auf der Heide for many discussions in the course*of*the development*of*the*lower**bound*argument; thanks to Ilan Newman for pointing out Szegedy's ...##
###
CREW PRAMS and decision trees

1989
*
Proceedings of the twenty-first annual ACM symposium on Theory of computing - STOC '89
*

This paper gives a full characterization

doi:10.1145/73007.73038
dblp:conf/stoc/Nisan89
fatcat:uzzmpa3ldbcdvl4mcj4l75z34q
*of*the time needed to compute a*boolean**function**on*a CREW*PRAM*with an unlimited number*of*processors. ... The characterization is given in terms*of*a new*complexity*measure*of**boolean**functions*: the "block sensitivity". ... Actually, [CDR] proved a more general result as they give a*lower**bound**on*the time needed to compute a*function**on*a CREW*PRAM*in terms*of*the function's "sensitivity". ...##
###
CREW PRAMs and Decision Trees

1991
*
SIAM journal on computing (Print)
*

This paper gives a full characterization

doi:10.1137/0220062
fatcat:umsq3apnvvcurhas76wu2wee4m
*of*the time needed to compute a*boolean**function**on*a CREW*PRAM*with an unlimited number*of*processors. ... The characterization is given in terms*of*a new*complexity*measure*of**boolean**functions*: the "block sensitivity". ... Actually, [CDR] proved a more general result as they give a*lower**bound**on*the time needed to compute a*function**on*a CREW*PRAM*in terms*of*the function's "sensitivity". ...##
###
Page 5544 of Mathematical Reviews Vol. , Issue 95i
[page]

1995
*
Mathematical Reviews
*

We further show that the (

*bounded*error) probabilistic time*complexity**of**Boolean**functions**on*CREW*PRAMs*differs at most by a constant factor from the deterministic time*complex*- ity. ... .], prove a general*lower**bound*valid for all*Boolean**functions*in the*bounded*er- ror model. ...##
###
Page 5448 of Mathematical Reviews Vol. , Issue 88j
[page]

1988
*
Mathematical Reviews
*

n))-

*lower**bounds*for Andreev’s*function*, which is now the largest*lower**bound*for the monotone circuit*complexity**of**functions*in NP. ... Razborov proved the first superpolynomial*lower**bounds**on*the monotone circuit*complexity**of**functions*in NP. Afterwards An- dreev proved even exponential*lower**bounds*for other*functions*in NP. ...##
###
Page 5373 of Mathematical Reviews Vol. , Issue 94i
[page]

1994
*
Mathematical Reviews
*

(ii) Some concrete

*Boolean**functions*cannot be approximated by polynomials with a low degree. This*lower**bound**technique*was extended by the author [“A note*on*a theorem*of*Razborov”, COINS Tech. ... The idea*of*his paper was to show a superpolynomial*lower**bound**on*the above-mentioned*Boolean*circuits with AND, OR, and XOR gates by proving the following two facts: (i) Each*Boolean**function*computable ...##
###
Limits on the Power of Parallel Random Access Machines with Weak Forms of Write Conflict Resolution

1996
*
Journal of computer and system sciences (Print)
*

We prove a

doi:10.1006/jcss.1996.0052
fatcat:quyrmsi5fbbjjhjpyyatmtyw4a
*lower**bound**on*the time required by the ROBUST*PRAM*to compute*Boolean**functions*in terms*of*the number*of*different values each memory cell*of*the*PRAM*can contain and the degree*of*the*function*... We extend our result to obtain a*lower**bound*, depending*on*the number*of*processors, for computing*Boolean**functions**on*the ROBUST*PRAM*, even with memory cells*of*unbounded size. A article no. 0052 ... This work was supported by the Natural Sciences and Engineering Research Council*of*Canada, the Information Technology Research Centre*of*Ontario, the Defense Advanced Research Projects Agency*of*the United ...##
###
New lower bounds for parallel computation

1989
*
Journal of the ACM
*

*Lower*

*bounds*are proven

*on*the parallel-time

*complexity*

*of*several basic

*functions*

*on*the most powerful concurrent-read concurrent-write

*PRAM*with unlimited shared memory and unlimited power

*of*individual ... The simulation

*technique*is

*of*independent interest since it can serve as a general tool to translate circuit

*lower*

*bounds*into

*PRAM*

*lower*

*bounds*. ... Q(log n)

*Lower*

*Bound*

*on*the Bit

*Complexity*

*of*ADDITION and Related

*Functions*Recently, Fich et al [9] and Meyer auf der Heide and Wigderson [22] proved several important

*lower*

*bounds*

*on*PRIORITY(w) ...

