## An Optimal Parallel Algorithm for Formula Evaluation (1992)

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Citations: | 43 - 6 self |

### BibTeX

@MISC{Buss92anoptimal,

author = {S. Buss and S. Cook and A. Gupta and V. Ramachandran},

title = {An Optimal Parallel Algorithm for Formula Evaluation },

year = {1992}

}

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### Abstract

A new approach to Buss’s NC¹ algorithm [Proc. 19thACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123-131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for NC¹ over AC¬ reductions. This approach is then used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits.

### Citations

240 | The parallel evaluation of generic arithmetic expressions
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- 1974
(Show Context)
Citation Context ...arithmic in the size of the formula. Early work on 2 BSVP was done by Spira [Spi71] who showed that any sentence of size n can be restructured into a formula of depth O(log n) and size O(n 2 ). Brent =-=[Bre74]-=- gave a restructured circuit of logarithmic depth and linear size to evaluate a given arithmetic formula. These results gave hope of obtaining a logarithmic depth circuit for formula evaluation by fin... |

217 |
A taxonomy of problems with fast parallel algorithms
- Cook
- 1985
(Show Context)
Citation Context ...btain an arithmetic NC 1 algorithm for arithmetic formula evaluation (over rings, fields and semi-rings). 2 Background 2.1 Boolean Circuit Complexity All unreferenced material in this section is from =-=[Coo85]-=- and we refer the reader to that paper for a more indepth discussion of Boolean circuit complexity. Definition: A Boolean circuit # on n inputs and m outputs is a finite directed acyclic graph with ea... |

152 |
On uniform circuit complexity
- Ruzzo
- 1981
(Show Context)
Citation Context ...e class of problems that have a logspace uniform family of circuits of depth O(log k n) and polynomial size, where n is the size of the input; for NC 1 a stronger notion of uniformity is usually used =-=[Ruz81]-=-. The class NC is the class of problems that have a logspace uniform family of circuits of polylog depth and polynomial size; this class is generally considered to characterize the class of problems w... |

127 | On uniformity within NC
- Barrington, Immerman, et al.
- 1990
(Show Context)
Citation Context ...duced by Sipser [Sip83] is the class of problems solvable by an ATM in log-time and O(1) alternations. Buss proposes LH as the definition of uniform AC 0 . Recently Barrington, Immerman and Straubing =-=[BIS88]-=- have shown that all 4 of the above characterizations give the same class thus suggesting that these may be the appropriate definition. We would like a circuit definition of AC 0 . We begin by definin... |

122 | On uniformity within NC1
- Barrington, Immerman, et al.
- 1990
(Show Context)
Citation Context ...duced by Sipser [Sip83] is the class of problems solvable by an ATM in log-time and O(1) alternations. Buss proposes LH as the definition of uniform AC 0 . Recently Barrington, Immerman and Straubing =-=[BIS88]-=- have shown that all 4 of the above characterizations give the same class thus suggesting that these may be the appropriate definition. We would like a circuit definition of AC 0 . We begin by definin... |

120 |
Parallel tree contraction and its applications
- Miller, Reif
- 1985
(Show Context)
Citation Context ...y implied the same space bound for BSVP. The result of Borodin [Bor77] that LOGSPACE # NC 2 once again placed this problem in NC 2 . The logarithmic time tree contraction algorithm of Miller and Reif =-=[MR85]-=- for arithmetic expression evaluation on a PRAM again translates into an NC 2 algorithm on arithmetic circuits. The first sub-NC 2 algorithm for BSVP was devised by Cook and Gupta [Gup85] and independ... |

97 | On relating time and space t o size and depth
- Borodin
- 1977
(Show Context)
Citation Context ...ognized in deterministic log space (LOGSPACE). Since the set of true Boolean sentences is an instance of these languages, this immediately implied the same space bound for BSVP. The result of Borodin =-=[Bor77]-=- that LOGSPACE # NC 2 once again placed this problem in NC 2 . The logarithmic time tree contraction algorithm of Miller and Reif [MR85] for arithmetic expression evaluation on a PRAM again translates... |

82 |
The Complexity of Computing
- Savage
- 1976
(Show Context)
Citation Context ...considered to be a very fast complexity class and many problems have been shown to be in NC 1 . Sum and product of 2 n–bit integers, sum of nn–bit integers and sorting nn–bit integers are all in NC 1 =-=[Sav76]-=-. Because of their shallow depth, NC 1 circuits can always be converted into equivalent circuits with fan-out 1, polynomial size and O(log n) depth. In this form, they can be expressed as formulas. Co... |

