In this thesis we study the application of a novel sampling technique to building fully-dynamic randomized graph algorithms. We present the following results: \begin{enumerate} \item A randomized algorithm to estimate the size of a cut in an undirected graph $G = (V, E)$ where $V$ is the set of nodes and $E$ is the set of edges and $n = |V|$ and $m = |E|$. Our algorithm processes edge insertions and deletions in $O(\log^2n)$ time. For a cut $(U, V\setminus U)$ of size $K$ for any subset $U$ of $V$, $|U| < |V|$ our algorithm returns an estimate $x$ of the size of the cut satisfying $K/2 \leq x \leq 2K$ with high probability in $O(|U|\log n)$ time. \item A randomized distributed algorithm for maintaining a spanning forest in a fully-dynamic synchronous network. Our algorithm maintains a spanning forest of a graph with $n$ nodes, with worst case message complexity $\tilde{O}(n)$ per edge insertion or deletion where messages are of size $O(\text{polylog}(n))$. For each node $v$ we require memory of size $\tilde{O}(degree(v))$ bits. This improves upon the best previous algorithm with respect to worst case message complexity, given by Awerbuch, Cidon, and Kutten, which has an amortized message complexity of $O(n)$ and worst case message complexity of $O(n^2)$. \end{enumerate} / Graduate / 0984 / b_mountjoy9@hotmail.com