From 9eeeeb88ee1476e2d24788c64f1a1bbbf686d730 Mon Sep 17 00:00:00 2001 From: Noah Zarro <noahzarro@gmail.com> Date: Thu, 29 Jul 2021 11:27:13 +0200 Subject: [PATCH] up to lecture 7 --- README.md | 5 ++++- images/arrow.png | Bin 0 -> 188087 bytes images/caching.png | Bin 0 -> 135931 bytes images/centralized.png | Bin 0 -> 96602 bytes images/home.png | Bin 0 -> 84298 bytes images/ivy.png | Bin 0 -> 83818 bytes images/luby.png | Bin 0 -> 94051 bytes images/mis.png | Bin 0 -> 65237 bytes images/pointer_forwarding.png | Bin 0 -> 56658 bytes summary.md | 22 ++++++++++++++++++++-- 10 files changed, 24 insertions(+), 3 deletions(-) create mode 100644 images/arrow.png create mode 100644 images/caching.png create mode 100644 images/centralized.png create mode 100644 images/home.png create mode 100644 images/ivy.png create mode 100644 images/luby.png create mode 100644 images/mis.png create mode 100644 images/pointer_forwarding.png diff --git a/README.md b/README.md index 5c2d971..c3dab41 100644 --- a/README.md +++ b/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. 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literal 0 HcmV?d00001 diff --git a/summary.md b/summary.md index b5cabf9..9a006f3 100644 --- a/summary.md +++ b/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||$`O(Tree Height)`$|$`2`$| |6-Color||$`O(\log*(n))`$|$`6`$| |Six-2-Three||$`O(1)`$|$`3`$| +|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|<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||$`O(D^2)`$|$`O(m+nD)`$| |Bellman-Ford BFS||$`O(D)`$|$`O(nm)`$| |GHS MST||$`O(n\log n)`$|$`O(m\log n)`$| +|Luby’s MIS||$`O(\log n)`$ w.h.p|$`O(m\log n)`$ w.h.p| + +### Sharing +|Name|Algorithm| +|---|---| +|Centralized Solution|| +|Home-Based Solution|| +|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|| +|Pointer Forwarding|| +|Ivy|| + ## Structures \ No newline at end of file -- GitLab