Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, January 1998. ACM Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of exclusive methods. The amount for each of the benchmarks above is 0.25, 17, 1035, and 137 microseconds, respectively. 8 Related Work Parallel Execution of Associative Operations. There is an extensive literature on the techniques that extract parallelism among associative exclusive operations [3, 5, 9, 12, 18]. In systems using the techniques, each thread executes associative exclusive operations in parallel and accumulates the contributions of # S#### ##### ##### o oo#### # S#### ##### ##### o oo#### # # S### #### #### o oo### # o oo### #### #### Fig. 4. Number of ....

Zhenjiang Hu, Masato Takeichi, and Wei-Ngan Chin. Parallelization in Calculational Forms. In Proceedings of the 25th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL '98), pages 316--328, 1998.

....of exclusive methods. The amount for each of the benchmarks above is 0.25, 17, 1035, and 137 microseconds, respectively. 8 Related Work Parallel Execution of Associative Operations. There is an extensive literature on the techniques that extract parallelism among associative exclusive operations [3, 5, 9, 12, 18]. In systems using the techniques, each thread executes associative exclusive operations in parallel and accumulates the contributions of # S#### ##### ##### o oo#### ##o oo#S# #OE## # S#### ##### ##### o oo#### ##o oo#S# #OE## # # S### #### #### o oo### ##o oo#S# #OE## # ....

Zhenjiang Hu, Masato Takeichi, and Wei-Ngan Chin. Parallelization in Calculational Forms. In Proceedings of the 25th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL '98), pages 316--328, 1998.

....parallel program to solve this problem. 3.1 Problems in Programming Ecient Skeletal Programs Programming with skeletons eciently is hard because it requires a proper choice of skeletons and an ecientcombination of them. Ecient programming with these skeletons is a well known area of research [Bir87, Ski94, Col95, GDH96, Gor96a, HIT97, HTC98]. Using skeletons, one often tries to solve a problem by composition of several passes so that each pass can be described in terms of a parallel skeleton. Considering the subproblem los1 whichjustchecks whether the points in a single ray ps are visible or not from the observation point p # , one ....

....have to be distributed to processors and each result must be gathered to the master processor. Another problem is that even for a simpler subproblem likePass 2 (a) solving it in terms of skeleton is actually not an easy task. One approach to cope with this problem is to derive a homomorphism [Col95, GDH96, Gor96a, HIT97, HTC98], resulting in skeletal parallel programs in the form of # = k#. Its theoretical foundation is the following homomorphism lemma. De nition 1 (Homomorphism) Function h is a homomorphism if it is de ned by h [ # h [a] f a h (x y) hx#hy where # is a binary operator whose unit is ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316{ 328, San Diego, California, USA, January 1998.

....parallel program to solve this problem. 3.1 Problems in Programming Ecient Skeletal Programs Programming with skeletons eciently is hard because it requires a proper choice of skeletons and an ecient combination of them. Ecient programming with these skeletons is a well known area of research [Bir87, Ski94, Col95, GDH96, Gor96a, HIT97, HTC98]. Using skeletons, one often tries to solve a problem by composition of several passes so that each pass can be described in terms of a parallel skeleton. Considering the subproblem los1 which just checks whether the points in a single ray ps are visible or not from the observation point p 0 , one ....

....have to be distributed to processors and each result must be gathered to the master processor. Another problem is that even for a simpler subproblem like Pass 2 (a) solving it in terms of skeleton is actually not an easy task. One approach to cope with this problem is to derive a homomorphism [Col95, GDH96, Gor96a, HIT97, HTC98], resulting in skeletal parallel programs in the form of = k . Its theoretical foundation is the following homomorphism lemma. De nition 1 (Homomorphism) Function h is a homomorphism if it is de ned by h [ h [a] f a h (x y) h x h y where is a binary operator whose unit is ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316{ 328, San Diego, California, USA, January 1998.

....(mutex) # OE## ### OE## o oo # # # o oo ww w w w (c) Amdahl (detach) # OE## ### OE## o oo # # # o oo ww w w w (d) Amdahl (detach fusion) Figure 4.13: Breakdowns of execution times of ImageViewer. 105 parallelism [FG94, HAM 95, Ope98] In addition, Hu et al. HTC98] proposed a formal and general technique for program transformations that make iterations expressed as recursions be executed efficiently in a manner similar to the execution of reductions. All these techniques can only be applied to the regular programming models in which it is obvious at what ....

