## Machine Models and Linear Time Complexity (1993)

Venue: | SIGACT News |

Citations: | 5 - 3 self |

### BibTeX

@ARTICLE{Regan93machinemodels,

author = {Kenneth W. Regan},

title = {Machine Models and Linear Time Complexity},

journal = {SIGACT News},

year = {1993},

volume = {24},

pages = {Fall}

}

### Years of Citing Articles

### OpenURL

### Abstract

wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constant-factor overhead ; if g = O(t log t) it has a factor-of-O(log t) overhead , and so on. The simulation is on-line if each step of M 1 i

### Citations

3830 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Motwani, et al.
(Show Context)
Citation Context ...d unit time per edit thereafter. Here "similar kind" means that the edits can be done by a finite-state machine, for which we take the deterministic generalized sequential machine (GSM) form=-=alized in [HU79]-=-. (A GSM S is essentially a Mealy machine which can output not just one but zero, two, three, or more symbols in a given transition.) Rule 1. In any move, P may mark locations a 1 ; b 1 ; a 2 ; b 2 , ... |

256 |
Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings
- Angluin, Valiant
- 1979
(Show Context)
Citation Context ...e "RAM with strongly polynomially compact memory" of Grandjean and Robson [GR91]. Gurevich and Shelah [GS89] show other models to belong (some with a small power of log t), including the RAC=-= model of [AV79] and "random-ac-=-cess Turing machines." Below the "Large Cluster" are Turing machines with d-dimensional worktapes (d-TMs), whose time classes we denote by DTIME d [t]. 4 For d = 1 we have the standard ... |

128 | A model for hierarchical memory - Aggarwal, Alpern, et al. - 1987 |

114 |
String matching and other products
- Fischer, Paterson
- 1974
(Show Context)
Citation Context ...t intersection: L int := f x 1 #x 2 # : : : #xm , y 1 #y 2 # : : : #ym : (9i; j) x i = y j g: (d) Triangle: L \Delta := fG : G is an undirected graph which contains a triangleg. L pat belongs to DLIN =-=[FP74]-=- (cf. [FMR72, GS83]), and was recently shown not to be recognizable by a one-way multihead DFA [JL93]. L dup and L int can be solved in linear time by a RAM which treats list elements as cell addresse... |

109 |
Hierarchical memory with block transfer
- Aggarwal, Chandra, et al.
- 1987
(Show Context)
Citation Context ...(n), and s(n) of input lengths n in the usual manner. 4. The Block Move Model The Block Move (BM) model can be regarded as an extension of the Block Transfer (BT) model of Aggarwal, Chandra, and Snir =-=[ACS87]-=-. The BT is a RAM with the special instructionscopy [a \Gamma m : : : a] into [b \Gamma m : : : b], which is charged m+ maxf(a); (b) g time units. The BT likewise sets no fixed limit on the block-size... |

97 |
Relations among complexity measures
- Pippenger, Fischer
- 1979
(Show Context)
Citation Context ... move a worktape head on some cell a to cell ba=2c, 2a, or 2a+1 in one step. A TC can simulate a log-cost RAM with constant-factor overhead [PR81], and can be simulated by a TM in time O(t 2 = log t) =-=[PF79]-=-. 3 Loui and Luginbuhl [LL92b] give an O(t log t= loglog t) simulation of a TC by a log-cost RAM M , and interestingly, prove that this is optimal for on-line simulation when t(n) = n. We note that th... |

78 |
Time-bounded random access machines
- Cook, Reckhow
- 1972
(Show Context)
Citation Context ... measure (RAM-TIME), which charges one time unit to execute any instruction. A standard RAM has addition and subtraction as arithmetic operations. 2 For our purpose, the arguments of Cook and Reckhow =-=[CR73]-=- that unit cost is unreasonable hold force. The SRAM model with arithmetic limited to increment and decrement lessens this objection, and here our interests begin. Cook and Reckhow instead advocated u... |

77 |
Emde Boas. Machine models and simulations
- van
- 1990
(Show Context)
Citation Context ... step of M 1 is simulated by step(s) of M 2 in some transparent manner---see [vEB90, LL92b] for discussion of how to formalize this. The most powerful sequential models in the survey by van Emde Boas =-=[vEB90]-=- are forms of the random access machine (RAM) with the unit cost time measure (RAM-TIME), which charges one time unit to execute any instruction. A standard RAM has addition and subtraction as arithme... |

72 | Nondeterminism within p - Buss, Goldsmith - 1993 |

64 |
Storage modification machines
- Schönhage
- 1980
(Show Context)
Citation Context ... "parallel" model in [vEB90, TLR92]. 1 We note the following "Large Cluster" of models known to simulate each other with overhead at most O(log t). The storage modification machine=-= (SMM) of Schonhage [Sch80]-=-, the programming model of Jones [Jon93], and the SRAM all simulate each other linearly [Sch80, Jon93]. The tree computer (TC) can be described as a TM which has worktapes in the form of infinite bina... |

