... @@ -126,3 +126,19 @@ A sorting network, that sorts two sorted lists into one sorted list. ... @@ -126,3 +126,19 @@ A sorting network, that sorts two sorted lists into one sorted list. A recursive sorting network that starts with single values ($n$ sorted lists) and merges two sorted lists to one in each step with a merging network A recursive sorting network that starts with single values ($n$ sorted lists) and merges two sorted lists to one in each step with a merging network It has dept of $\mathcal{O}({\log^2 n})$ It has dept of $\mathcal{O}({\log^2 n})$ # 9. Lecture It covers the calculation of the network diameter ## Naïve Approach Every node calculates its maximum radius by flooding echo. The largest radius is then chosen as the diameter. This leads to message congestion and has round complexity of $\mathcal{O}(D)$ ## BFS Based As the naïve approach, but each node only starts the flooding if it receives the pebble. The pebble is forwarded in a BFS way and pauses at each token one round to ensure that the flooding messages do not overlap. It has round complexity of $\mathcal{O}(n) + \mathcal{O}(D) = \mathcal{O}(n)$ ## Fooling Sets A matrix which has the same entries on two opposite corners, but another entry in either one of the remaining corners. It is defined recursively, i.e. it consists itself out of fooling sets ## Communication Complexity To test equality of to bit strings of length $k$ (a function with a fooling set as solution set) $\mathcal{O}(\log 2^k)$ bits have to be exchanged. After this, the solution set can be narrowed down to a single value. If fewer bits get exchanged, the solution set can still be a fooling set, which is not monochromatic (same values anywhere) and is therefore ambiguous.