Logarithmic Lower Bounds in the Cell-Probe Model

TL;DR

This paper introduces a novel technique for proving cell-probe lower bounds, establishing optimal bounds for various dynamic data structure problems and breaking longstanding barriers in the field.

Contribution

It presents a new method for proving lower bounds in the cell-probe model, achieving optimal bounds for several dynamic problems and extending results to external-memory models.

Findings

Proved an amortized Omega(log n) lower bound for dynamic data structures.

Established the first Omega(log_B n) lower bound in external-memory models.

Matched upper and lower bounds for partial sums based on word size and update magnitude.

Abstract

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.