Time-complexity. In the simulation, ’s overhead is dominated by com- puting (σi,j, Ri, Ai) for j = i and hj. Computing hj requires qH exponentiations in G (the overhead to sample a random element in G is about one exponenti- O O B ation). Computing σi,j requires (n2) exponentiations in G. Note that can compute Ai by an exponentiation in GT rather than a pairing computation. Computing Ri, Ai requires (n) exponentiations in G and GT , respectively. Let τExp denote the time complexity to compute one exponentiation without differ- entiation of exponentiations in different groups. Hence, the time complexity of B is τj = τ + O((qH + n2)τExp). H
Time-complexity. The main strength of the baseline method is its extremely cheap computation cost. The time complexity of the algorithm is O(n3r) for well-structured K, as derived in this section. To begin the derivation, we start by explicitly stating the steps in the algorithm. These can vary between implementations, but here we discuss what is currently known to be the most efficient general method for computing the TSVD approximation (3.1).
Time-complexity. In this section we show that the truncation method takes O(n3 +(l+m)n2r +k2r +k3) time to compute for image deconvolution problems. In practice, this is not better than the baseline, but it is nonetheless fast enough to be feasible for very large matrices. The steps of the truncation-based Kroncker TSVD approximation algorithm are as follows, with the first step the same as the baseline method: Σ
Time-complexity. In this section we show that the reordering method is computationally feasible, with an O(n3r + k2r + k3) running time for the image deconvolution problem. As with the basic truncation method, this is not as fast as the baseline, but is computationally feasible. This time complexity masks that care should be taken in certain steps to optimize performance for modern computer architectures; see Subsection 5.2.1 for details. Recall the steps of the reordering Kronecker-based TSVD algorithm:
Time-complexity. Consider an algorithm that solves consensus in an asyn- chronous system equipped with an eventual failure detector such as AΩ. In every execution, a correct leader process eventually emerges, but there is no bound on the time at which a correct process is elected. Obviously, the worst- case number of reads, or writes, performed by a process is unbounded. Thus, we measure the time complexity of asynchronous consensus algorithms in solo executions. Specifically, the individual write complexity (respectively the individ- ual step complexity) is the worst-case number of write operations (respectively the total number of read aand write operation) that occur in solo executions in which only one process participates and this process is the leader output by the failure detector from the beginning of the execution.
Time-complexity. The time complexity of our distance estimation methods are O(nms) where n is the number of participants, m is the number of targets, and s is the number of points (sampling points when fi is continuous) in each participant’s cloaked area. Algorithm 1 runs in O(nm) for the centroid-point method because the probabilities are either 0 or 1, so the algorithm finds the answer in one pass through all participant-target pairs. For the expected-probabilistic approach, the algorithm will run no more than R rounds for all participant-target pairs, so the the time complexity is O(Rnm).
Time-complexity. Algorithm 2 runs no more than R0 rounds of passing through all targets in ⌧ , so having the number of all targets as m, the time complexity is O(R0m).