72 |
Borel sets and circuit complexity
- Sipser
- 1983
(Show Context)
Citation Context ...h of the induction is constant and shows that all these characterizations are equivalent. An alternate definition is proposed by Buss [Bus87]. The log-time hierarchy (denoted LH) introduced by Sipser =-=[Sip83]-=- is the class of problems solvable by an ATM in log-time and O(1) alternations. Buss proposes LH as the definition of uniform AC 0 . Recently Barrington, Immerman and Straubing [BIS88] have shown that... |

66 | The Boolean formula value problem is in ALOGTIME
- Buss
- 1987
(Show Context)
Citation Context ...esis context-free grammars can be recognized in depth O(log n log log n) while Ramachandran showed that arithmetic formulas over semi-rings can be evaluated within the same time bound. Recently, Buss =-=[Bus87]-=- devised an alternating log time algorithm for both BSVP and the recognition problem for parenthesis context-free grammars. Since alternating log time is equivalent to NC 1 [Ruz81] under a strong noti... |

62 | Expressibility and parallel complexity
- Immerman
- 1989
(Show Context)
Citation Context ...n (proposition 2.3) would have to place further resource restrictions on the machine since a straightforward extension of the proposition would imply AC 0 = NL (non-deterministic log space). Immerman =-=[Imm89]-=- proposes defining AC k (k # 0) in terms of a CRAM (a CRCW PRAM which is strengthened slightly to allow a processor to shift a word left or right by log n bits in unit time). This modification does no... |

39 |
On time hardware complexity tradeoffs for boolean functions
- Spira
- 1971
(Show Context)
Citation Context ....g., [Coo85, KR90, Ruz81]. Simple fan-in arguments show that any circuit for formula evaluation must have depth at least logarithmic in the size of the formula. Early work on 2sBSVP was done by Spira =-=[Spi71]-=- who showed that any sentence of size n can be restructured into a formula of depth O(log n) and size O(n2 ). Brent [Bre74] gave a restructured circuit of logarithmic depth and linear size to evaluate... |

35 |
zur Gathen, Parallel arithmetic computations: A survey. Mathematical foundations of computer science
- von
- 1986
(Show Context)
Citation Context ...ame as BSVP, -- given an arithmetic formula over an algebra and the value of each variable in the formula, determine the value of the formula. We use the arithmetic-Boolean circuits of von zur Gathen =-=[vzG86]-=- as our model and the corresponding arithmetic complexity theory. Once again, the goal is to obtain a circuit of small depth that solves this problem for all inputs of a given depth. We assume that ea... |

30 |
Some exact complexity results for straight-line computations over semirings
- Jerrum, Snir
- 1982
(Show Context)
Citation Context ...nce in the usual way, where 0 and 1 represent F alse and T rue respectively. Definition: The Boolean sentence value problem (BSVP) is: Given a Boolean sentence, A, what is the value of A? Definition:(=-=[JS82]-=-) A semi-ring S is a 5-tuple(S, #, #, 0, 1) where 0, 1 # S such that: 1. (S, #, 0) is a commutative monoid 2. (S, #, 1) is a monoid 3. # distributes over # 4. for every a # S, a # 0 = 0 = 0 # a For co... |

30 | and Vijaya Ramachandran, “Parallel algorithms for shared-memory machines - Karp - 1990 |

24 | Log space recognition and translation of parenthesis languages, manuscript
- LYNCH
(Show Context)
Citation Context ...irect implementation of these algorithms seems to require #(log 2 n) depth for the restructuring. This result placed BSVP in NC 2 . The BSVP can be shown to be in NC 2 through other techniques. Lynch =-=[Lyn77]-=- showed that parenthesised context-free languages can be recognized in deterministic log space (LOGSPACE). Since the set of true Boolean sentences is an instance of these languages, this immediately i... |

19 |
Speedups of deterministic machines by synchronous parallel machines
- Dymond, Tompa
- 1983
(Show Context)
Citation Context ...them is of interest. In this paper, we present a simple NC 1 algorithm for BSVP which incorporates Buss's original ideas into a two person pebbling game similar to that introduced by Dymond and Tompa =-=[DT85]-=-. This algorithm is designed to give insight into the mechanism of Buss's algorithm. We show that our result is optimal by proving that the problem is complete for NC 1 under AC 0 reductions. We then ... |

14 |
Memory bounds for recognition of context-free and context-sensitive languages
- Lewis, Stearns, et al.
- 1965
(Show Context)
Citation Context ...e sentence has value 1, then the Pebbler can force a win in O(log n) moves by the strategy used by Tompa [Tom85] to e#ciently pebble a tree. (The basic idea goes back to Lewis, Stearns, and Hartmanis =-=[LSH65]-=- in their proof that context free languages can be recognized in space O(log 2 n).) This strategy can be described as follows: Consider the challenged subsentencesA k to be scarred by replacing each o... |

