Add all my notes

imperfect_notes/pragy/Graphs 1.md

@@ -0,0 +1,166 @@
+Graphs 1
+--------
+
+
+G = (V, E)
+
+- directed / undirected
+- edge describes some relationship
+- edge can have value - which can be some property of that relationship
+- edit distance of pan --- pen is 1
+- bi-directional: a->b implies b->a
+- weighted/unweighted - when 1 property associated with edge - some number
+- when all weights are same, we don't represent the weights
+- ask students for examples of undirected graphs (fb friends) and directed (twitter follows / company hierarchy) and weighted graphs (city distances)
+
+Trees vs graphs
+- exactly 1 path b/w any pair of nodes in a tree
+- because acyclic. If 2 paths, cycle occurs A-B-A or A-x-B-x
+
+Simple graph
+- no self loops
+- no multiple edges b/w any 2 nodes
+- max edges = $n \choose 2$
+- such is a clique $K_{n}$
+
+
+Disconnected vs Connected
+connected = can go from any node to any node
+
+a b c
+a-b c
+
+a-b-c
+a-b
+\c/
+
+Connected Components
+largest subgraphs which are connected
+
+connected graph has exactly 1 connected component
+
+directed needs strong connections to be connected
+a->b is not connected
+-- --
+
+Representations
+---------------
+
+Adjacency Matrix
+----------------
+does path exist?
+
+for undirected, $A$ is symmetric across the major diagonal.
+Diagonal is 0 for a simple graph
+
+Space: $O(n^2)$
+Checking edge: $O(1)$
+
+Can puts weights/cost/distance as well.
+
+$A^n_{ij} =$ number of $k$ length walks b/w $i, j$
+
+Adjacency List
+--------------
+
+List of lists
+
+Space: efficient for sparse graphs
+Time: $O(n)$ lookup in the worst case
+$O(1)$ degree count in worst case
+better if we only want to list down the neighbors
+
+-- --
+
+Traversals
+----------
+
+- same as for trees, just remember visited to prevent infinite loops
+- start from any node - because no root
+- tree defined using graph - rooted, unrooted
+- BFS - $O(|V| + |E|)$
+
+
+DFS using stack
+- remember to keep the count of children explored and the parent, so that we can resume exploring from the next child
+
+
+Check Connected
+---------------
+
+undirected - just run DFS from 1. If all visited, connected
+
+DFS over disconnected graph - must start from all nodes one by one
+
+BFS - 
+
+![dc2d0b12.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/dc2d0b12.png)
+
+start from some node.
+levels, min distance from that node
+
+BFS for single source shortest path, when edge weights are EQUAL
+
+Doesn't work when a-b-c < a-c
+
+-- --
+
+Undirected graph with weights 1 or 2
+Can still NOT apply BFS
+
+-- --
+
+pseudo/dummy nodes
+- can't always create if weights are large
+- how to add them virtually? counter - inefficient
+- heap - efficient
+
+-- --
+
+![93c9f38f.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/93c9f38f.png)
+
+1 = land, 0 = water
+find the number of islands
+diagonals allowed
+
+find the number of connected components via DFS
+
+![5c8b1184.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/5c8b1184.png)
+-- --
+
+grid with 0s and 1s
+
+capture all 0s surrounded by 1s on all 4 sides
+
+![6f39934a.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/6f39934a.png)
+
+apply DFS to find components of 0s
+
+if reaches boundary, don't capture
+if not, we've surrounded, capture all
+
+-- --
+
+reach from source to destination
+
+![151b8e6a.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/151b8e6a.png)
+
+-- --
+
+shortest distance of all 0s from any 1
+
+![f8a8702a.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/f8a8702a.png)
+
+BFS from all 1s
+
+inspire from 1 and 2 bombs
+![ad3d4993.png](:storage/30547dac-c72e-46b3-86e3-479b31df3a42/ad3d4993.png)
+
+
+-- --
+
+any shortest distance?
+
+BFS from all 0s
+or prev and find min
+