Lower Bounds

Abstract

This book presents efficient algorithms for a large amount of important problems in ad hoc and sensor networks. Consider a problem $\mathcal{P}$ and the most efficient algorithm $\mathcal{A}$ (known so far) solving it with running time O(g(n)), where n is the number of nodes in the network. O(g(n)) is called upper bound on the running time for $\mathcal{P}$. After the problem is solved, the next step is to ask: Can the problem $\mathcal{P}$ be solved more efficiently? Is the running time of the algorithm $\mathcal{A}$ optimal?

If somebody devises a faster algorithm for $\mathcal{P}$, then the answers to both questions are obvious. However, it can also happen that no faster algorithm is suggested for some time. Then the only way to answer these questions is to establish a lower bound Ω(f(n)) on the running time of the algorithms for the problem $\mathcal{P}$. This task is by no means trivial, as it involves showing that even algorithms which have not been invented yet cannot be faster than Ω(f(n)).