9 |
Input-driven languages are in log n depth
- Dymond
- 1988
(Show Context)
Citation Context ...hine. Buss also showed that his result is optimal in a very strong sense --- he showed that BSVP is complete for alternating log time under reductions from any level of the log-time hierarchy. Dymond =-=[Dym88]-=- extended Buss's result for parenthesis grammars to showing that all input-driven languages can be recognized in NC 1 . His technique 3 generalizes the game described by Buss. More recently, Muller an... |

8 |
On time hardware complexity tradeo#s for Boolean functions
- Spira
- 1971
(Show Context)
Citation Context ....g., [Coo85, KR90, Ruz81]. Simple fan-in arguments show that any circuit for formula evaluation must have depth at least logarithmic in the size of the formula. Early work on 2 BSVP was done by Spira =-=[Spi71]-=- who showed that any sentence of size n can be restructured into a formula of depth O(log n) and size O(n 2 ). Brent [Bre74] gave a restructured circuit of logarithmic depth and linear size to evaluat... |

7 |
Complexity theory for unbounded fan-in parallelism
- CHANDRA, STOCKMEYER, et al.
- 1982
(Show Context)
Citation Context ... families. The n th member of the family recognizes all strings in the language of length n. A generalization of the uniform Boolean circuit families are the unbounded fan-in uniform circuit families =-=[CSV82]-=-. These circuit families are allowed arbitrary fan-in at the # and # gates. We need a new uniformity condition. Definition: The direct connection language for an unbounded fan-in family of circuits ##... |

3 |
A fast parallel algorithm for recognition of parenthesis languages
- Gupta
- 1985
(Show Context)
Citation Context ...ller and Reif [MR85] for arithmetic expression evaluation on a PRAM again translates into an NC 2 algorithm on arithmetic circuits. The first sub-NC 2 algorithm for BSVP was devised by Cook and Gupta =-=[Gup85]-=- and independently by Ramachandran [Ram86]. Their circuit family for the problem was logspace uniform and had a depth of O(log n log log n) and this gave new hope that the problem had an NC 1 algorith... |

3 | The taxonomy ofproblems with fast parallel algorithms - CooK - 1985 |

3 | AND E PREPARATA, Parallel restructuring and evaluation of expressions - MULLER - 1988 |

2 |
Restructuring formula trees. Unpublished manuscript
- Ramachandran
- 1986
(Show Context)
Citation Context ...sion evaluation on a PRAM again translates into an NC 2 algorithm on arithmetic circuits. The first sub-NC 2 algorithm for BSVP was devised by Cook and Gupta [Gup85] and independently by Ramachandran =-=[Ram86]-=-. Their circuit family for the problem was logspace uniform and had a depth of O(log n log log n) and this gave new hope that the problem had an NC 1 algorithm. Cook and Gupta also showed that parenth... |

2 |
The Complexity of Computing. Wiley--Interscience
- Savage
- 1976
(Show Context)
Citation Context ...dered to be a very fast complexity class and many problems have been shown to be in NC 1 . Sum and product of 2 n--bit integers, sum of n n--bit integers and sorting n n--bit integers are all in NC 1 =-=[Sav76]-=-. Because of their shallow depth, NC 1 circuits can always be converted into equivalent circuits with fan-out 1, polynomial size and O(log n) depth. In this form, they can be expressed as formulas. Co... |

2 |
A pebble game that models alternation. Unpublished manuscript
- Tompa
(Show Context)
Citation Context ...ubsentence A k . The maximal subsentences of A k play the role of the inputs to the node. If the sentence has value 1, then the Pebbler can force a win in O(log n) moves by the strategy used by Tompa =-=[Tom85]-=- to e#ciently pebble a tree. (The basic idea goes back to Lewis, Stearns, and Hartmanis [LSH65] in their proof that context free languages can be recognized in space O(log 2 n).) This strategy can be ... |

2 | Input-driven languages are in log n depth, Information Process - DYMOND - 1988 |

1 | fastparallel algorithmfor recognition ofparenthesis languages - GUPTA, A - 1985 |

1 | Parallel algorithmsfor shared-memory machines - RAMACHANDRAN - 1990 |

1 | Log space recognition and translation ofparenthesis languages - LYNCH - 1977 |

1 | Restructuringformula trees, unpublished manuscript - RAMACHANDRAN - 1986 |