New techniques and tighter bounds for local computation algorithms

• We introduce d-light graphs, a new family that includes graphs of constant bounded degree and Erdos–Renyi graphs.
• We develop new techniques for bounding the time and space requirements of LCAs.
• We use these techniques to develop algorithms for a large family of problems on d-light graphs.
• For example, we construct an LCA that requires O(log⁡nlog⁡log⁡n)O(log⁡nlog⁡log⁡n) space and O(log2⁡n)O(log2⁡n) time for MIS.
• We show that the above LCA requires O(log⁡log⁡n)O(log⁡log⁡n) time and O(log⁡n)O(log⁡n) space in expectation.

Given an input x and a search problem F, local computation algorithms (LCAs) implement access to specified locations of y   in a legal output y∈F(x)y∈F(x), using polylogarithmic time and space. Parnas and Ron [27] and Mansour et al. [19] showed how to convert certain distributed and online algorithms to LCAs, respectively. In this work, we expand on those lines of work and develop new techniques for designing LCAs and bounding their space and time complexity.