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Complexity Lower Bounds for Approximation Algebraic Computation Trees

1999
*
Journal of Complexity
*

The goal of this paper is to prove

doi:10.1006/jcom.1999.0519
fatcat:2pvxuqrnhzcrncvkh4hbbondxq
*lower**bounds**for**approximated**computations*. As it is customary*for**lower**bounds*, we consider some form of*algebraic**tree*as our*computational*model (cf. ... We prove*lower**bounds**for**approximate**computations*of piecewise polynomial functions which, in particular, apply*for*round-o*computations*of such functions. ... , and Yao 1998 ]*for*the discussion and the exponential*lower**bound**for*ternary rathen than the usual binary*computation**trees*). ...##
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Page 2762 of Mathematical Reviews Vol. , Issue 99d
[page]

1991
*
Mathematical Reviews
*

(E-PAMP-I; Pamplona)

*Lower**bounds**for*Diophantine*approximations*. (English summary) Algorithms*for**algebra*(Eindhoven, 1996). J. Pure Appl.*Algebra*117/118 (1997), 277-317. ...*lower**bound*is shown*for*the depth of randomized*algebraic*decision*trees*that recognize the knapsack problem and an Q(nlogn)*lower**bound**for*the element distinctness problem. ...##
###
Page 7812 of Mathematical Reviews Vol. , Issue 97M
[page]

1997
*
Mathematical Reviews
*

The transforma- tion of comparison model

*lower**bounds*, which are usually easier to obtain, to the parallel-random-access-machine, unifies some known*lower**bounds*and gives new*lower**bounds**for*several ... In this latter case, a*lower**bound*of the*complexity*is given versus the total number of vertices in all obstacles. ...##
###
Page 514 of Mathematical Reviews Vol. , Issue 99a
[page]

1991
*
Mathematical Reviews
*

Symbolic

*Comput*. 13 (1992), no. 3, 231- 233; MR 93b:13020] strongly reduces the gap between the upper*bound*and the*lower**bound**for*this problem. ... An Ackermannian upper*bound**for*PIMP over the integers has been given vy G. Gallo and B. Mishra [Appl.*Algebra*Engrg. Comm.*Comput*. 5 (1994), no. 6, 343-370; MR 95i:13026]. ...##
###
Lower bounds for algebraic computation trees

1983
*
Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC '83
*

A topological method is given

doi:10.1145/800061.808735
dblp:conf/stoc/Ben-Or83
fatcat:lwhhw5siejedjagvnnvadm2oeq
*for*obtaining*lower**bounds**for*the height of*algebraic**computation**trees*, and*algebraic*decision*trees*. ... Using this method we are able to generalize, and present in a uniform and easy way, almost all the known nonlinear*lower**bounds**for**algebraic**computations*. ... Since our*algebraic**computation**tree*model can handle the operation of taking square roots most of the*lower**bounds*from Shamos's work can be extended to*lower**bounds*on the*complexity*of solving the problems ...##
###
Page 7109 of Mathematical Reviews Vol. , Issue 97K
[page]

1997
*
Mathematical Reviews
*

Such a

*lower**bound*result*for*the depth of*algebraic*decision*trees*was proved before by J. L. Montafia, J. E. Morais and L. M. Pardo [Appl.*Algebra*Engrg. Comm. ... N., Jr.] (1-PAS-C; University Park, PA)*Complexity**lower**bounds**for**computation**trees*with elementary transcendental function gates. (English summary) Theoret.*Comput*. ...##
###
On the complexity of approximating Euclidean traveling salesman tours and minimum spanning trees
[chapter]

1996
*
Lecture Notes in Computer Science
*

In the

doi:10.1007/3-540-62034-6_38
fatcat:uirglv2cm5dkhom3sljlzy3qly
*algebraic**computation**tree*model, the*complexities*of both these problems are shown to be (n log n=r),*for*all n and r such that r < n and r is larger than some constant. ... We consider the problems of*computing*r-*approximate*traveling salesman tours and r-*approximate*minimum spanning*trees**for*a set of n points in IR d , where d 1 is a constant. ... This algorithm ts in the*algebraic**computation**tree*model. Hence, the*lower**bound*of Theorem 1 is tight. ...##
###
On the Complexity of Approximating Euclidean Traveling Salesman Tours and Minimum Spanning Trees

