Lower Bounds for Computations with the Floor Operation

We prove an Ω(√log n) lower bound on the depth of any decision tree with operations {+, −, *, /, ⌊·⌋, <}, that decides whether an integer is a perfect square, for any n-bit integer. We then extend the arguments to obtain the same lower bound on the time complexity of any RAM program with operations {+, −, *, /, ⌊·⌋, <} that solves the problem. Our proof technique can be used to derive lower bounds for many other problems. Another related result is described in a companion paper ([IBM Research Report RC 14271], [Proceedings of the 29th FOCS]).

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