58 | The computational complexity of universal hashing - Mansour, Nisan, et al. - 1993 |

54 | Two-tape simulation of multitape Turing machines - Hennie, Stearns - 1966 |

38 |
On determinism versus nondeterminism and related problems
- Paul, Pippenger, et al.
- 1983
(Show Context)
Citation Context ...al vicinity. So do all the other models in the "Large Cluster," even that of [GR91] (consider t much less than the overall runtime). TMs with d-dimensional tapes have vicinity O(t d ). The t=-=heorem of [PPST83] that NLIN-=- 6= DLIN is not known to carry over to any model of super-linear vicinity. Our "BM" model below has a parameterswhich calibrates its vicinity. 2. Linear-Time Problems The following natural d... |

34 | Satisfiability is quasilinear complete in NQL
- Schnorr
- 1978
(Show Context)
Citation Context ...often change its head direction, but in going to a BM this is a break in pipelining and subject to a memory-access charge. Let us abbreviate quasi-linear time by qlin := n(log n) O(1) , and following =-=[Sch78], DTIME[ql-=-in] to DQL, NTIME[qlin ] to NQL. All models in the "Large Cluster" accept the same class---call it DNLT---of languages in time qlin . Gurevich and Shelah [GS89] also proved that the nondeter... |

32 |
A density control algorithm for doing insertions and deletions in a sequentially ordered in good worst-case time
- Willard
- 1992
(Show Context)
Citation Context ...not have an insert/delete mode or allow P to "cut" z and/or "paste" S(z) at location a 2 . In practice, a large number of block insertions and deletions can "fragment" fi=-=les and file systems. Willard [Wil92]-=- gives efficient ways of simulating these operations on a realistic sequential file system model which lacks them. However, information in [a 2 : : : b 2 ] may be preserved under the following convent... |

31 | Why is boolean complexity theory difficult?, in
- Valiant
- 1992
(Show Context)
Citation Context ... b 2 for a single pass, making O(log n)-many deterministic passes otherwise. Proving non-membership in log-pass TLIN, or generally with R(n) = o(n), may be a leg up on the problem stressed by Valiant =-=[Val92]-=- of finding natural problems which don't have simultaneously linear-size, log-depth circuits. Some related open problems have come up in this work: 7 ffl Does every language in TLIN have linear-sized ... |

28 |
Spatial machines: a more realistic approach to parallel computation
- Feldman, Shapiro
- 1992
(Show Context)
Citation Context ... to a matter of n vs. n 2 . From reasonable RAMs to d-TMs it is a matter of n vs. n 1+1=d . A focal point for the problem is the concept of vicinity of a machine model M raised by Feldman and Shapiro =-=[FS92], which we-=- formalize as follows. Let C be a data configuration, and let HC stand for the finite set of memory locations [or data items] designated as "scanned" in C. For all t ? 0, let I t denote the ... |

27 | A general sequential time-space tradeoff for finding unique elements - Beame - 1991 |

25 | Near-optimal time-space tradeoff for element distinctness - Yao - 1988 |

20 | Superlinear lower bounds for bounded-width branching programs - Barrington, Straubing - 1991 |

20 | Time-space optimal string matching - GALIL, SEIFERAS - 1983 |

20 |
On the power of multiplication in random-access machines
- Hartmanis, Simon
- 1974
(Show Context)
Citation Context ...r's own work reported here was partly supported by NSF Grant CCR-9011248 2 Having unit-cost multiplication yields the MRAM, which can accept QBF and other PSPACE-complete languages in polynomial time =-=[HS74], and is regarded as-=- a "parallel" model in [vEB90, TLR92]. 1 We note the following "Large Cluster" of models known to simulate each other with overhead at most O(log t). The storage modification machi... |

14 | Real-time simulation of multihead tape units - Fischer, Meyer, et al. - 1972 |

14 |
Nearly-linear time
- Gurevich, Shelah
- 1989
(Show Context)
Citation Context ...tent i used is bounded by a polynomial in the running time t, which is the defining property of the "RAM with strongly polynomially compact memory" of Grandjean and Robson [GR91]. Gurevich a=-=nd Shelah [GS89] show other models t-=-o belong (some with a small power of log t), including the RAC model of [AV79] and "random-access Turing machines." Below the "Large Cluster" are Turing machines with d-dimensional... |

