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    Sorry, I know this is an old question/answer, but I've been trying to understand this last point you make. A hash table has O(1) time complexity. However, once you use a sparse array, aren't you left needing to do a binary search to find your value? At that point doesn't the time complexity become O(log n)?
    – herbrandson
    Commented Apr 28, 2016 at 14:50
  • @herbrandson: no... a sparse array simply means relatively few indices have been populated with values - you can still index directly to the specific array element for the hash value you've calculated from your key; still, the sparse array implementation Simon describes is only sane in very limited circumstances: when bucket sizes are of the order of memory page sizes (vs. say int keys at 1-in-1000 sparseness and 4k pages = most pages touched), and when the OS treats all-0 pages efficiently (so all-unused-bucket pages don't need backing memory), when address space is plentiful....
    – Tony Delroy
    Commented May 2, 2016 at 2:39
  • @TonyDelroy - that's true it is oversimplification but the idea was to give a overview of what they are and why, not a practical implementation. The details of the latter are more nuanced, as you nod to in your expansion.
    – simon
    Commented Feb 13, 2019 at 21:10