Abstract
An accumulation is a higher-order operation over structured objects of some type; it leaves the shape of an object unchanged, but replaces each element of that object with some accumulated information about the other elements. Upwards and downwards accumulations on trees are two instances of this scheme; they replace each element of a tree with some function—in fact, some homomorphism—of that element's descendants and of its ancestors, respectively. These two operations can be thought of as passing information up and down the tree.
We introduce these two accumulations, and show how together they solve the so-called prefix sums problem.
The author is grateful to Richard Bird, Geraint Jones, Lambert Meertens, David Skillicorn, and the anonymous referees, all of whom have made suggestions to improve the presentation of this paper.
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© 1993 Springer-Verlag Berlin Heidelberg
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Gibbons, J. (1993). Upwards and downwards accumulations on trees. In: Bird, R.S., Morgan, C.C., Woodcock, J.C.P. (eds) Mathematics of Program Construction. MPC 1992. Lecture Notes in Computer Science, vol 669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56625-2_11
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DOI: https://doi.org/10.1007/3-540-56625-2_11
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