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 |
|---|---|---|---|---|
Element-wise operations | \({\mathcal {O}} (1)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (1)\) | |
Rotation | \({\mathcal {O}} (1)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (1)\) | |
Sum/max/min | \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}}(P)\) | |
Cumulative sum | \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}}(P)\) | |
Normalising the weights | \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}}{(P)}\) | |
Minimum variance resampling | \({\mathcal {O}} \left (\frac {N}{P}\log N\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}}(P)\) | |
(Bitonic) sort | \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{2}\right)\) | \({\mathcal {O}}{(N)}\) | \({\mathcal {O}}(P)\) | |
Redistribution from [9] | \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{3}\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (P)\) | |
Improved redistribution | \({\mathcal {O}} \left (\frac {N}{P}(\log N)^{2}\right)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (P)\) | |
Naïve redistribution | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (N)\) | \({\mathcal {O}} (1)\) |