... ... @@ -142,3 +142,30 @@ A set where all nodes have the same value and each two nodes can span up a recta ## 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. # 11. Lecture It covers the topic of wireless transport protocols to avoid collisions ## Slotted Aloha In this protocol, each node sends in a slot with probability $1/n$. The probability that in a slot any node successfully transmits is $1/e$. But $n$ must be known. ## Collision Detection - CD If a receiver can distinguish receiving nothing from receiving from more than one peer. ## Initialization The process of obtaining ids $1 \dots n$ is called initialization. It can be achieved in the following ways: ### Without CD, n known Just do slotted aloha, each node that transmitted successfully gets the next ID. ### With CD, n unknown Sort peers into a binary tree, where each peer ends in a leaf. First all nodes are in the root node. In a node each peer selects either 1 or 0, and then sends either i n the slot 1 or 0. If a collision happens, peers move to the child node, corresponding to the slot they selected. If nobody transmitted in a slot, the corresponding child node can be ignored. If only one peer transmitted, it gets the next free ID. The tree is traversed until there are no collisions anymore. ### Without CD, n unknown Same as above, but each slot is split into two transmissions, one where only the leader $l$ transmits, and one where everybody who wants to transmit in this slot $S$ and $l$ transmit. Like this it can be distinguish if $S$ is empty or contains more than two peers. But a leader has to be determined first. ## Leader Election ### Without CD Transmit in each round $k$ transmit $ck$ times with probability $1/k$, the first one to transmit alone becomes the leader. ### With CD Every node transmit in each round with probability $1/2$, if more than one node transmits, all nodes that did not transmit quit the protocol.