Modular smoothed analysis

Spielman’s smoothed complexity - a hybrid between worst and average case complexity measures - relies on perturbations of input instances to determine where average-case behavior turns to worst-case. The paper proposes a method supporting modular smoothed analysis. The method, involving a novel permutation model, is developed for the discrete case, focusing on randomness preserving algorithms. This approach simplifies the smoothed analysis and achieves greater precession in the expression of the smoothed complexity, where a recurrence equation is obtained as opposed to bounds. Moreover, the approach addresses, in this context, the formation of input instances–an open problem in smoothed complexity. To illustrate the method, we determine the modular smoothed complexity of Quicksort

Modular average case analysis: Language implementation and extension

Motivated by accurate average-case analysis, MOdular Quantitative Analysis (MOQA) is developed at the Centre for Efficiency Oriented Languages (CEOL). In essence, MOQA allows the programmer to determine the average running time of a broad class of programmes directly from the code in a (semi-)automated way. The MOQA approach has the property of randomness preservation which means that applying any operation to a random structure, results in an output isomorphic to one or more random structures, which is key to systematic timing. Based on original MOQA research, we discuss the design and implementation of a new domain specific scripting language based on randomness preserving operations and random structures. It is designed to facilitate compositional timing by systematically tracking the distributions of inputs and outputs. The notion of a labelled partial order (LPO) is the basic data type in the language. The programmer uses built-in MOQA operations together with restricted control flow statements to design MOQA programs. This MOQA language is formally specified both syntactically and semantically in this thesis. A practical language interpreter implementation is provided and discussed. By analysing new algorithms and data restructuring operations, we demonstrate the wide applicability of the MOQA approach. Also we extend MOQA theory to a number of other domains besides average-case analysis. We show the strong connection between MOQA and parallel computing, reversible computing and data entropy analysis

Modular smoothed analysis of median-of-three Quicksort

Spielman’s smoothed complexity - a hybrid between worst and average case complexity measures - relies on perturbations of input instances to determine where average-case behavior turns to worst-case. This approach simplifies the smoothed analysis and achieves greater precession in the expression of the smoothed complexity, where a recurrence equation is obtained as opposed to bounds. Moreover, the approach addresses, in this context, the formation of input instances–an open problem in smoothed complexity. In [23], we proposed a method supporting modular smoothed analysis and illustrated the method by determining the modular smoothed complexity of Quicksort. Here, we use the modular approach to calculate the median of three variant and compare these results with those in [23]