Introduction to Algorithms LECTURE 16 (Chap. 22) Elementary Graph Algorithm • 22.2 Breath-first search • 22.3 Depth-first search • 22.4 Topological sort Breadth first search animation

Breadth first search animation Get ahold of a network, and use the same network to illustrate the shortest path problem for communication newtorks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.) Breadth first search animation

Initialize 1 2 4 5 3 6 9 7 8 1 2 4 5 3 6 9 7 8 1 1 pred(1) = 0 next := 1 order(next) = 1 LIST:= {1} Unmark all nodes in N; Mark node s LIST 1 next 1

Select a node i in LIST 1 2 4 5 3 6 9 7 8 1 1 1 In breadth first search, i is the first node in LIST LIST 1 next 1

If node i is incident to an admissible arc… 2 4 2 2 8 1 1 1 1 5 7 Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i Select an admissible arc (i,j) 9 3 6 LIST 1 2 2 1 next

If node i is incident to an admissible arc… 2 4 2 2 8 1 1 1 1 5 5 7 3 Mark Node j pred(j) := i Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST 9 3 6 LIST 1 2 5 3 2 next

If node i is incident to an admissible arc… 2 4 2 2 8 1 1 1 1 5 5 7 3 Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i 9 3 3 6 4 LIST 1 2 5 3 4 2 3 next

If node i is not incident to an admissible arc… 2 4 2 2 8 1 1 1 1 1 5 5 7 3 9 Delete node i from LIST 3 3 6 4 LIST 1 2 5 3 2 4 3 next

Select Node i 2 4 2 2 2 8 1 1 1 1 1 5 5 7 3 9 The first node on LIST becomes node i 3 3 6 4 LIST 1 2 5 3 2 4 3 next

If node i is incident to an admissible arc… 5 2 4 4 2 2 2 8 1 1 1 5 5 7 3 Select an admissible arc (i,j) Mark Node j pred(j) := i Next := Next + 1 order(j) := next add j to LIST 9 3 3 6 4 LIST 1 2 5 3 4 5 2 4 3 next

If node i is not incident to an admissible arc… 5 2 4 4 2 2 2 2 8 1 1 1 5 5 7 3 Delete node i from LIST 9 3 3 6 4 LIST 1 2 5 3 4 5 2 3 4 next

Select a node 5 2 4 4 2 2 2 8 1 1 1 5 5 5 7 3 The first node on LIST becomes node i 9 3 3 6 4 LIST 1 2 5 3 4 5 2 3 4 next

If node i is incident to an admissible arc… 5 2 4 4 2 2 2 8 1 1 1 5 5 5 7 3 Mark Node j pred(j) := i Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST 9 3 3 6 6 4 6 LIST 1 2 5 3 4 6 3 2 4 5 next

If node i is not incident to an admissible arc… 5 2 4 4 2 2 2 8 1 1 1 5 5 5 5 7 3 Delete node i from LIST 9 3 3 6 6 4 6 LIST 1 2 5 3 4 6 6 3 2 4 5 next

Select node 3 5 2 4 4 2 2 2 8 1 1 1 5 5 5 5 7 3 node 3 is not incident to any admissible arcs delete node 3 from LIST 9 3 3 3 3 6 6 4 6 LIST 1 2 5 3 4 6 2 3 5 4 6 next

Select a node 5 2 4 4 4 2 2 8 1 1 1 5 5 7 3 i : = 4 9 3 3 6 6 4 6 LIST 1 2 5 3 4 6 6 2 3 4 5 next

If node i is incident to an admissible arc… 5 2 4 4 4 7 2 2 8 8 1 1 1 5 5 7 3 Mark Node j pred(j) := i Next := Next + 1 order(j) := next add j to LIST Select an admissible arc (i,j) 9 3 3 6 6 4 6 LIST 1 2 5 3 4 6 8 3 7 2 4 6 5 next

If node i is not incident to an admissible arc… 5 2 4 4 4 4 7 2 2 8 8 1 1 1 5 5 7 3 Delete node i from LIST 9 3 3 6 6 4 6 LIST 1 2 5 3 4 6 8 7 3 2 6 4 5 next

Select node i 5 2 4 4 7 2 2 8 8 1 1 1 5 5 7 3 i := 6 9 3 3 6 6 6 4 6 LIST 1 2 5 3 4 6 8 7 3 2 6 4 5 next

