Vassilevska W. [STOC 13] show that inÕ (m √ n) time, one can compute for each v ∈ V in an undirected graph, an estimate e (v) for the eccentricity (v) such that max {R, 2 /3 • (v)} ≤ e (v) ≤ min {D, 3 /2 • (v)} where R = minv (v) is the radius of the graph. Here we improve the approximation guarantee by showing that a variant of the same algorithm can achieve estimates (v) with 3 /5 • (v) ≤ (v) ≤ (v).

A classic result in the analysis of data structures is that path compression with linking by rank solves the disjoint set union problem in almost-constant amortized time per operation. Recent experiments suggest that in practice, a naïve linking method works just as well if not better than linking by rank, in spite of being theoretically inferior. How can this be? We prove that randomized linking is asymptotically as efficient as linking by rank. This result provides theory that matches the experiments, which implicitly do randomized linking as a result of the way the input instances are generated.

The theory community has proposed several new heap variants in the recent past which have remained largely untested experimentally. We take the field back to the drawing board, with straightforward implementations of both classic and novel structures using only standard, well-known optimizations. We study the behavior of each structure on a variety of inputs, including artificial workloads, workloads generated by running algorithms on real map data, and workloads from a discrete event simulator used in recent systems networking research. We provide observations about which characteristics are most correlated to performance. For example, we find that the L1 cache miss rate appears to be strongly correlated with wallclock time. We also provide observations about how the input sequence affects the relative performance of the different heap variants. For example, we show (both theoretically and in practice) that certain random insertion-deletion sequences are degenerate and can lead to misleading results. Overall, our findings suggest that while the conventional wisdom holds in some cases, it is sorely mistaken in others.

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