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Revision #1 to TR17-019 | 21st February 2017 12:06

A Unified Method for Placing Problems in Polylogarithmic Depth

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Revision #1
Authors: Andreas Krebs, Nutan Limaye, Michael Ludwig
Accepted on: 21st February 2017 12:06
Downloads: 113
Keywords: 


Abstract:

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem etc.. Many variants of the problems where the algebra is well behaved have been studied. For example, the problem over the Boolean semiring or over the semiring (Z, +, ×). Here, we allow the algebra to be completely general and obtain a bound for the term evaluation problem. We consider the problem of deriving upper bounds in terms of polylogarithmically deep circuits. To that end we present a generic term evaluation algorithm that works in polylogarithmic depth.

This efficient term evaluation algorithm over a very general algebra then
serves as a tool for showing polylogarithmic time upper bounds for various well-studied problems. To underline the utility of our result we show new bounds and reprove known results using our approach and thereby present a unified proof approach for problems of this nature. The spectrum of problems for which we apply our term evaluation algorithm is wide: in particular, the application of the algorithm we consider include (but are not restricted to) arithmetic formula evaluation, word problems for tree and visibly pushdown automata, and various problems related to bounded tree-width and clique-width graphs.



Changes to previous version:

Multiple typos fixed.


Paper:

TR17-019 | 8th February 2017 16:32

A Unified Method for Placing Problems in Polylogarithmic Depth





TR17-019
Authors: Andreas Krebs, Nutan Limaye, Michael Ludwig
Publication: 10th February 2017 01:37
Downloads: 211
Keywords: 


Abstract:

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem etc.. Many variants of the problems where the algebra is well behaved have been studied. For example, the problem over the Boolean semiring or over the semiring (Z, +, ×). Here, we allow the algebra to be completely general and obtain a bound for the term evaluation problem. We consider the problem of deriving upper bounds in terms of polylogarithmically deep circuits. To that end we present a generic term evaluation algorithm that works in polylogarithmic depth.

This efficient term evaluation algorithm over a very general algebra then
serves as a tool for showing polylogarithmic time upper bounds for various well-studied problems. To underline the utility of our result we show new bounds and reprove known results using our approach and thereby present a unified proof approach for problems of this nature. The spectrum of problems for which we apply our term evaluation algorithm is wide: in particular, the application of the algorithm we consider include (but are not restricted to) arithmetic formula evaluation, word problems for tree and visibly pushdown automata, and various problems related to bounded tree-width and clique-width graphs.



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