If node i is incident to an admissible arc… 5 2 4 4 7 2 2 8 8 1 8 1 1 5 5 7 7 3 Select an admissible arc (i,j) Mark Node j pred(j) := i Next := Next + 1 order(j) := next add j to LIST 9 3 3 6 6 6 4 6 LIST 1 2 5 3 4 6 8 7 2 8 3 5 6 4 7 next

If node i is incident to an admissible arc… 5 2 4 4 7 2 2 8 8 1 8 1 1 5 5 7 7 3 Select an admissible arc (i,j) Mark Node j pred(j) := i Next := Next + 1 order(j) := next add j to LIST 9 9 3 3 6 6 6 4 9 6 LIST 1 2 5 3 4 6 8 7 9 8 9 2 4 5 3 7 6 next

If node i is not incident to an admissible arc… 5 2 4 4 7 2 2 8 8 1 8 1 1 5 5 7 7 3 Delete node i from LIST 9 9 3 3 6 6 6 6 4 9 6 LIST 1 2 5 3 4 6 8 7 9 9 8 7 5 6 2 3 4 next

Select node 8 5 2 4 4 7 2 2 8 8 8 8 1 8 1 1 5 5 7 7 3 node 8 is not incident to an admissible arc; delete it from LIST 9 9 3 3 6 6 6 6 4 9 6 LIST 1 2 5 3 4 6 8 7 9 3 9 2 8 6 4 7 5 next

Select node 7 5 2 4 4 7 2 2 8 8 1 8 1 1 5 5 7 7 7 7 3 node 7 is not incident to an admissible arc; delete it from LIST 9 9 3 3 6 6 6 6 4 9 6 LIST 1 2 5 3 4 6 8 7 9 3 9 2 8 6 4 7 5 next

Select node 9 5 2 4 4 7 2 2 8 8 1 8 1 1 5 5 7 7 3 node 9 is not incident to an admissible arc; delete it from LIST 9 9 9 9 3 3 6 6 6 6 4 9 6 LIST 1 2 5 3 4 6 8 7 9 3 9 2 8 6 4 7 5 next

Breadth first search animation Depth First Search Get ahold of a network, and use the same network to illustrate the shortest path problem for communication newtorks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.) Breadth first search animation

Initialize 1 2 4 5 3 6 9 7 8 1 2 4 5 3 6 9 7 8 1 1 pred(1) = 0 next := 1 order(next) = 1 LIST:= {1} Unmark all nodes in N; Mark node s LIST 1 next 1

Select a node i in LIST 1 2 4 5 3 6 9 7 8 1 1 1 In depth first search, i is the last node in LIST LIST 1 next 1

If node i is incident to an admissible arc… 2 4 2 2 8 1 1 1 1 5 7 Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i Select an admissible arc (i,j) 9 3 6 LIST 1 2 2 1 next

Select the last node on LIST 2 4 2 2 2 8 1 1 1 1 1 5 7 9 3 6 Node 2 gets selected LIST 1 2 2 1 next

If node i is incident to an admissible arc… 2 4 4 2 2 2 3 8 1 1 1 1 1 5 7 Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i 9 3 6 LIST 1 2 4 3 2 1 next

Select 2 4 4 4 2 2 2 2 3 8 1 1 1 1 1 5 7 Select the last node on LIST 9 3 6 LIST 1 2 4 2 3 1 next

If node i is incident to an admissible arc… 2 4 4 4 2 2 2 2 3 8 8 4 1 1 1 1 1 5 7 Mark Node j pred(j) := i Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST 9 3 6 LIST 1 2 4 8 3 4 2 1 next

Select 2 4 4 4 2 2 2 2 3 8 8 8 4 1 1 1 1 1 5 7 Select the last node on LIST 9 3 6 LIST 1 2 4 8 3 2 1 4 next

If node i is not incident to an admissible arc… 2 4 4 4 2 2 2 2 3 8 8 8 8 4 1 1 1 1 1 5 7 Delete node i from LIST 9 3 6 LIST 1 2 4 8 1 3 2 4 next

