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Table 1 Theoretical complexities (in terms of time, space and total data transfers per unit time) of various algorithmic components of the particle filter with N data and P processors

From: MapReduce particle filtering with exact resampling and deterministic runtime

Section Algorithmic component Time Space Data transfers
4.1 Element-wise operations \({\mathcal {O}} (1)\) \({\mathcal {O}} (N)\) \({\mathcal {O}} (1)\)
4.2 Rotation \({\mathcal {O}} (1)\) \({\mathcal {O}} (N)\) \({\mathcal {O}} (1)\)
4.3 Sum/max/min \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}}(P)\)
4.4 Cumulative sum \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}}(P)\)
4.5 Normalising the weights \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}}{(P)}\)
4.6 Minimum variance resampling \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}}(P)\)
4.7 (Bitonic) sort \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{2}\right)\) \({\mathcal {O}}{(N)}\) \({\mathcal {O}}(P)\)
4.8.1 Redistribution from [9] \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{3}\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}} (P)\)
4.8.2 Improved redistribution \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{2}\right)\) \({\mathcal {O}} (N)\) \({\mathcal {O}} (P)\)
6.1.1 Naïve redistribution \({\mathcal {O}} (N)\) \({\mathcal {O}} (N)\) \({\mathcal {O}} (1)\)