The Sudakov veto algorithm reloaded
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We perform a careful analysis of the main Monte Carlo algorithm used in parton shower simulations, the Sudakov veto algorithm. We prove a general version of the algorithm, directly including the dependence on the infrared cutoff. Taking this as a starting point, we then consider non-positive definite splitting kernels, as encountered when dealing with sub-leading colour correlations or splitting kernels beyond leading order. New algorithms suited for these situations are developed.