# Problem 55: Applications of Clifford Algebras in Graph Streams

It is interesting to compare these results to the framework of designing randomized algorithms for computing the permanent. Let $A$ be a 0-1 matrix, and $B$ be the matrix obtained from $A$ by replacing each 1 uniformly and randomly with an element from a finite set $D$. With suitable choices of the set $D$, the determinant of $B$ can be used to approximate the permanent of $A$. As shown by Chien et al. [ChienRS-03] and discussed by Muthukrishnan [Muthukrishnan-06], by choosing elements of $D$ from $\mathbb Z$, $\mathbb C$, or a Clifford algebra, the variance of the estimator drops significantly each time when we move to a more “complex” algebra. It seems plausible that similar techniques can be used to improve the space complexity of graph streaming algorithms which are based on complex-valued random variables.