Zhenjiang Hu, Masato Takeichi, and Wei-Ngan Chin. Parallelization in Calculational Forms. In Proceedings of the 25th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL '98), pages 316--328, San Diego, USA, January 1998. ACM. 151

....we can only get a subset of the shapely functions resp. expressions. For the analysis of higher order functions we propose a higher order analysis where shapes as functions are possible. A related approach to parallel programming is the Bird Meertens Formalism (BMF) Ski94,Ski92,Gor97,HTC98] Here a program is transformed from an abstract specification to an efficient parallel program, where the parallelization is done via homomorphisms 2 defined over the data structures, i.e. the structure of algebraic or categorical data types. To analyse real program libraries we have to ....

Zhenjiang Hu, Masato Takeichi, and Wei-Ngan Chin. Parallelization of calculational forms. In Proc. POPL'98. ACM Press, Jan 1998.

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Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, January 1998. ACM Press.

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Z. Hu, M. Takeichi, and W.N. Chin, "Parallelization in calculational forms", In Proceedings of the 25th ACM Symposium on Principles of Programming Languages, San Diego, California, USA, January 1998, pages 316--328.

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Hu, Z., Takeichi, M., Chin, W.: Parallelization in calculational forms. In: 25th ACM Symposium on Principles of Programming Languages, San Diego, California, USA (1998) 316--328

....f 3 (a : x ) let v = a (f 3 x) in v v Function definitions in this paper are written in Haskell syntax. For the rest of the paper, we shall discuss detecting parallelism for recursive functions of the form where f is inductively defined on a list. This form was first described in [11]. E [ denotes an expression context with three groups of holes ##. It contains no occurrence of references to a , x and f . #t i=1 is a group of m terms, each of which is allowed to contain occurrences of a, but not those of references to (f x ) q i x j =1 denotes a group of n ....

....occurrence of references to a , x and f . #t i=1 is a group of m terms, each of which is allowed to contain occurrences of a, but not those of references to (f x ) q i x j =1 denotes a group of n function applications, each of which is a mutumorphism (aka. parallelized function, c.f. [11]) Lastly, f x# is the self recursive call. For ease of presentation, we consider the following simplifications to our language that can be overcome in our full implementation. Each recursive function has only one recursion parameter located at position p0 ; the rest of the parameters are ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, January 1998. ACM Press.

.... of some optimization passes of compilers [GLJ93, OHIT97] Particularly, it has been shown that many important program transformations suchasdeforestation (or fusion) tupling transformation, parallelization and accumulation can effectively and elegantly be formalized in calculational form [TM95, HIT96, HITT97, HTC98, HIT99]. Why to Code Calculation We believe that it is both worthwhile and challenging to provide a flexible mechanism to code program calculations, and to make such calculations be part of programs. There are two main reasons. ffl Coding calculation can help programmers to document and reuse their ....

.... tupling = x: f x; g x) e 1 = f [ a 8 (f x; g x) f (a : x) e 2 = g [ f x; g x) g (a : x) in let a fi (x; y) a 8 (x; in foldr (fi) e 1 ;e 2 ) Following this way, it should not be difficult to code other interesting calculation laws such as the parallelizing theorem [HTC98], the accumulation calculation theorem [HIT99] and the diffusion calculation law [HTI99] 4.2 Coding Calculation Strategies Valid transformations on program can be described by a set of calculation rules; while calculation strategies are applied to obtain the desired optimization effects ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, USA, January 1998.

....to optimization of functional programs. It was first proposed as the theory of lists [5] and was then extended to be a general theory of datatypes [7] It has proved to be very useful not only in deriving variant e#cient programs [23] but also in constructing optimization passes in compilers [36, 21, 22]. Our formulating program analysis in this framework enables utilization of existing calculation techniques. There are several works on catamorphic approach to computation on graphs. For instance, 15] treats graphs with embedded functions, i.e. graphs are treated as functions that generates all ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, USA, January 1998.

....our use of correctnesspreserving transformation steps. We are working on a survey of other existing algorithms for the frequent set problem. This work is a continuation of our effort to apply calculational transformation techniques [THT98] to the development of efficient programs [OHIT97, HITT97, HTC98] Our previous work put emphasis on mechanical implementation of the transformation techniques, while this paper aims to show that modular transformation strategy is also very helpful for guiding programmers researchers in development of new algorithms. Acknowledgments This paper owes much to ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316-- 328, San Diego, California, USA, January 1998.

....equations. At this point, two questions may puzzle the reader : How do we obtain such snoc based equations When should we use them The snoc based equations can be obtained as a by product of parallelization. Given a cons based equation, the inductive parallelization method presented in [HTC98] is capable of (automatically) deriving a based parallel equation. This can subsequently be instantiated to the snoc based equation. As an example, consider the cons based version of inits function given in Sec 1. Using the method of [HTC98] it is possible to derive the following based ....

