up to lecture 7

README.md

@@ -48,6 +48,9 @@ If one iteration is complete, the remaining colors are regrouped again in the sa
 
 # 4. Lecture
 
+A 2-coloring of a n-node directed path requires at least $`\Omega(n)`$
+
+Any deterministic algorithm for 3-coloring n-node directed paths needs at least $`\frac{\log∗ n}{2}− 2`$
 
 # 5. Lecture
 
@@ -74,7 +77,7 @@ The goal is to develop algorithms that allow read and write access to a shared r
 
 ## Basic Algorithms
 
-In the centralized solution, the object is stored on a single node ath the root of a spanning tree. This root handles all accesses.
+In the centralized solution, the object is stored on a single node at the root of a spanning tree. This root handles all accesses.
 
 The home-based solution works as above, but with a separate home for each object. (I guess)
 

summary.md

@@ -6,6 +6,7 @@
 - $`\Delta`$ is the maximum degree of any node in the tree
 - $`\Chi`$ is the Chromatic Number, i.e. the least number of colors 
 - $`\log*(x)`$ how many times $`\log`$ has to be applied to x until it becomes 2
+- $`\Delta`$ Cover Free Families: A set of sets, where a set $`i`$ is not a a subset of the union of $`\Delta`$ other sets
 
 ### Graphs
 - Radius (Node): The maximum distance between a node and the one that is the furthest away from it
@@ -13,8 +14,11 @@
 - Diameter: The maximum distance between any two nodes in the graph 
 
 ## Theorems
--  1.13: $`\Chi(Tree)\leq2`$
-
+- 1.13: $`\Chi(Tree)\leq2`$
+- 4.1: Any LOCAL algorithm for 2-coloring an n-node directed path requires at least $`\Omega(n)`$
+rounds.
+- 4.2: Any deterministic algorithm for 3-coloring n-node directed paths needs at least $`\frac{\log∗ n}{2}− 2`$
+rounds.
 ## Algorithms
 
 ### Coloring
@@ -26,6 +30,8 @@
 |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>|$`O(1)`$|$`O()`$|
+|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
 
@@ -36,5 +42,17 @@
 |Dijkstra BFS|![dijkstra_bfs](images/dijkstra_bfs.png)|$`O(D^2)`$|$`O(m+nD)`$|
 |Bellman-Ford BFS|![bellman_ford_bfs](images/bellman_ford_bfs.png)|$`O(D)`$|$`O(nm)`$|
 |GHS MST|![ghs](images/ghs.png)|$`O(n\log n)`$|$`O(m\log n)`$|
+|Luby’s MIS|![luby](images/luby.png)|$`O(\log n)`$ w.h.p|$`O(m\log n)`$ w.h.p|
+
+### Sharing
+|Name|Algorithm|
+|---|---|
+|Centralized Solution|![centralized](images/centralized.png)|
+|Home-Based Solution|![home](images/home.png)|
+|Arrow|<ul><li>object is saved at node that last accessed it, this is the parent</li><li>find requests can be sent</li><li>find request reverses parent child relationships</li><li>any node that has the object sends it to its parent</li></ul>|
+|Caching|![caching](images/caching.png)|
+|Pointer Forwarding|![pointer_forwarding](images/pointer_forwarding.png)|
+|Ivy|![ivy](images/ivy.png)|
+
 
 ## Structures
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