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Sampling Arborescences in Parallel
For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75].
Point Processes
Data Structures and Algorithms
Combinatorics
Probability
Algorithms and Hardness for Linear Algebra on Geometric Graphs
We investigate whether or not it is possible to solve the following problems in $n^{1+o(1)}$ time for a $\mathsf{K}$-graph $G_P$ when $d < n^{o(1)}$: $\bullet$ Multiply a given vector by the adjacency matrix or Laplacian matrix of $G_P$ $\bullet$ Find a spectral sparsifier of $G_P$ $\bullet$ Solve a Laplacian system in $G_P$'s Laplacian matrix For each of these problems, we consider all functions of the form $\mathsf{K}(u, v) = f(\|u-v\|_2^2)$ for a function $f:\mathbb{R} \rightarrow \mathbb{R}$.
