Connected (reachable)
Connected Component: maximal set of vertices that are connected (connected individual graph)
Search Problem: finding a witness (more complex, for cryptography)
Verification Problem: checking a witness
Definition of Reachable
Depth First Search with Adjacency List
Bad DFS that has infinite loop
Fixing DFS using mark array
Looper
Heap header
alloc costs O(v)
dfs_helper costs O(e)
free costs O(1)
together O(v+e)
but if you use hashset or with sparse graph, they cost O(e)
Stack header (only use when graph is small)
Implementation with Adjacency Matrix
Costs: O(v^2)
Breath First Search
BFS using queue
remember to free queue, mark array
Code
Invariants
Invariants to Prove no Path: every path from start to target goes through frontier
If we use queue: breath first
If we use stack or other: depth first
If we use priority: A* heuristic (but insertion and removal from priority queue is not O(1))
