more definitions for dsa2

data_struct2/README.md

@@ -2,7 +2,7 @@
 layout: page
 title: Advanced Data Structures and Algorithms (DSA2)
 description: Linked Lists, Trees, Queues, Heaps, Stacks, etc. 
-date: '2024-03-17 12:24:11 -0400'
+date: '2024-03-17'
 permalink: /data_struct2/
 image: /static/post-image/bigo.png
 categories: DSA
@@ -14,13 +14,36 @@ tags:
 
 ## Advanced Data Structures and Algorithms (DSA2)
 
-Adjacency representation (for graphs)
+#### My (incomplete) notes for DSA2 
+
+### Recurrences 
+
+Differential equation
+- K = 2
+- Only positive coefficients (b…bk)
+- A*T(n-1) + B*T(n-2)
+
+Master theorem:
+- Constants: a >= 1, b > 1
+- F(n) > 0 
+- Check n^log(b,a) _ f(n)
+- Case 3: 
+  - a*f(n/b) <= c*f(n) then t(n) = Θ(f(n))
+    - for c < 1
+    - regularity check 
+
+
+### Adjacency representation (for graphs)
 
 | Representation | Find Edge | Insert Edge | Delete Edge | Memory     |
 | -------------- | --------- | ----------- | ----------- | ---------- |
 | List           | `O(|V|)`    | `O(1)`        | `O(|V|)`      | `Θ(|V|+|E|)` |
 | Matrix         | `O(1)`      | `O(1)`        | `O(1)`        | `Θ(|V|^2)`   |
 
+- In (binary) matrix form, every row/column cooresponds to a vertex. The index will be switched from 0 to 1 if there is an edge between those wo vertices. 
+
+- If we are using a (linked) list, we insert an edge at the head of the list. Otherwise, we just go to the respective row and column in the matrix and flip the bit from 0 to 1
+
 ### Vertex colors
 
 - **White:** unprocessed/undiscovered
@@ -29,10 +52,13 @@ Adjacency representation (for graphs)
 
 ### Edge types
 
+- Let U be the parent, and V be the descendent 
+
 - **Tree Edge**
   - Parent to a child
   - Goes to undiscovered vertex
   - V finishes before U
+  - U is grey, V is white
 
 - **Back Edge**
   - To an ancestor
@@ -49,12 +75,47 @@ Adjacency representation (for graphs)
     - To a finished vertex discovered after the current vertex
     - Indirect descendant (not child)
   - V finishes before U
+  - U is grey and V is black
 
 - **Cross Edge**
   - Everything else
     - To a vertex finished before the current vertex's discovery
-  - One branch to another tree, or one tree to another
+  - One branch to another tree, or one tree to another (ie 1 component to another)
   - V finishes before U
+  - U is grey and V is black
+
+White Path Lemma
+- \( V \) descends \( U \) iff at time \( d[u] \) there exists a path from \( u \) to \( v \) composed completely of white vertices.
+
+DAG (Directed Acyclic Graph)
+- Directed graph G is acyclic iff DFS produces no back edges
+
+Topological Sort in DAG
+- Order relation
+
+Sparse Graph
+- \( |E| = O(|V|) \)
+
+Dense Graph
+- \( |E| = \Θ(|V|^2) \)
+- more vertices than edges (by an order of magnitude)
+
+Complete Graph
+- Edge between every 2 vertices (in undirected graph)
+
+Connected Components
+- Maximal, connected subgraph
+- Maximal means no more vertices can be added to the subgraph to still be connected 
+
+Tree
+- Acyclic and connected graph
+- Connected components of a forest = a tree
+
+Forect
+- Acyclic graph
+- a tree is a forest, but a forest isn't always a tree 
+- its a graph and not a set of trees 
+- a subgraph of a forest = always a forest
 
 
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