....the inductive parallelization method presented in [HTC98] is capable of (automatically) deriving a based parallel equation. This can subsequently be instantiated to the snoc based equation. As an example, consider the cons based version of inits function given in Sec 1. Using the method of [HTC98], it is possible to derive the following based parallel equation. inits(xs ys) inits(xs) map( xs ) inits(xs) By instantiating ys to [y] we can obtain the following snoc based equation. inits(xs [y] inits(xs) xs [y] The second question is when should we use such snoc based ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316-328, San Diego, California, January 1998. ACM Press.

....have to be distributed to processors and each result must be gathered to the master processor. As a matter of fact, even for a simpler subproblem like Pass 2 (a) solving it in terms of skeleton is actually not an easy task. One approach for coping with this problem is to derive a homomorphism [Col95,Gor96,HTC98], resulting in skeletal parallel programs in the form of ( f ) Though homomorphisms may deal well with programming using map and reduce skeletons, they cannot directly support programming with the skeleton of scan [Ble89] which uses an accumulating parameter. 3.2 Skeleton Composition As ....

....found in Section 4.4. 7 4.2 Parallelizability of accumulate To see that accumulate is indeed a parallel skeleton, we will show that it can be implemented eciently in parallel. As a matter of fact, the recursive de nition for accumulate belongs to the class of parallelizable recursions de ned in [HTC98]. The following theorem gives the resulting parallel version for accumulate. Theorem 1 (Parallelization) The function accumulate de ned in De nition 1 can be parallelized to the following divide and conquer program. accumulate [ e = g e accumulate x e = fst (accumulate 0 x e) accumulate ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316-328, San Diego, California, USA, January 1998.

....sets from processor P i 1 to P i for all i, to see whether these single item frequent sets in P i 1 could be merged with frequent sets computed in P i . Note that this parallel algorithm could be formally obtained from the sequential program tab in Figure 1 by parallelization calculation [HTC98] which is omitted here. 4.5 Implementation The derived algorithm can be used practically to win over the existing algorithms, because of the single traversal of database and much less use of the costly operation for checking of subset relationship. To be able to compare our results more ....

....fusion, accumulation, lter promotion and tabulation. These calculation techniques are quite well known in the functional programming community. This work is a continuation of our e ort to apply calculational transformation techniques [THT98] to the development of ecient programs [OHIT97, HITT97, HTC98] Our previous work put emphasis on mechanical implementation of the transformation techniques, while this paper shows that this calculation strategy is also very helpful for guiding programmers researchers in the development of new algorithms. Our derived frequent set al..gorithm compares very ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316-328, San Diego, California, USA, January 1998.

No context found.

) Hu, Z., Takeichi, M., and Chin, W. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages (San Diego, California, USA, Jan. 1998), pp. 316--328.

....using the same technique to develop a new piece of code, there is no way he can reuse the development recorded as a comment. Second, coding calculation can help to mechanize derivation of efficient programs. Many calculation laws and theorems such as fusion [32] tupling [18] and parallelization [20] have been developed, but few of them have been fully implemented in practical compilers. There are two major difficulties. First, even for a simple calculation law like the cheap fusion in [15, 32] one cannot code it as naturally as expressed in the paper. Rather one has to take pain to design ....

.... a a simpleIf if Ture then e 1 else e 2 = e 1 simpleIf if F lase then e 1 else e 2 = e 2 which performs static evaluation of if expression. Following this way, it should not be difficult to code other interesting calculation laws such as the tupling law [18] parallelizing theorem [20], the accumulation calculation theorem [17] and the diffusion calculation law [21] 4.2 Coding Calculation Strategies Valid transformations on program can be described by a set of calculation rules; while calculation strategies are applied to obtain the desired optimization effects [33] In ....

Z. Hu, M. Takeichi, and W. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, USA, Jan. 1998.

....recursion, and to bridge the gap between natural definitions using recursions and definitions using parallel primitives. It includes as its special case the well known homomorphism lemma [Bir87] which has served as the basis for deriving parallel programs on lists [Col95, GDH96, Gor96b, HIT97, HTC98] The key idea to establish our theorem is an essential use of scans to memoize intermediate results in parallel computation. ffl Our polytypic framework can provide both explicit and implicit way to describe parallelism, supporting both mechanical implementation and flexible programming. In ....

....2. 2 Parallel Programming in BMF Besides the work [Ski90, Ski94] on looking for architecture independent parallel implementation of some specific catamorphisms, studies on parallel programming in BMF are actually quite recently, focusing mainly on list functions as in [Col95, GDH96, Gor96b, HIT97, HTC98] The main idea is to derive the so called List homomorphisms [Bir87] which are nothing more than catamorphisms on join lists as defined above. The relevance of homomorphisms to parallel programming is basically from the homomorphism lemma [Bir87] cata JList 8 k (8) 8= ffi (k3) where 8 ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316-- 328, San Diego, California, USA, January 1998.