13 |
Some combinatorial game problems require V(n k ) time
- Adachi, Iwata, et al.
- 1984
(Show Context)
Citation Context ...RAM which treats list elements as cell addresses. L \Delta is not believed recognizable in linear time on a RAM at all---the best method known is to square the adjacency matrix of G. Adachi and Iwata =-=[AI84]-=- showed that every language in DTIME[O(n k )] many-one reduces in O(n log n) time to a language L k defined by a pebble game with 4k + 1 pebbles; thus L k is not in DTIME[n k\Gammaffl ]. Grandjean [Gr... |

13 |
On the power of one-way communication
- Chang, Ibarra, et al.
- 1988
(Show Context)
Citation Context ...ss TLIN. Whether the language D 2 of balanced (; ); [; ] is in TLIN at all is open. The reduced form is comparable to the "sweeping automata" and linear iterative arrays studied by Ibarra et=-= al. (see [CIV88]-=-). 5. Complexity Classes Like the RAM but unlike what is known for the multitape TM (see [CR73, HU79]), the BM under anysd has only a constant-factor overhead for universal simulation. 7 One consequen... |

13 |
On time versus space II
- Paul, Reischuk
- 1979
(Show Context)
Citation Context ...f infinite binary trees, or which has standard tapes but can move a worktape head on some cell a to cell ba=2c, 2a, or 2a+1 in one step. A TC can simulate a log-cost RAM with constant-factor overhead =-=[PR81]-=-, and can be simulated by a TM in time O(t 2 = log t) [PF79]. 3 Loui and Luginbuhl [LL92b] give an O(t log t= loglog t) simulation of a TC by a log-cost RAM M , and interestingly, prove that this is o... |

12 |
k one-way heads cannot do string matching
- Jiang, Li
- 1993
(Show Context)
Citation Context ...angle: L \Delta := fG : G is an undirected graph which contains a triangleg. L pat belongs to DLIN [FP74] (cf. [FMR72, GS83]), and was recently shown not to be recognizable by a one-way multihead DFA =-=[JL93]-=-. L dup and L int can be solved in linear time by a RAM which treats list elements as cell addresses. L \Delta is not believed recognizable in linear time on a RAM at all---the best method known is to... |

12 | Two time-space tradeoffs for element distinctness - Karchmer - 1986 |

11 | New real-time simulations of multihead tape units - Leong, Seiferas - 1981 |

9 |
L.G.: On time versus space and related problems
- Hopcroft, Paul, et al.
- 1975
(Show Context)
Citation Context ...ime O(t + (n + u) loglog s). 5 The inclusion DTIME[t] ` RAM-TIME[t= log t] implies a log t separation, which is hardly felt to be the true power of the unit-cost RAM over the TM, but it was proved by =-=[HPV75]-=- only for t(n) = \Omega\Gamma n log n). Sudborough and Zalcberg [SZ76] prove DLIN ae RAM-TIME[O(n)]. 2 physics as we know it. However the RAM, even under log-cost, has exponential vicinity. So do all ... |

9 | Tape versus queue and stacks: The lower bounds - Li, Vitányi - 1988 |

9 |
An information-theoretic approach to time bounds for on-line computation
- Paul, Seiferas, et al.
- 1981
(Show Context)
Citation Context ...tearns simulation [HS66, HU79, WW86] is memory-efficient unders1 . We suspect that for d ? 1 the converse simulation in (c) requires notably more than the O(log t) overhead of the d = 1 case (a); see =-=[PSS81]-=- for related matters. The intuitive reason is that a d-TM may often change its head direction, but in going to a BM this is a break in pipelining and subject to a memory-access charge. Let us abbrevia... |

8 |
Data structures for distributed counting
- Fürer
- 1984
(Show Context)
Citation Context ... results point out the difficulty of obtaining even such a separation of TLIN from linear-time on a TC or SRAM: 7 TMs with a fixed number ks2 of worktapes do have constant factor universal simulation =-=[Fur84]-=-. The constant factor in Jones' universal simulation theorem is independent of the program being simulated [Jon93]---this yields a sharper result than Theorem 5.1 for his model. Sudborough and Zalcber... |

8 | V.: Multiplication, division, and shift instructions in parallel random-access machines - Trahan, Loui, et al. - 1992 |

7 | Nondeterministic linear-time tasks may require substantially nonlinear deterministic time in the case of sublinear work space - Gurevich, Shelah - 1990 |

7 | Separating complexity classes related to certain input oblivious logarithmic space bounded Turing machines - Krause, Meinel, et al. - 1989 |

6 | Real-time simulation of concatenable double-ended queues by double-ended queues - Kosaraju - 1979 |

6 | On Oblivious Branching Programs of Linear Length - Krause, Waack |

6 | The power of the queue - Li, Longpr'e, et al. - 1992 |

5 | Two nonlinear lower bounds for on-line computations - Duris, Galil, et al. - 1984 |