##
###
On separating the EREW and CREW PRAM models

1989
*
Theoretical Computer Science
*

This problem defines a partial

doi:10.1016/0304-3975(89)90169-2
fatcat:gwzkjumgnjfeliogznkgiddde4
*function*, that is,*one*that is defined only*on*a restricted set*of*inputs. ... Here we solve this problem by generalizing the Selection Problem to a derision tree problem which is defined*on*a full domain and to which Snir's*lower**bound*applies. ... It is easy to see that the*complexity**of*problem (S)*on*a CREW*PRAM*is O(1), and Snir [6] proved a O(x/~g n) upper ar_d ',,r~er*bound**on*solving the problem in the EREW model. ...##
###
Optimal bounds for decision problems on the CRCW PRAM

1987
*
Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
*

We prove optimal R(log n/log log n)

doi:10.1145/28395.28405
dblp:conf/stoc/BeameH87
fatcat:kepq6msierfdtixy4vvshmnzlm
*lower**bounds**on*the time for CRCW PRAM's with polynomially*bounded*numbers*of*processors or memory cells to compute parity and a number*of*related problems. ... We also exhibit a strict time hierarchy*of*explicit*Boolean**functions**of*n bits*on*such machines which holds up to O(log n/ log log n) time. ... We also prove a tight O(logn)*lower**bound**on*the time to compute almost all n-bit*Boolean**functions**on*CRCW PRAM's with polynomial numbers*of*processors. ...##
###
Optimal bounds for decision problems on the CRCW PRAM

1989
*
Journal of the ACM
*

We prove optimal R(log n/log log n)

doi:10.1145/65950.65958
fatcat:tja3jvt6yrckncrkqzouzi7vxi
*lower**bounds**on*the time for CRCW PRAM's with polynomially*bounded*numbers*of*processors or memory cells to compute parity and a number*of*related problems. ... We also exhibit a strict time hierarchy*of*explicit*Boolean**functions**of*n bits*on*such machines which holds up to O(log n/ log log n) time. ... We also prove a tight O(logn)*lower**bound**on*the time to compute almost all n-bit*Boolean**functions**on*CRCW PRAM's with polynomial numbers*of*processors. ...##
###
Page 1964 of Mathematical Reviews Vol. , Issue 2001C
[page]

2001
*
Mathematical Reviews
*

*Lower*

*bound*

*techniques*are investigated which behave multiplica- tively for

*functions*defined

*on*direct sums

*of*sets. ... It may be

*of*interest especially for re- searchers working in proving

*lower*

*bounds*

*on*

*complexity*. ...

##
###
Page 3948 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

{For the entire collection see MR 91¢:68005.}
92g:68047 68Q10 68Q25
Kutylowski, Mirostaw (PL-WROC-C)
Time

*complexity**of**Boolean**functions**on*CREW*PRAMs*. SIAM J. Comput. 20 (1991), no. 5, 824-833. ... It follows that the time required by the logical ‘or’*of*n variables is at least log,n. We present an essentially different method*of*estimating*PRAM**complexity**of**Boolean**functions*. ...##
###
Separating the power of EREW and CREW PRAMs with small communication width
[chapter]

1993
*
Lecture Notes in Computer Science
*

We prove that evaluating a

doi:10.1007/3-540-57155-8_245
fatcat:2sxnhzzbnjfd7aw2wb4xgi3aea
*Boolean*decision tree*of*height h requires 0(hÂ(m+log*h)) time*on*any EREW*PRAM*with communication width m and any number*of*processors. ... Since this*function*can be easily computed in time O(-h)*on*a CREW*PRAM*with communication width 1 using 2 O(h) processors, this gives a separation between the two models whenever the EREW*PRAM*has communication ... For example, the*technique*in the*lower**bound*result for the OR*function**on*CREW(1)*PRAMs*[VW85, Bea86] is very similar to the*technique*that Kuty*owski [Kut91] eventually used in his optimal*bound*...
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