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Competitive programming in Haskell: better binary search
- 082349872349872 4y agocf EWD1293: https://www.cs.utexas.edu/users/EWD/ewd12xx/EWD1293.PDF https://www.cs.utexas.edu/users/EWD/ewd12xx/EWD1293.PDF (note that binary search is an example of computer scientists acknowledging the existence of practitioners: they won't go so far as to do a k-ary search, for that would require engineering a [necessarily ephemeral] suitable value of k, but they have demonstrated goodwill in stepping beyond the much-cheaper-to-prove 1-ary search.)
- vanderZwan 4y agoYou're making me wonder if it would make sense to have k scale with r (where r the narrowed down range). In practice binary search tends to perform better if we switch to linear search for small enough r, right? Kind of like how insertion sort is faster when r is less than 20-40 integers on modern hardware (this is assuming we're dealing with dense arrays in either scenario, obviously). One could say that this is effectively switching from k = 2 to k = r at some threshold value. What if instead we increase k gradually as some proportion of r as the latter gets smaller, and sample in ascending order (Although I bet that even if it works, interpolation search is probably faster anyway) [0] https://en.wikipedia.org/wiki/Interpolation_search https://en.wikipedia.org/wiki/Interpolation_search
- moonchild 4y ago> we can also do exact binary search on the bit representations of floating-point numbers! That is, we do binary search as if the bit representations of l and r were unsigned integers. This is possibly more efficient than “continuous” binary search, and lets us find the two precisely adjacent floating-point numbers where our predicate switches from False to True. Why is it better to use integer operations than with floating-point operations? Assuming everything is inlined and type-inferred away, float math should be faster, and just as precise. One may argue that the convergence rate is less predictable; but this may be a feature, if the predicate under consideration is not quite monotonic (monotonicity is hard to guarantee when negative numbers are involved--though '\x -> x^2 >= 150' is straight up not monotonic, both mathematically and numerically). > It even works when l is negative and r is positive (it seems like in that case the bit representation of l would correspond to a larger unsigned integer than r, but somehow it all works anyway!). It's because of underflow: ghci> (unsafeCoerce (-100.0) :: Word) - (unsafeCoerce 100.0 :: Word) 9223372036854775808 For Word, the condition 'r - l > 1' is true in all cases where l>r.
- tromp 4y ago> For Word, the condition 'r - l > 1' is true in all cases except when r=l. Simplest counterexample: (l,r) = (0,1) > For Word, the condition 'r - l > 1' is true in all cases where l>r. Simplest counterexample: (l,r) = (1,0), although one should always have l <= r.
- babel_ 4y agoThe "black magic" of binaryFloat is a known type pun to a sign-magnitude integer, so because it searched for positive values it always worked, albeit it took a few steps to converge, however for negative values it is not correctly handling it and will explode --- it also won't return NaN for sqrt of negative x, though it explodes well before that becomes a problem. Negative l positive r works because the first binary step will halve it, thereby losing the top sign bit (being equivalent to a right shift), and basically restarting the search with an l of 0.0 (or rather some very small float near it). I recommend printing each step out with binary representation, to see the working. The spec covers this binary shennanigans under comparison and total ordering, though it relies on some NaN behaviour being followed (754-2008 recommended's is_quiet flag behaviour). See Herf's old post for the proper bit twiddling https://web.archive.org/web/20220326204603/http://stereopsis.com/radix.html https://web.archive.org/web/20220326204603/http://stereopsis... (wayback as their webserver is down). (This is an expanded version of my own comment on the page, since it seems stuck in "awaiting moderation" limbo, just in case it shows up later while I'm not here.)