until lecture 13

README.md

@@ -152,7 +152,23 @@ To test equality of to bit strings of length $`k`$ (a function with a fooling se
 # 10. Lecture
 
 TODO
+
 # 11. Lecture
+
+TODO
+
+# 12. Lecture
+
+## Setup
+Messages have to be routed in a fully connected graph with $`n`$ nodes. Each edge can only deliver one message per round.
+
+## Solution
+Construct graph with all nodes once on the left and once on the right side. For each message that has to be sent, add an edge that connects the sender edge on the left and the receiver on the right. Do a $`2\Delta +1`$ coloring of the edges, and use the node corresponding to the color as a relay for this message.
+
+## Coloring
+The coloring can be randomized. However, since the colors are random, it is possible that edges from sender to relay or relay to target are double assigned. Therefore $`5\log n`$ copies of each message are sent. Assuming each node only sends and receives at max $`\frac{n}{20\log n}`$ messages, all messages are successfully transmitted with probability $`1-\frac{1}{n^3}`$.
+
+# 13. Lecture
 It covers the topic of wireless transport protocols to avoid collisions
 
 ## Slotted Aloha
@@ -168,14 +184,14 @@ The process of obtaining ids $`1 \dots n`$ is called initialization. It can be a
 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.
+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 in 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.
+In each round $`k`$, transmit $`ck`$ times with probability $`\frac{1}{2^k}`$, 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.

summary.md

@@ -31,7 +31,7 @@ rounds.
 |Slow Tree Coloring|![slow_tree_coloring](images/slow_tree_coloring.png)|$`O(Tree Height)`$|$`2`$|
 |6-Color|![6_color](images/6_color.png)|$`O(\log*(n))`$|$`6`$|
 |Six-2-Three|![shift down](images/shift_down.png)![six-2-three](images/six_2_three.png)|$`O(1)`$|$`3`$|
-|Linial|<ul><li>Assign a set of a $`\Delta+1`$ cover free family to each node</li><li>Choose an ID for this node that is not contained in the sets of its neighbors</li></ul>|$`O(1)`$|$`O()`$|
+|Linial|<ul><li>Assign a set of a $`\Delta+1`$ cover free family to each node</li><li>Choose an ID for this node that is not contained in the sets of its neighbors</li></ul>|$`O(1)`$|$`\Delta+1`$|
 |MIS|![mis](images/mis.png)<lu><li>Create $`\Delta+1`$ copies of the graph</li><li>Calculate MIS</li><li>If $`i^{th}`$ copy is in MIS, color vertex with color $`i`$</li></lu>|$`O(n)`$|$`\Delta+1`$|
 
 ### Trees
@@ -74,7 +74,7 @@ rounds.
 |---|---|---|
 |Naive Diameter Construction|![naive_diameter](images/naive_diameter.png)|$`O(D)`$|
 |APSP Diameter Construction|![diameter_synchronous](images/diameter_synchronous.png)|$`O(n)`$|
-|Borůvka's Algorithm|<ul><li>Start with empty spanning forest</li><li>Each fragment computes minimal outgoing edge</li><li>Fragments get merged over minimal outgoing edges, and they get added to the MST</li><li>Repeat</li></ul>|$`O(\log n)`$|
+|Borůvka's Algorithm|<ul><li>Start with empty spanning forest</li><li>Each fragment computes minimal outgoing edge</li><li>Fragments get merged over minimal outgoing edges, and they get added to the MST</li><li>Repeat</li></ul>|$`O((D+\sqrt{n})\log n)`$|
 
 ### Definitions
 |Name|Definition|
@@ -83,6 +83,20 @@ rounds.
 |Monochromatic|![monochromatic](images/monochromatic.png)|
 |Fooling Set|![](images/fooling_set.png)|
 
+### Wireless Protocols
+|Name|Algorithm|CD|n known|
+|---|---|---|---|
+|Slotted Aloha|![slotted_aloha](images/slotted_aloha.png)||:heavy_check_mark:|
+|Uniform Initialization with CD|![uniform_initialization_with_cd](images/uniform_initialization_with_cd.png)|:heavy_check_mark:|:x:|
+|Fake CD|![fake_CD](images/fake_CD.png)|:x:|:x:|
+|Uniform Leader Election|![uniform_leader_election](images/uniform_leader_election.png)|:x:|:x:|
+|Uniform Leader Election with CD|![uniform_leader_election_CD](images/uniform_leader_election_CD.png)|:heavy_check_mark:|:x:|
+
+### Labeling
+|Name|Algorithm|Label Size|
+|Naive Distance Labeling|![naive_distance_labeling](images/naive_distance_labelin.png)|$`O(n\log n)`$|
+||||
+
 ## Structures
 - TODO: Local Model
 
@@ -93,5 +107,14 @@ rounds.
 
 |Symbol|Meaning|
 |---|---|
-|$`f \in O(g)`$||
+|$`f \in O(g)`$|f "smaller/equal" than g|
+|$`f \in \Theta(g)`$|g and f are similar|
+|$`f \in \Omega(g)`$|f is "larger/equal" than g|
 
+### Inequalities
+
+|Name|Inequality|
+|---|---|
+|Boole|$`Pr[\bigcup_i E_i] \leq \sum_i Pr[E_i]`$, where $`E_i`$ are events|
+|Markov|$`Pr[\mid X \mid \geq a] \leq \frac{E[X]}{a}`$|
+||$`1-p\leq(1-p/k)^k`$|