Meetup summary

2024-12-20 - Special topic - pairing heaps

Today we’ll be covering pairing heaps, a probabilistic heap data structure which is simple and easy to implement, but which performs amazingly well in practice and is surprisingly difficult to analyze.

Agenda:

  • Introduce pairing heaps and their motivation.
  • Briefly discuss their appealing characteristics (simple to implement, practical in a purely applicative (but sadly, not fully persistent) setting, blazing fast).
  • Discuss some theoretical downsides, e.g., with respect to the optimal real-time asymptotic runtimes of Brodal queues or full persistence with leftist heaps.
  • Walk through a simple implementation and performance comparison with binary heaps.
  • Discuss the performance characteristics (both worst-case and amortized). Note that it’s the amortized performance that makes this data structure interesting, but the analysis is surprisingly tricky. We’ll go over an intuitive explanation of why it works out the way it does, but we won’t go into rigorous detail. Note that it took many years from the original publication to get the current best-bound on some operations. (For example, see this paper for a decent overview; I saw a handful of different paper over the decades claiming iterative improvement, but none of these have a strong practical implication given the operations we typically care about.)
  • Compare pairing heaps with splay trees (possibly a topic for a future session) to better understand the motivation behind pairing heaps and—in particular—why the merge-left operation is asymmetric with the pair-right operation.

Notes

  • We started out with an impromptu knots session and discussed:
    • The standard “right” bowline (ABoK 1010) and its virtues and vices (jam-resistant but vulnerable to cyclic loading).
    • The cowboy bowline (ABoK 1034.5); similar to standard bowline, but slightly more resistant to ring loading. People seem to like it because the tail “doesn’t get in the way” of the eye loop.
    • The virtues of the zeppelin bend.
    • How to encode arbitrary (starting with simple) knots as braids, with the end goal of programmatically enumerating all knots of a certain class and detecting duplicates under permutation, rotation, and/or reflection.
      • This was a very rudimentary discussion where we brainstormed some approaches but did not come up with any concrete or satisfactory algorithms. Planning to come back to this in a future session to iron things out.
      • My long-run hope is to connect this with AC by establishing some generating functions for the knot classes of interest. As I understand it, this is currently an unsolved problem in the general case. However, we may be able to exploit symmetries or asymptotic to get some interesting answers.
  • Went over the motivations for priority queues in general and the minimal subset of operations we care about for most purposes (insert and deleteMin, for a min-heap).
  • Discussed the memory layout for pairing heaps and implemented insert and deleteMin on the whiteboard.
  • Despite some literature claiming that binary heaps outperform pairing heaps when deleteMin is not used, I did not find this to be the case. In the experiments we ran, pairing heaps dramatically outperformed binary heaps for large datasets with interleaved inserts/deletions. This held even for small datasets.
  • tl;dr In your next programming competition, you should probably reach for a pairing heap any time you’re doing optimization with graph search. On top of the being performant, they’re very easy to implement correctly.
  • A vocal subset of the group apparently hates Haskell 😭. I might try to avoid it next time even though this and ML are probably the easiest/clearest way to spell out purely applicative data structures. I’m open to suggestions for other languages or concrete pseudo-code shapes. It’s good to at least have some fixed set of conventions to clearly convey the ideas.

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