@@ -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.