From Shapes to Amortized Complexity

Abstract

We propose a new method for the automated resource bound analysis of programs manipulating dynamic data structures built on top of an underlying shape and resource bound analysis. Our approach first constructs an integer abstraction for the input program using information gathered by a shape analyser; then a resource bound analyzer is run on the resulting integer program. The integer abstraction is based on shape norms -numerical measures on dynamic data structures (e.g., the length of a linked list). In comparison to related approaches, we consider a larger class of shape norms which we derive by a lightweight program analysis. The analysis identifies paths through the involved dynamic data structures, and filters the norms which are unlikely to be useful for the later bound analysis. We present a calculus for deriving the numeric changes of the shape norms, thereby generating the integer program. Our calculus encapsulates the minimal information which is required from the shape analysis. We have implemented our approach on top of the Forester shape analyser and evaluated it on a number of programs manipulating various list and tree structures using the Loopus tool as the underlying bounds analyser. We report on programs with complex data structures and/or using complex algorithms that could not be analysed in a fully automated and precise way before.

Complexity-guided container replacement synthesis

“…Checking the complexity superiority requires precise reasoning of program complexity. However, deriving a tight bound of program complexity is stunningly difficult [Wilhelm et al 2008] and far from practical for a real-world program [Gulwani et al 2009a;Xie et al 2016], especially for programs involving sophisticated manipulations of data structures [Fiedor et al 2018;Gulwani et al 2009c;Lu et al 2021;Srikanth et al 2017]. Although the container complexity superiority does not imply the complexity superiority, it provides the effective guidance to find the opportunity of synthesizing the replacements to improve program efficiency, as evidenced by our evaluation in ğ 7.…”

Rely-guarantee bound analysis of parameterized concurrent shared-memory programs

“…To make use of the wide range of existing sequential bound analyzers for integer programs (e.g., [4,6,9,12,20,23,36]), our method translates the pointer program I into an equivalent integer program I : Using the technique of [8], our algorithm translates the interleaved program with pointers I = P Instr(R) and predicate Inv ∧ g 0 into a bisimilar integer programÎ and predicate Inv ∧ g 0 . Alternatively, one could directly compute bounds on the pointer program I using techniques such as described in [3,17,37].…”

Type-Based Analysis of Logarithmic Amortised Complexity

“…We now overview alternatives to conducting amortised cost analysis through the means of a type-and-effect system. The line of work [16,[49][50][51]58] has focused on identifying abstractions resp. abstract program models that can be used for the automated resource analysis of imperative programs.…”

“…Some of the cited approaches are able to conduct an automated amortised analysis in the sense of Sleator and Tarjan: The work on type-based cost analysis by Martin Hofmann and his collaborators [22,24,26,[30][31][32][33][34][37][38][39], which we discuss in more detail below, directly employs potential functions as proposed by Sleator and Tarjan [52,53]. For imperative programs, a line of work infers cost bounds from lexicographic ranking functions using arguments that implicitly achieve an amortised analysis [16,[49][50][51] (for details we refer the reader to [51]). The connection between ranking functions and amortised analysis has also been discussed in the context of term rewrite systems [33].…”