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Complexity Analysis via Approach Spaces

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Abstract

Complexity of a recursive algorithm typically is related to the solution to a recurrence equation based on its recursive structure. For a broad class of recursive algorithms we model their complexity in what we call the complexity approach space, the space of all functions in X =  ]0, ∞ ]Y, where Y can be a more dimensional input space. The set X, which is a dcpo for the pointwise order, moreover carries the complexity approach structure. There is an associated selfmap Φ on the complexity approach space X such that the problem of solving the recurrence equation is reduced to finding a fixed point for Φ. We will prove a general fixed point theorem that relies on the presence of the limit operator of the complexity approach space X and on a given well founded relation on Y. Our fixed point theorem deals with monotone selfmaps Φ that need not be contractive. We formulate conditions describing a class of recursive algorithms that can be treated in this way.

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Authors and Affiliations

  1. Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussel, België

    E. Colebunders & S. De Wachter

  2. CEOL, University College Cork, Western Gateway Building, floor 2, Western Road, Cork, Ireland

    M. Schellekens

Authors
  1. E. Colebunders
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  2. S. De Wachter
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  3. M. Schellekens
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Corresponding author

Correspondence to E. Colebunders.

Additional information

S. De Wachter has a PhD fellowship of the Research Foundation Flanders (FWO).

M. Schellekens was supported by the Science Foundation Ireland (SFI) Grant, under Grant number: 07/IN.1/I977.

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Colebunders, E., De Wachter, S. & Schellekens, M. Complexity Analysis via Approach Spaces. Appl Categor Struct 22, 119–136 (2014). https://doi.org/10.1007/s10485-013-9302-2

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  • DOI: https://doi.org/10.1007/s10485-013-9302-2

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