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This thesis investigates parallel algorithms for a small, but representative, subclass of graph and matrix problems. In some cases, we develop new algorithms which we analyze for theoretical efficiency. In other cases, we modify and implement existing algorithms which we analyze for practical efficiency. We show how n-node, e-edge graphs can be contracted in a manner similar to the parallel tree contraction algorithm due to Miller and Reif. We give an O((n+e)/lgn)-processor deterministic algorithm that contracts a graph in O(lg squared n) time in the EREW PRAM model.