Irregular algorithms are organized around pointer-based data structures such as graphs and trees, and they are ubiquitous in applications. Recent work by the Galois project has provided a systematic approach for parallelizing irregular applications based on the idea of optimistic or speculative execution of programs. However, the overhead of optimistic parallel execution can be substantial. In this paper, we show that many irregular algorithms have structure that can be exploited and present three key optimizations that take advantage of algorithmic structure to reduce speculative overheads. We describe the implementation of these optimizations in the Galois system and present experimental results to demonstrate their benefits. To the best of our knowledge, this is the first system to exploit algorithmic structure to optimize the execution of irregular programs.