This work concerns the problem of broadcasting a large message efficiently when each processor has partial prior knowledge about the contents of the broadcast message. The partial information held by the processors might be out of date or otherwise erroneous, and consequently, different processors may hold conflicting information. The problem of Broadcast with Partial Knowledge (BPK) was introduced in the context of Topology Update - the task of updating network nodes about the network topology after topological changes. Awerbuch, Cidon, and Kutten gave a message optimal solution for BPK, yielding a message optimal Topology Update algorithm. However, the time complexity of both algorithms was not optimal. The time complexity was subsequently improved in two follow up papers, but the best known time complexity was still higher than optimal by at least a logarithmic factor. tn this paper we present a time-optimal, communication-optimal algorithm for BPK. The algorithm is randomized, and, similar to previous randomized algorithms, it does not require the additional knowledge assumptions essential for deterministic solutions. In addition to the theoretical interest in optimality, a logarithmic factor is often important in practice, especially when using the procedure as a component within a periodically activated Topology Update algorithm.