AbstractWe give a number of new lower bounds in the cell probe model with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations.

We study the signed prefix sum problem: given a string of lengthnof 0s and signed 1s, compute the sum of itsith prefix during updates. We show a lower bound of Omega(logn/log logn) time per operations, even if the prefix sums are bounded by logn/log lognduring all updates. We also show that if the update time is bounded by the product of the worst-case update time and the answer to the query, then the update time must be Omega(sqrt(logn/log logn)).

These results allow us to prove lower bounds for a variety of seemingly unrelated dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of Omega(logn/log logn) per operation. We give a lower bound for dynamic transitive closure on upward planar graphs with one source and one sink of Omega(logn/(log logn)^{2}) per operation. We give a lower bound of Omega(sqrt(logn/log logn)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same lower bound for the dynamic word problem for the free group withkgenerators. We also give lower bounds for certain range searching variants and for the dynamic prefix majority and prefix equality problems.

Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of Computation; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems

Additional Key Words and Phrases: cell probe model, lower bounds, dynamic graph algorithms, Dyck languages

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