# 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

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 $${\mathcal {O}} (N)$$ $${\mathcal {O}} (N)$$ $${\mathcal {O}} (1)$$