We study the algorithmic complexity of computing persistent homology of a randomly generated filtration. We prove upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on \v{C}ech, Vietoris--Rips and Erd\H{o}s--R\'enyi filtrations after matrix reduction, which in turn provide bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes, which we link to the fill-in of the boundary matrix. Our fill-in bounds for \v{C}ech and Vietoris--Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an Erd\H{o}s--R\'enyi filtration realising the worst-case fill-in and computation.

翻译：我们研究了计算随机生成滤集的持久同调的算法复杂度。我们证明了在矩阵约简后，\v{C}ech、Vietoris--Rips 和 Erd\H{o}s--R\'enyi 滤集的边界矩阵的平均填充量（非零条目数）的上界，这进而为条形码计算的期望复杂度提供了界限。我们的方法基于先前关于相应复形期望贝蒂数的结果，我们将其与边界矩阵的填充量联系起来。我们对于 \v{C}ech 和 Vietoris--Rips 复形的填充量界限在渐近意义下（至多差一个对数因子）是紧的。特别地，我们的填充量和计算量界限均优于最坏情况估计。我们还提供了一个实现最坏情况填充量和计算量的 Erd\H{o}s--R\'enyi 滤集。