Lecture 023

Graphs

Vertices: node (v) Edge: lines (e=[0, v(v-1)/2]) Endpoints: at the end of edges Un-directed: (A, B) = (B, A) No Self-Edge: no (V, V) Neighbors: vertices' connected to Subgraph: a portion of the graph Complete graph: every vertex connects

Lights Out

Board Configuration: 2^{n^2} Possible Moves: n^2 Vertices: 2^{n^2} Edges: n^2 * 2^{n^2} / 2

• 2^{n^2} vertices, each has n^2 neighbors, since un-directed, divide by 2.

Implicit graph: We don't need a graph data structure

• it will be too big to graph all possibilities

• we explore graph, we never build graph

Explicit graph: data structure in memory

• iterate through neighbors of a vertex without repeating

Implementation

Vertices: node (v) Edge: lines (e=[0, v(v-1)/2]) Neighbors: [0, min(v, e)] Cost: $e \in O(v^2)$, but might be lower if we use e too.

	Linked list of edges	Hash set of edges	Matrix	List
graph_hasedge	O(e)	O(1) avg+amt	O(1)	O(min(v,e))
graph_addedge	O(1)	O(1)	O(1)	O(1)
graph_get_neighbors	O(e)	O(e)	O(v)	O(1) cuz grab
space			O(v^2)	O(v+e)\O(max(v,e))

iterate neighbors: O(1)*O(min(v,e)) dense graph: $e \in O(v^2)$ sparse graph: $e \in O(v) \lor e \in O(vlog(v))$ (we typically work with sparse graphs)

Representation

• adjacency matrix mirrors

• nothing in the middle

  

  

Code

Next Neighbors

- Adjacency list costs O(v+e)

- Adjacency matrix costs O(v^2) - for dense graph they are the same, for sparse graph Adjacency list is better