A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to polynomial-time computable functions ƒr: {1, … , 2k} → {1, … , 2k}. These ƒr's cannot be distinguished from random functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory.

How to construct random functions

A constructive theory of randomness for functions, based on computational complexity, is developed, and a pseudorandom function generator is presented. This generator is a deterministic polynomial-time algorithm that transforms pairs (g, r), where g is any one-way function and r is a random k-bit string, to polynomial-time computable functionsf,: { 1, . . . , 2') + { 1, . . . , 2kl. Thesef,'s cannot be distinguished from random functions by any probabilistic polynomial-time algorithm that asks and receives the value of a function at arguments of its choice. The result has applications in cryptography, random constructions, and complexity theory. Categories and Subject Descriptors: F.0 (Theory of Computation): General; F. 1.1 (Computation by Abstract Devices): Models of Computation-computability theory; G.0 (Mathematics of Computing): General; G.3 (Mathematics of Computing): Probability and Statistics-probabilistic algorithms; random number generation

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Almost everywhere high nonuniform complexity

We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized $\Omega(\lg n)$ lower bound per operation for several data structural problems on $n$ elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of $\Omega(\lg n\,/\lg\lg n)$ for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first $\Omega(\log_B n)$ lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update.

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Logarithmic Lower Bounds in the Cell-Probe Model

We develop a new technique for proving cell-probe lower bounds on dynamic data structures. This technique enables us to prove an amortized randomized Omega(lg n) lower bound per operation for several data structural problems on n elements, including partial sums, dynamic connectivity among disjoint paths (or a forest or a graph), and several other dynamic graph problems (by simple reductions). Such a lower bound breaks a long-standing barrier of Omega(lg n / lglg n) for any dynamic language membership problem. It also establishes the optimality of several existing data structures, such as Sleator and Tarjan's dynamic trees. We also prove the first Omega(log_B n) lower bound in the external-memory model without assumptions on the data structure (such as the comparison model). Our lower bounds also give a query-update trade-off curve matched, e.g., by several data structures for dynamic connectivity in graphs. We also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update.

Two methods are given for obtaining lower bounds on the cost of accessing a sequence of nodes in a symmetrically ordered binary search tree, in a model where rotations can be done on the tree and the entire sequence is known before accessing begins (but the accesses must be done in the order given). For example, it can be proven that the bit-reversal permutation requires $\Theta (n\log n)$ time to access in this model. It is also shown that the expected cost of accessing random sequences in this model is the same as it is for the case where the tree is static.

Lower bounds for accessing binary search trees with rotations

A language L is random with respect to a given complexity class C if for all ′ ∈ C L and ′ disagree on half of all strings. It is known that for any complexity class there are recursive languages that are random with respect to that class. Here it is shown that there are tight space and time hierarchies of random languages, and that EXPTIME contains P-isomorphism classes containing only languages that are random with respect to polynomial-time computations. The technique used is extended to show that for any constructible bound on time or space it is possible to deterministically generate binary sequences that appear random to all prediction algorithms subject to the given resource bound. Furthermore, the generation of such a sequence requires only slightly more resources than the given bound.

Randomness and the density of hard problems

Lower Bounds for Accessing Binary Search Trees With Rotations (Preliminary Version)

We describe two methods for obtaining lower bounds on the cost of accessing a sequence of nodes of a symmetrically ordered binary search tree, where rotations can be done on the tree. The bounds apply to offline as well as online algorithms.

Robert E. Wilber

Papers

Lower bounds for accessing binary search trees with rotations

Randomness and the density of hard problems

Lower Bounds for Accessing Binary Search Trees With Rotations (Preliminary Version)

Lower bounds for accessing binary search trees with rotations