1997
*
Algorithmica
*

In the

doi:10.1007/pl00009183
fatcat:ly5rdng3ebgefasnd7vxzvd3dm
*algebraic**computation**tree*model, the*complexities*of both these problems are shown to be (n log n=r),*for*all n and r such that r < n and r is larger than some constant. ... We consider the problems of*computing*r-*approximate*traveling salesman tours and r-*approximate*minimum spanning*trees**for*a set of n points in IR d , where d 1 is a constant. ... This algorithm ts in the*algebraic**computation**tree*model. Hence, the*lower**bound*of Theorem 1 is tight. ...##
###
Page 4807 of Mathematical Reviews Vol. , Issue 86j
[page]

1986
*
Mathematical Reviews
*

We show an ex- ponential

*lower**bound*on the decision*tree**complexity*of some Boolean function having linear formula size and linear one-time- 68Q Theory of*computing*86j:68048 only branching program*complexity*...*Complexity*sets, 10. Matrix multiplication, 11. Bilinear transformations, 12.*Complex*- ity of*algebras*. 13. The exponent of matrix multiplication, 14. The rank and*approximate*rank, 15. ...##
###
Computing Complex Iceberg Cubes by Multiway Aggregation and Bounding
[chapter]

2004
*
Lecture Notes in Computer Science
*

Our

doi:10.1007/978-3-540-30076-2_11
fatcat:fjigbfl2zvdt7fve5rrrpylxpm
*bounding*techniques are as general as the Divide and*Approximate*pruning techniques*for**complex*constraints and yet our multiway aggregation is as efficient as Star-cubing. ... Citation: Chou, P and Zhang, X 2004, '*Computing**complex*iceberg cubes by multiway aggregation and*bounding*', in Y Kambayashi et al. (ed.) Data Warehousing and ... The*bounding*techniques are general and can prune*for**complex*nonmonotone constraints defined with distributive and*algebraic*functions. ...##
###
A lower bound for the integer element distinctness problem

1991
*
Information and Computation
*

A

doi:10.1016/0890-5401(91)90034-y
fatcat:kikxwteekjfvhmlrav7zcj6y7y
*lower**bound*of Q(n log n) is proved*for*the integer element distinctness problem-Given ...*ALGEBRAIC**COMPUTATION**TREES*In this section we carry over Theorem 5-*lower**bounds**for*rational version of problems-to the*algebraic**computation**tree*model. THEOREM 6. ... One of the motivations*for*proving*lower**bounds*on decision/*computation**trees*is the hope of carrying these*lower**bounds*over to RAM? ...##
###
Page 2810 of Mathematical Reviews Vol. , Issue 93e
[page]

1993
*
Mathematical Reviews
*

Frederic Green (1-CLRK)
93e:68026 68Q20 68725
Hirsch, Michael D. (1-PRIN-CS)

*Lower**bounds**for*the nonlinear*complexity*of*algebraic**computation**trees*with integer inputs.*Comput*. ...*Complexity*1 (1991), no. 3, 257-268. Summary: “A. C. C. Yao [SIAM J.*Comput*. 20 (1991), no. 4, 655-668; MR 92g:68042] proved some*lower**bounds**for**algebraic**computation**trees*with integer inputs. ...##
###
Page 5370 of Mathematical Reviews Vol. , Issue 94i
[page]

1994
*
Mathematical Reviews
*

Some

*lower*and upper*bounds**for**algebraic*decision*trees*of various degrees are found. It is shown that over GF(2) decision*trees*of degree d are more powerful than*trees*of degree < d. ... Summary: “We consider the*complexity*of*computing*Boolean functions with*algebraic*decision*trees*over GF(2) and R. ...##
###
Page 1848 of Mathematical Reviews Vol. , Issue 98C
[page]

1998
*
Mathematical Reviews
*

Yao [“An exponential

*lower**bound*on the size of the*algebraic*decision*trees**for*MAX”, Preprint; per bibl.] have recently shown that if the degree of the query polynomials is*bounded*by a constant, then ... As*for*the*lower**bound*, the problem is shown to be I}-hard. ...##
###
Page 2536 of Mathematical Reviews Vol. , Issue 83f
[page]

1983
*
Mathematical Reviews
*

This elegant concept is strictly connected with obtaining the

*lower**bounds**for*time*complexity*in models of decision*trees*. ... There are many techniques to derive*lower**bounds*in models of linear decision*trees*. ...
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