Graph Defination:

• Directed edges are ordered edges
• Undirected edges are unorded edges (i.e. (u,v) and (v,u) are different)
• $E=O(V^2)$

Neighbor Sets:

• outgoing neighbor set
• incoming neighbor set
• out-degree
• in-degree

Graph Representation:

• A set to map u to Adj(u)
  • DAA or hash table (lookup fast by vertices)
• Adjacency list to store Adj(u)
  • array or linked list(usuallt to insert)

Hand shaking lemma:

$\sum_{u \in V} deg(u)\le 2|E|$

Path:

• the lengh l(p): the edges in the path
• distance $\delta (u,v)$ is the minimum lengh of the path from u to v.

Graph Path Problem:

• Single pair reachbility: Is there a path from s to v
• single pair shortest path: return the shortest path from s to v
• single source shortest path: return $\delta(s,v)$ for all v in graph

If we solve one of them,we will find we could solve all them absoloutly.

BFS:

def bfs(Adj, s):
    parent = [None for x in Adj]
    parent[s] = s
    level = [[s]]
    while len(level[-1]) > 0:
        level.append([])  # remember to augment the new level for new items
        for u in level[-2]:
            for v in level[-1]:
                if parent[v] is None:
                    parent[v] = u
                    level[-1].append(v)
    return parent


def unweighed_shortest_path(Adj, s, t):
    parent = bfs(Adj, s)
    # if we can reach t from s
    if parent[t] is None:
        parent[t] = None
    i = t
    path = [t]
    while i != s:
        i = parent[i]
        path.append(i)
    return path[::-1]

Running time:

$O(|V|+|E|)$ ,because we will go through all the vertices,every node,we will go the edges.