Abstract. Integer time series are often subject to constraints on the aggregation of the integer features of all occurrences of some pattern within the series. For example, the number of inflexions may be constrained, or the sum of the peak maxima, or the minimum of the peak widths. It is currently unknown how to maintain domain consistency efficiently on such constraints. We propose parametric ways of systematically deriving glue constraints, which are a particular kind of implied constraints, as well as aggregation bounds that can be added to the decomposition of time-series constraints [5]. We evaluate the beneficial propagation impact of the derived implied constraints and bounds, both alone and together.
We introduce the concept of regular expression characteristics as a unified way to concisely express bounds on time-series constraints. This allows us not only to define time-series constraints in a compositional way, but also to deal with their combinatorial aspect in a compositional way, without developing ad-hoc bounds for each time-series constraint separately. IntroductionA time series is here a sequence of integers, corresponding to measurements taken over a time interval. Time series are common in many application areas, such as the output of electric power stations over multiple days [9], or the manpower required in a call-centre [5], or the daily capacity of a hospital clinic over a period of years. Time series are constrained by physical or organisational limits, which restrict the evolution of the series.We showed in [6] that many constraints γ( X 1 , X 2 , . . . , Xn , N ) on an unknown time series X = X 1 , X 2 , . . . , Xn can be specified in a compositional way by a triple σ, f, g , where σ is a regular expression over the alphabet Σ = {'<', '=', '>'} (we assume the reader is familiar with regular expressions [13]), while f ∈ {max, min, one, surf, width} is called a feature function, and g ∈ {Max, Min, Sum} is called an aggregator function. Volume II of the global constraint catalogue [4] contains 266 such functional time-series constraints. This is an extended version of parts of the CP 2016 paper [3] which involves a subset of the original authors. This paper introduces 6 new characteristics of regular expressions to express generic bounds on time-series constraints, which were not discussed in the original paper [3].1 are regular expressions that respectively describe the regular languages Lr 1 ∪ Lr 2 , Lr 1 ∩ Lr 2 , and L * r1 .Example 1 Consider the alphabet Σ = {'<', '=', '>'}.• Decreasing = '>' is a regular expression on Σ. The word v = '>' is a word of length 1 on Σ that belongs to L Decreasing , and it does not have any proper factors. The word '>>' is a word of length 2 on Σ, which does not belong to L Decreasing .• Inflexion = '<(<|=)*> | >(>|=)*<' is a regular expression on Σ. The word v = '>=<' is a word of length 3 on Σ that belongs to L Inflexion . The word υ has multiple proper factors, e.g., '>', '<'. The word '>=<<' does not belong to L Inflexion .Definition 2 A regular expression r is a non-fixed length regular expression if not all words of Lr have the same length.Example 2 We give two examples of regular expressions, a first one with a fixed length and a second one with a non-fixed length.• The Decreasing = '>' regular expression has a fixed length since L Decreasing contains a single word.• The Inflexion = '<(<|=)*> | >(>|=)*<' regular expression does not have a fixed length since L Inflexion contains words of different length.Definition 3 A regular expression over an alphabet A is disjunction-capsuled if it is in the form of 'r 1 r 2 . . . rp', where every r i (with i ∈ [1, p]) is, either a letter of the alphabet A, or a regular expression whose regular language contains the ...
We propose a systematic approach for generating linear implied constraints that link the values returned by several automata with accumulators after consuming the same input sequence. The method handles automata whose accumulators are increased by (or reset to) some non-negative integer value on each transition. We evaluate the impact of the generated linear invariants on conjunctions of two families of timeseries constraints. E. Arafailova is supported by the EU H2020 programme under grant 640954 for the GRACeFUL project. N. Beldiceanu is partially supported by GRACeFUL and by the Gaspard-Monge programme. H. Simonis is supported by Science Foundation Ireland (SFI) under grant numbers SFI/12/RC/2289 and SFI/10/IN.1/I3032.
We consider, for an integer time series, two families of constraints restricting the max, and the sum, respectively, of the surfaces of the elements of the sub-series corresponding to occurrences of some pattern. In recent work these families were identified as the most difficult to solve compared to all other time-series constraints. For all patterns of the time-series constraints catalogue, we provide a unique per family parameterised among implied constraint that can be imposed on any prefix/suffix of a time-series. Experiments show that it reduces both the number of backtracks/time spent by up to 4/3 orders of magnitude.
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