Lecture 01-27-17

Directed acyclic graphs

• Applications
  • Scheduling
    • git logs
  • Gantt chart / Pert chart
    • schedule with things that have to be done before other things
  • Critical path: “blockage”
    • most time intensive path
    • basics of computing?

Longest path

• Longest path in a graph
  • no good/”fast” algorithm to do so
  • find in directed acyclic graphs “very quickly”

Topological sort

• Sort order of vertices
  • x₁, x₂ .. x_n
• Definition
  • Order of the verticies such that all edges go from the left to the right
• No graph with a cycle can have a topological sort
• Every graph without a cycle does have a topological sort

Proof that every graph without a cycle has a topological sort

• Claim that vertices with no in-edges first
• Must not work because at some point no vertex has 0 in-edges
• Say at some point you get to no vertices with 0 and then run out of vertices because it’s finite

Topological sort part 2

• Time complexity: O(n + m)
• Transitive reduction
  • only include edges that are not implied by transitivity
• Transitive closure
  • include all things implied by transitivity
• Method:
  • store using adjacency lists
    • Using an adjacency matrix: cannot do this in N² time
  • keep track of all “in-degrees”
  • Make lists by in degree, then when visiting 0, traverse adjacency list
• Sidenotes
  • Does not make sense on undirected
  • Becomes an efficient algorithm for detecting cycles in a digraph
  • Issues with ordering
    • transferring and setup times between unrelated things
    • “not here, I’ll shutup”
• Use
  • Going to show that easy to find longest path in a acyclic digraph

Homework

• Take home exam 2: 2/6