5 |
with compact memory: a robust and realistic model of computation
- RAM
- 1991
(Show Context)
Citation Context ...l address a and register content i used is bounded by a polynomial in the running time t, which is the defining property of the "RAM with strongly polynomially compact memory" of Grandjean a=-=nd Robson [GR91]. Gurevich and Shela-=-h [GS89] show other models to belong (some with a small power of log t), including the RAC model of [AV79] and "random-access Turing machines." Below the "Large Cluster" are Turing... |

5 |
Nontrivial lower bound for an NP problem on automata
- Grandjean
- 1990
(Show Context)
Citation Context ...84] showed that every language in DTIME[O(n k )] many-one reduces in O(n log n) time to a language L k defined by a pebble game with 4k + 1 pebbles; thus L k is not in DTIME[n k\Gammaffl ]. Grandjean =-=[Gra90] showed that ev-=-ery language in NLIN reduces to a natural NP-complete problem called "RISA" by DLIN-reductions; thus RISA is not in DLIN. However, we know of no language in NLIN " P which is not in DLI... |

5 | Fast simulation of Turing machines by random access machines
- Katajainen, Leeuwen, et al.
- 1988
(Show Context)
Citation Context ...e83, vEB90]. 4 A d-TM is simulated linearly by an SMM [Sch80], in time O(t \Delta 5 d log t ) by a TC [Rei82], and in time O(t(log t) 1\Gamma1=d (loglog t) 1=d ) by a log-cost RAM [LL92a]. For d = 1, =-=[KvLP88]-=- give more detail: if the TM runs in time t, space s, and gives output length u, the log-cost RAM takes time O(t + (n + u) loglog s). 5 The inclusion DTIME[t] ` RAM-TIME[t= log t] implies a log t sepa... |

5 | A new parallel vector model, with exact characterizations of NC k
- Regan
- 1994
(Show Context)
Citation Context ...ery L 2 DLIN has O(n log n)-sized circuits.) ffl Is log-pass TLIN ` NC 1 ? (Generally, NC k ` BM poly work, O(log k n) passes ` NC k+1 (ks1), and some restrictions on GSMs give equality in the former =-=[Reg93]-=-.) ffl Consider sorted lists which have p n elements, each of size p n. For any d ? 1, two such lists can be merged in linearsd -time. But can this be done in linears1 -time? ffl Does every language a... |

4 | The complexity of matrix transposition on one-tape off-line Turing machines - Dietzfelbinger, Maass, et al. - 1991 |

4 |
Constant factors do matter
- Jones
- 1993
(Show Context)
Citation Context ...e note the following "Large Cluster" of models known to simulate each other with overhead at most O(log t). The storage modification machine (SMM) of Schonhage [Sch80], the programming model=-= of Jones [Jon93]-=-, and the SRAM all simulate each other linearly [Sch80, Jon93]. The tree computer (TC) can be described as a TM which has worktapes in the form of infinite binary trees, or which has standard tapes bu... |

4 |
Optimal on-line simulations of tree machines by random access machines
- Loui, Luginbuhl
- 1992
(Show Context)
Citation Context ... cell a to cell ba=2c, 2a, or 2a+1 in one step. A TC can simulate a log-cost RAM with constant-factor overhead [PR81], and can be simulated by a TM in time O(t 2 = log t) [PF79]. 3 Loui and Luginbuhl =-=[LL92b]-=- give an O(t log t= loglog t) simulation of a TC by a log-cost RAM M , and interestingly, prove that this is optimal for on-line simulation when t(n) = n. We note that their M has the property that ev... |

4 |
A fast implementation of multidimensional storage into a tree storage
- Reischuk
- 1982
(Show Context)
Citation Context ...y under 3 This improves the O(t 2 ) simulation of log-cost RAM by TM given in [CR73]; see also [Wie83, vEB90]. 4 A d-TM is simulated linearly by an SMM [Sch80], in time O(t \Delta 5 d log t ) by a TC =-=[Rei82]-=-, and in time O(t(log t) 1\Gamma1=d (loglog t) 1=d ) by a log-cost RAM [LL92a]. For d = 1, [KvLP88] give more detail: if the TM runs in time t, space s, and gives output length u, the log-cost RAM tak... |

4 |
On families of languages defined by time-bounded random access machines
- Sudborough, Zalcberg
- 1976
(Show Context)
Citation Context ... t] implies a log t separation, which is hardly felt to be the true power of the unit-cost RAM over the TM, but it was proved by [HPV75] only for t(n) = \Omega\Gamma n log n). Sudborough and Zalcberg =-=[SZ76] prove DLI-=-N ae RAM-TIME[O(n)]. 2 physics as we know it. However the RAM, even under log-cost, has exponential vicinity. So do all the other models in the "Large Cluster," even that of [GR91] (consider... |