Graphs

Graph problems are plentiful but the core data structure does have some unique techniques that can be applied to it

Most of the algorithms used for graph problems are found under Graph instead

Runtime analysis

• DFS/BFS/Topological sort: $O(|V| + |E|)$

• Number of vertices is $O(V)$

• Number of edges is $O(E)$

Graph representations

1. Adjacency matrix

2. Adjacency list

3. Hash table of hash tables

Take note…

• Tree like diagrams could be a graph with cycles so clarify before assuming

• Correctly track visited nodes and not visit each node more than once

Corner cases

1. Empty graphs

2. Graph with one or two nodes

3. Disconnected graphs

4. Graphs with cycles

Techniques

Shifting the goalpost

Rather than starting from the original problem, try reversing the problem and start with the end state in mind or the other set of elements

"Virus" traversal

This method of traversal is also known as "level-order BFS" but I like to name is as a "virus" traversal as you can imagine it as a virus spreading across the graph.

The infected nodes spread to their neighbors at once, must like how level-order BFS works

• Queue all elements that fit a criteria and BFS from that queue at once

• Ensure that visited set is maintained so no duplicate elements are queued

• Useful for propagating an operation to all neighbors

Each batch of processing of processing is often seen as "1 day" of virus spreading of one step.