....equations. At this point, two questions may puzzle the reader How do we obtain such snoc based equations And when should we use them The snoc based equations can be obtained as a by product of parallelization. Given a cons based equation, the inductive parallelization method presented in [HTC98] is capable of (automatically) deriving a based parallel equation. This can subsequently be instantiated to the snoc based equation. As an example, consider the cons based version of inits function given in Sec 1. Using the method of [HTC98] it is possible to derive the following based ....

....the inductive parallelization method presented in [HTC98] is capable of (automatically) deriving a based parallel equation. This can subsequently be instantiated to the snoc based equation. As an example, consider the cons based version of inits function given in Sec 1. Using the method of [HTC98], it is possible to derive the following based parallel equation: inits(xs ys) inits(xs) map( xs ) inits(xs) By instantiating ys to [y] we can now obtain the following snoc based equation: inits(xs [y] inits(xs) xs [y] 11 Our second question was when should we use such ....

Z. Hu, M. Takeichi, and WN. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, San Diego, California, January 1998. ACM Press (to appear).

....calls to constructively transform programs to match these schemes, but these proposals [Roe91, GDH96] often require deep intuition and the support of ad hoc lemmas making automation difficult. Another approach is to provide more specialised schemes, either statically [PP91] or via a procedure [HTC98] that can be directly matched to sequential specification. Though cheap to operate, the generality of this approach is often called into question. On the imperative language (e.g. Fortran) front, there have been significant interests in the parallelization of reduction style loops. A work ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, San Diego, California, January 1998. ACM Press (to appear).

....which a solution to the bracket matching problem can be specialized, but it has linear time behavior in the worst case. It would be more convincing if we could derive an optimal homomorphic algorithm. This paper shows how these two problems can be well solved based on the parallelization theorem [15]. In particular, we will propose a systematic and formal derivation of, to the best of our knowledge, a novel optimal homomorphic algorithm for bracket matching. 2 List Homomorphisms List homomorphisms, an important concept in Bird Meertens Formalisms (BMF for short) 2] play a central role in ....

....for the singleton list with element a, and x y for the concatenation of two lists x and y. The term [1] 2] 3] denotes a list with three elements, often abbreviated to [1; 2; 3] We also write a : xs for [a] xs. List homomorphisms become more and more attractive in parallel programming [2, 6, 9, 10, 11, 13, 15], mainly because of their distinguished properties of simplicity, clear parallelism, and manipulability. ffl First, they are the simplest recursive skeletons on join lists; function h is a list homomorphism 1 , if there exist a function k and an associative binary operator 8 so that h is ....

[Article contains additional citation context not shown here]

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, USA, January 1998.

....calls to constructively transform programs to match these schemes, but these proposals [Roe91,GDH96] often require deep intuition and the support of adhoc lemmas making automation difficult. Another approach is to provide more specialised schemes, either statically [PP91] or via a procedure [HTC98] that can be directly matched to sequential specification. Though cheap to operate, the generality of this approach is often called into question. On the imperative language (e.g. Fortran) front, there have been interests in parallelization of reduction style loops. A work similar to ours was ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, January 1998. ACM Press.

....equations of depart, uH dep and uG dep satisfy the auxiliary requirement of Theorem 2, and may themselves be used as auxiliary calls for other functions. The key idea of sharing auxiliary recursive calls via parameter abstraction (during inductive derivation) is an innovation first introduced in [19]. We refine this technique by addressing the type of accumulative parameters allowable for such auxiliary functions. 6 Sample Preservable R Contexts We have presented two theorems which highlight sufficient conditions for deriving efficient parallel programs. This result is quite general as it ....

....from first principle (e.g. via context preservation modulo replication) another approach for improving parallelization is to develop more specialised program schemes that could provide a better match up with common sequential programs. A step in this direction has recently been formalised in [19]. By relying on a novel synthesis lemma (which combines the requirements of both generalization and inductive derivation) a good set of commonly used schemes can be recognized by a parallelization algorithm in calculating parallel programs. The main advantage of this approach is a cheap and ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th Annual ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, January 1998. ACM Press.

....frequent sets from processor P i 1 to P i for all i, to see whether these single item frequent sets in P i 1 could be merged with frequent sets computed in P i . Note that this parallel algorithm can be obtained directly from the sequential program tab in Figure 1 by parallelization calculation [HTC98] which is omitted here. Practical Issues The derived algorithm can be used practically to win over the existing algorithms. To be able to compare our results more convincingly with those in data mining field, we are mapping the algorithm to a C program and testing it on the popular benchmark ....

Z. Hu, M. Takeichi, and W.N. Chin. Parallelization in calculational forms. In 25th ACM Symposium on Principles of Programming Languages, pages 316--328, San Diego, California, USA, January 1998.

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