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Parallel Complexity and P-Complete Problems

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Parallel Complexity and P-Complete Problems 九州大学学術情報リポジトリ Kyushu University Institutional Repository Parallel Complexity and P-Complete Problems Miyano, Satoru Research Institute of Fundamental Information Science Kyushu University http://hdl.handle.net/2324/3112 出版情報：RIFIS Technical Report. 6, 1988-10-24. Research Institute of Fundamental Information Science, Kyushu University バージョン： 権利関係： cal Report Parallel Complexity and P-Complete Problems Satoru Miyano October 24, 1988 Research Institute of Fundamental Information Science Kyushu University 33 hkuoka812, Japan E-rnaii: [email protected] Phone: 092(641)1101 Ext.4471 PARALLEL GOnllFLEXfTY AND P-COMPLETE, FROBLEhllS Reseascll Irlstitute of F~~ndamenltalInformation Science Kyushu University 33, Fuli~asla812, Japan ABSTRACT The ciass NC is a subclass of P but, unfortunately, r~o!iody I:as succecdwl irl proving P#NC althougll it The class IVC consists of probiem solvable by par- is strongly believed that PfNG Like NPfP question. allel algorithms running in time O((log n)v using poly- If P#N@ is approved, a proof that a problem is P- nomial number of processors for some k 2 (3. It is complete iviil be a social proof of nonparallelizability. strongly believed that PfNC. If P#N@ is assumed, This paper gives very general P-complete theorems the P-completeness of a problem implies that no efi- concerning graph algorithms. These theorem yield a cient parallel algorithm exists for the problem. This neiai series of P-cornpiete problems arising from graph paper presents very general P-comp!eteness tileorems optixr~ization probienss including the lexicographically which yield a new series of P-complete psoblenls includ- first ~naxirrlalindependent set problem (Cook 1983). Po- ing the lexicographically first maxima! independent set te~~tially,inftnitely rnanj- nontrivial P-complete prob- problem (Cook 1983). We also give a rneliiod of firding 1c111scalr he derived, parallelisrn in sorrle kind of sequential aigoritfiil~s. This paper is also concerned with a method of find- ing parallelism in problems. Parallelism can be con- sidered in two ways. One is paralleiisrn in existing se- quential algorithms (Kuck 1973). Tlie other is inherent An important role of the parailel complexity theory parallelism in problerns themselves that requires more is to gain insights into inherent parallelism in various mathematical analysis. A method we present in this pa- types of computing problem. It is intended to provide per iocate between these two parallelisms. This method knowledges for answering the following basic questions: is very helpful io show that some problems are in PJC. We exemplify the method through some applications. 1. What kind of problems allow fast eflicient parallel algorithms? 2 EFFICENT PARALLEL ALGORITHhTS 2. What kind of problerns are inherently sequential? AND NC Namely, it aims at understanding of the range of prob- NC is a mnemonics for Nick's Class (Nick Pippenger lems which allow fast emcient parallel algorithms. Si- 1979) named by S.A. Cook in recognition of his contribu- multaneously, it tries ts capture theoretical limits of tion. Tlre purpose of this section is to convince that NC parallel comput- t'ions. is a reasonable class which provides a yardstick mmsur- ing parallel complexity of problems. It has been observed that there are some probienls which can be solved by easy polynomial ti~riesecjnenlial 2.1 Efliciel~tParallel Algoritllms algorikhnls but do not seem to allow any fast parallel al- gorithms. Recent researches show that these probtems hiany formal parallel cornputation models have been are P-complete. On the otller hand, the class NC, idern- considered (Cook 1981). A broadly accepted model is tified by Pippenger (19791, is understood as the class of the paraliel RAhl (PItAhi) model in which a number problem which allow fast efficient parallel algorithm, of processors work together synchronously and cornmu- nicate with a common random access memory. The 2.2 Uniform Circuits and Parallel RAM PRAM model is further classified according to read and The class NC is defined by the uniform circuit model write access abilities as CREW PRAM (Concurrent Read that is suited for classifying problems precisely. Exclusive Write), EREW PRAM and CRCW PRAM. a. n n Problems for which we consider parailel complexity Defirlition A circuit wit11 inputs and outputs is a finite labelled directed acyclic graph satisfying the fol- are formulated as follows: lowings: There are exactly n nodes of indegree 0 called Definition A problem (or search problem) S with a size inputs which are