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Lower bounds for solving linear diophantine equations on random access machines

Author:

Friedhelm Meyer auf der HeideAuthors Info & Claims

Journal of the ACM (JACM), Volume 32, Issue 4

Pages 929 - 937

https://doi.org/10.1145/4221.4250

Published: 01 October 1985 Publication History

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Abstract

The problem of recognizing the language L_n(L_{n, k}) of solvable Diophantine linear equations with n variables (and solutions from {O, …, k}ⁿ) is considered. The languages ∪_nϵN L_n, ∪_nϵN L_{n, l}, the knapsack problem, are NP-complete. The Ω(n² lower bound for L_n,1 on linear search algorithms due to Dobkin and Lipton is generalized to an Ω(n²log(k + 1)) lower bound for L_{n, k}. The method of Klein and Meyer auf der Heide is further improved to carry over the Ω(n²) lower bound for L _n,1 to random access machines (RAMS) in such a way that it holds for a large class of problems and for very small input sets. By this method, lower bounds that depend on the input size, as is necessary for L_n, are proved. Thereby, an Ω(n²log(k + 1)) lower bound is obtained for RAMS recognizing L_n or L_{n, k}, for inputs from {0, …, (nk)^0(n²)}ⁿ.

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Cited By

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Meer KNaif A(2015)Generalized finite automata over real and complex numbersTheoretical Computer Science10.1016/j.tcs.2015.05.001591:C(85-98)Online publication date: 2-Aug-2015
https://dl.acm.org/doi/10.1016/j.tcs.2015.05.001
Maass W(2014)On the use of inaccessible numbers and order indiscernibles in lower bound arguments for random access machinesThe Journal of Symbolic Logic10.1017/S002248120002795X53:04(1098-1109)Online publication date: 12-Mar-2014
https://doi.org/10.1017/S002248120002795X
Meer KNaif A(2014)Generalized Finite Automata over Real and Complex NumbersTheory and Applications of Models of Computation10.1007/978-3-319-06089-7_12(168-187)Online publication date: 2014
https://doi.org/10.1007/978-3-319-06089-7_12
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Index Terms

Lower bounds for solving linear diophantine equations on random access machines
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming
  2. Probability and statistics
    1. Probabilistic reasoning algorithms
      1. Random number generation
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming

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Reviews

Reviewer: Garry J Tee

For integers a 1, . . . , a n and b, the problem of deciding whether non-negative integers &agr; 1, . . . ,&agr; n exist such that a 1&agr; 1 + . . . + a n&agr; n =:- 9T b :9F(*):Y (for all positive integers n) is known to be NP-complete. If the coefficients &agr; 1, . . . ,&agr; n are each restricted to 0 or 1, then the problem reduces to the knapsack problem, which is NP-complete. A random access machine is an abstract device, which is convenient for estimating complexity of computations. Each step of a random access machine consists either of a storage access (direct or indirect), an addition, a subtraction, or an if-question. Theorem 5 shows that if each element of a is bounded by 0 ?9T a i ?9T((2 k + 1) 3( n:9- T+ 1) 2) ( n + 1) 2 and the coefficients of &agr; are each bounded by 0 ?9T&agr; i ?9T k, then each random access machine which solves the problem (*) needs at least 1-2- n( n ? - 1)log( k + 1) ? n steps. Theorem 6 shows that the same lower bound for the number of steps holds for the case where the coefficients &agr; :- Ci are not bounded above, and the upper bound for the a i is twice that for Theorem 5.

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Association for Computing Machinery

New York, NY, United States

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Published: 01 October 1985

Published in JACM Volume 32, Issue 4

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Meer KNaif A(2015)Generalized finite automata over real and complex numbersTheoretical Computer Science10.1016/j.tcs.2015.05.001591:C(85-98)Online publication date: 2-Aug-2015
https://dl.acm.org/doi/10.1016/j.tcs.2015.05.001
Maass W(2014)On the use of inaccessible numbers and order indiscernibles in lower bound arguments for random access machinesThe Journal of Symbolic Logic10.1017/S002248120002795X53:04(1098-1109)Online publication date: 12-Mar-2014
https://doi.org/10.1017/S002248120002795X
Meer KNaif A(2014)Generalized Finite Automata over Real and Complex NumbersTheory and Applications of Models of Computation10.1007/978-3-319-06089-7_12(168-187)Online publication date: 2014
https://doi.org/10.1007/978-3-319-06089-7_12
Just BMeyer auf der Heide FWigderson A(2011)On computations with integer divisionRAIRO - Theoretical Informatics and Applications10.1051/ita/198923010101123:1(101-111)Online publication date: 8-Jan-2011
https://doi.org/10.1051/ita/1989230101011
Ben-Amram AGalil Z(2005)Lower bounds on algebraic random access machinesAutomata, Languages and Programming10.1007/3-540-60084-1_88(360-371)Online publication date: 7-Jun-2005
https://doi.org/10.1007/3-540-60084-1_88
Lürwer-Brüggemeier KMeyer F(2005)Capabilities and complexity of computations with integer divisionSTACS 9310.1007/3-540-56503-5_46(463-472)Online publication date: 27-May-2005
https://doi.org/10.1007/3-540-56503-5_46
Gröger HTurán G(2005)On linear decision trees computing Boolean functionsAutomata, Languages and Programming10.1007/3-540-54233-7_176(707-718)Online publication date: 8-Jun-2005
https://doi.org/10.1007/3-540-54233-7_176
Karpinski M(2003)Randomized complexity of linear arrangements and polyhedra?Fundamentals of Computation Theory10.1007/3-540-48321-7_1(1-12)Online publication date: 3-Jun-2003
https://doi.org/10.1007/3-540-48321-7_1
Ben-Amram AGalil Z(2002)Topological Lower Bounds on Algebraic Random Access MachinesSIAM Journal on Computing10.1137/S009753979732939731:3(722-761)Online publication date: 1-Mar-2002
https://dl.acm.org/doi/10.1137/S0097539797329397
(1999)Lower Bounds for the Complexity of Functions in a Realistic RAM ModelJournal of Algorithms10.1006/jagm.1998.099432:1(1-20)Online publication date: 1-Jul-1999
https://dl.acm.org/doi/10.1006/jagm.1998.0994
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Distributed Lower Bounds for Ruling Sets

Super-linear time-space tradeoff lower bounds for randomized computation

Lower Bounds for Randomized Exclusive Write PRAMs

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