Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(n 3/2 ) in the matrix model and in Θ( √ nm) in the array model, while the complexity of Connectivity is also in Θ(n 3/2 ) in the matrix model, but in Θ(n) in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

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We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Høyer, and Tapp, and imply an O(N 3/4 log N ) quantum upper bound for the element distinctness problem in the comparison complexity model (contrasting with Θ(N log N ) classical complexity). We also prove a lower bound of Ω( √ N ) comparisons for this problem and derive bounds for a number of related problems.