CS252 Algorithms
Wednesday, 28 September 2022

+ Questions about homework, reading, whatever

+ In-class test on 10/7
- There will be a lighter-weight homework assignment this week
- On Friday 9/30 I'll talk through the structure
    and topics for the exam

+ BFS & DFS
- Intuition is weird
- Sample graph & let's run BFS & DFS by hand
- Look at the algorithms
    - analyze running times
- Look again: memory usage
    - stack, queue
    - recursion and the system stack
- Building the tree

+ Topological sort
- Why?
- DAGs
- Algorithm
    - try it on some examples
        - example that results in no-such-sorting
        - example that succeeds
    - termination
    - arguing that a DAG has a no-incoming-edge node


==========

BFS(s):
[Data structures]
1   T: tree
2   L: list of lists ("layers")
3   Visited: list of bool, one per node
4   current_layer


[Initialization]
5   L[0] = [s] -- no additional layers yet
6   current_layer = 0
7   Visited[s] = true
8   Visited[v] = false for all v != s
9   T = ∅

[Traversal]
10  while L[current_layer] is not empty:
11      initialize L[current_layer + 1] = []
12      for each node u ∈ L[current_layer]:
13          for each edge (u, v):
14              if not Visited[v]:
15                  Visited[v] = true
16                  add edge (u, v) to T
17                  add v to L[current_layer+1]
18      current_layer = current_layer + 1

====

DFS(s):
[Data structures]
1   S: stack of nodes
2   Visited: list of bool, one per node

[Initialization]
3   S = [s]

[Traversal]
4   while S is not empty
5       u = S.pop()
6BROKEN!!!!       if not Visited[u]:
7           Visited[u] = true
8           for each edge (u, v):
9               S.push(v)

====

Topological sort
Given: directed graph G

TopologicalSort(G):
    ''' Return None if there is no topological sort '''
    v = any node with no incoming edges
    if no such v exists:
        return None
    else:
        return [v] + TopologicalSort(G - {v})