TY - JOUR

T1 - A short cut to parallelization theorems

AU - Morihata, Akimasa

PY - 2013/9/1

Y1 - 2013/9/1

N2 - The third list-homomorphism theorem states that if a function is both foldr and foldl , it has a divide-and-conquer parallel implementation as well. In this paper, we develop a theory for obtaining such parallelization theorems. The key is a new proof of the third list-homomorphism theorem based on shortcut deforestation. The proof implies that there exists a divide-and-conquer parallel program of the form of h (x 'merge' y) = h1 x - h2 y, where h is the subject of parallelization, merge is the operation of integrating independent substructures, h1 and h2 are computations applied to substructures, possibly in parallel, and - merges the results calculated for substructures, if (i) h can be specified by two certain forms of iterative programs, and (ii) merge can be implemented by a function of a certain polymorphic type. Therefore, when requirement (ii) is fulfilled, h has a divide-and-conquer implementation if h has two certain forms of implementations. We show that our approach is applicable to structure-consuming operations by catamorphisms (folds), structure-generating operations by anamorphisms (unfolds), and their generalizations called hylomorphisms.

AB - The third list-homomorphism theorem states that if a function is both foldr and foldl , it has a divide-and-conquer parallel implementation as well. In this paper, we develop a theory for obtaining such parallelization theorems. The key is a new proof of the third list-homomorphism theorem based on shortcut deforestation. The proof implies that there exists a divide-and-conquer parallel program of the form of h (x 'merge' y) = h1 x - h2 y, where h is the subject of parallelization, merge is the operation of integrating independent substructures, h1 and h2 are computations applied to substructures, possibly in parallel, and - merges the results calculated for substructures, if (i) h can be specified by two certain forms of iterative programs, and (ii) merge can be implemented by a function of a certain polymorphic type. Therefore, when requirement (ii) is fulfilled, h has a divide-and-conquer implementation if h has two certain forms of implementations. We show that our approach is applicable to structure-consuming operations by catamorphisms (folds), structure-generating operations by anamorphisms (unfolds), and their generalizations called hylomorphisms.

KW - Divide-and-conquer Parallelism

KW - Shortcut Deforestation

KW - Third List-homomorphism Theorem

UR - http://www.scopus.com/inward/record.url?scp=84888785987&partnerID=8YFLogxK

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M3 - Article

AN - SCOPUS:84888785987

VL - 48

SP - 245

EP - 256

JO - ACM SIGPLAN Notices

JF - ACM SIGPLAN Notices

SN - 1523-2867

IS - 9

ER -