labelled with XI,..., zn, respectively. parameter h(n) is a family (Sn),ll of relations S, C The nodes of indegree 2 (resp. indegree 1) are labelled (0,l)" x (0,13"") for n 2 1. For n 2 1, s E (0, 11, is with either V (or) or i\ (and) (resp. 7 (not)). Exactly called an instance and an object y E (0, I)~(")satisfying m nodes labelled with yl,..., y, are specified as outputs. S,(x, y), if any, is called a solution for z. We denote by size(a) (resp. depi'h(a)) the number of node in rr (resp. the length of the longest path from Example 2.1 A nasirna! independent set (XIIS) of an some input to some output). undirected graph is a maximal set U of vertices such that no two vertices in CT are adjacent. The problem of find- Definition We say that a circuit family {a,),>l- with ing a MIS is formulated in the following way: .A graph output size h(n) compute a function J = (Jn)nll, if with n vertices is represented by an n x n-adjacency ma- each a, computes fn, where a, is a circuit, with n inputs trix and a subset of vertices is represented by an n-bit and h(n) outputs and j, : (O,1jn -+ {O, l)'In). For a vector, Then MIS=(MTS,) is defined only for Integers search problem S = (S,),>1, wesay that solves of the form n2 as S if the function {fn],zl- computed by {an),LI satisfies MIS,? (0,1)"' x (0, l)", S,(s,f,fx)) for all n 2 1. where for (s,yj EMIS,=, 3; is a symmetric matrix repre- Several kinds of uniformities of circuits have been senting an undirected graph with n vertices and y a bit proposed (Ruzzo 1981, Cook 1951, 1983, 1985). One of vector representing a MIS in the graph. Definition A parallel algorithm on a PRAM solving a problem is efficient if, given an input of size n, it runs (1) in time O((log njk) for some constant k 2 0, (2) with a polynomial number of processors. This definition is based on the observation that the time Ofjlog n)k)is very fast and a polynomial number of pro- cessors is feasible. The PRAM model may not seem very realistic in the practical sense. A more realistic model is, for example, a network model. But there are some works showing that PRAM is a good model. For example, Alt et al. Fig. 2.1 Overview of Complexity Classes (1987) showed that the shared memory of any EREW PRAM with n processors and m cells of shared mem- them defines that a circuit family (a,),2l is "uniform" ory can be emulated by an n-processor module parallel (called log uniform) if, given 12 in unary, the description computer with a slow down of O(log nz). Ranade (1987) of the nth circuit a, is log space computable" But NC proved that one step of an n-processor CRCW PRAM defined below does not change by the choice of unifor- can be emulated on an n-processor FFT network in time mity (Ruzzo 1981)- The location of NC is shown in O(Log n) with high probability, using an FIFO queue of Fig. 2.1 together with its neighbors. size 0(1) at each node. Hence the gap is only O(lognj 'A function f is log space computable if it is computed by while keeping the number of processors. a deterministic off-line Turing machine using O(logn3 worktape space. Definition the hardness of parallelization. (1) NCk=(S 1 S is solvable by a uniform circuit The knowledge that a problem is P-complete pro- family such that sizejru,) vides algorithm designers valuable information about is boilnded by some polynomiai the approaches they should choose. It would relieve and simultaneously depih(ccn) = wasting efforts for devising drastically fast paraliel ai- ?zIk)). Q(Qog gorithms and, instead, direct toward ways which lead to useful algorithnw. Proofs of P-completeness may also tell us which parts of problems are hard to parallelize. From the above arguments and the following result, we see that NC is a reasonable class. The P-completeness due to Cook (1985) adopted the NC1-reducibility that uses NC1-computable O(1og n) Theorem 2.1 (Stockmeyer and Vishkin 1984) depth uniform circuits with oracle gates. However, we use an extended version of the many-one log space re- ducibility (Hopcroft and Ull~nan1979) instead of the where CRCW-PRAM(~~(",O((log n)O("))) is the class NC1-reducibility because of the following reasons. It of problems solvable on a CRCW PRAM with polyno- is easier to find required log space computable functions mial number of processors in time O(jlogn)k) for some which are also computable in NC2. Above all, all known constant k 2 O . P-completeness of natural problems can also be showr, via log space reductions. In the arguments above, we have ignored constants appearing in 0-notation and the degree of poiynomi- Definition Let S and T be problems of size param- als. In parallel computations, constants affect very seri- eters t~(123 and h(nj, respectively.
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