Select 2 4 4 4 4 2 2 2 2 3 8 8 8 8 4 1 1 1 1 1 5 7 Select the last node on LIST 9 3 6 LIST 1 2 4 8 1 4 2 3 next

If node i is incident to an admissible arc… 2 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 7 Mark Node j pred(j) := i Next := Next + 1 order(j) := next add j to LIST Select an admissible arc (i,j) 9 3 6 LIST 1 2 4 5 8 1 5 2 4 3 next

Select 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 7 Select the last node on LIST 9 3 6 LIST 1 2 4 8 5 1 2 5 3 4 next

If node i is incident to an admissible arc… 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 7 Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i 9 3 6 6 6 LIST 1 2 4 8 5 6 6 3 4 2 5 1 next

Select the last node on LIST 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7 Select node 6 9 3 6 6 6 6 LIST 1 2 4 8 5 6 3 4 6 5 1 2 next

If node i is incident to an admissible arc… 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7 Mark Node j pred(j) := i Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST 9 9 3 6 6 6 7 6 LIST 1 2 4 5 8 6 9 1 7 2 4 5 3 6 next

Select the last node on LIST 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 1 1 1 1 5 5 5 5 7 Select node 9 9 9 9 3 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 1 2 4 6 5 3 next

If node i is incident to an admissible arc… 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 Select an admissible arc (i,j) Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i 9 9 9 3 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 8 4 3 1 6 2 5 7 next

Select the last node on LIST 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 Select node 7 9 9 9 9 3 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 4 8 1 6 2 3 5 7 next

If node i is not incident to an admissible arc… 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7 Delete node 7 from LIST 9 9 9 9 3 6 6 6 6 7 6 LIST 1 2 4 8 5 6 9 7 1 2 8 4 6 3 7 5 next

Select node 9 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7 Delete node 9 from LIST But node 9 is not incident to an admissible arc. 9 9 9 9 9 9 3 6 6 6 6 7 6 LIST 1 2 4 8 5 6 9 7 8 4 2 1 7 3 5 6 next

Select node 6 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 7 7 7 7 But node 6 is not incident to an admissible arc. Delete node 6 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 4 3 2 6 7 8 1 5 next

Select node 5 2 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7 But node 5 is not incident to an admissible arc. Delete node 5 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6 LIST 1 2 4 8 5 6 9 7 3 2 1 4 5 6 8 7 next

Select node 4 2 4 4 4 4 4 4 4 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7 Delete node 4 from LIST But node 4 is not incident to an admissible arc. 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 7 1 2 5 8 4 6 3 next

Select node 2 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 5 5 5 5 5 5 7 7 7 7 But node 2 is not incident to an admissible arc. Delete node 2 from LIST 9 9 9 9 9 9 3 6 6 6 6 6 6 7 6 LIST 1 2 4 5 8 6 9 7 6 3 1 4 5 7 2 8 next

Select node 1 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7 Next := Next + 1 order(j) := next add j to LIST Mark Node j pred(j) := i Select an admissible arc (i,j) 9 9 9 9 9 9 3 3 6 6 6 6 6 6 7 9 6 LIST 1 3 2 4 8 5 6 9 7 2 6 9 7 3 8 5 1 4 next

Select node 3 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7 Delete node 3 from LIST But node 3 is not incident to an admissible arc. 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6 LIST 1 3 2 4 5 8 6 9 7 9 8 5 3 4 2 1 7 6 next

Select node 1 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7 Delete node 1 from LIST But node 1 is not incident to an admissible arc. 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6 LIST 1 2 3 4 8 5 6 9 7 9 5 2 1 3 4 7 6 8 next

LIST is empty 2 4 4 4 4 4 4 4 2 2 2 2 2 2 3 8 8 8 8 4 5 1 8 1 1 1 1 1 1 1 1 5 5 5 5 5 5 7 7 7 7 The algorithm ends! 9 9 9 9 9 9 3 3 3 3 6 6 6 6 6 6 7 9 6 LIST 1 2 3 4 5 8 6 9 7 9 4 5 3 6 2 1 7 8 next

The depth first search tree 1 3 2 9 8 7 5 4 6 Note that each induced subtree has consecutively labeled nodes

Breadth first search animation Topological Ordering Get ahold of a network, and use the same network to illustrate the shortest path problem for communication newtorks, the max flow problem, the minimum cost flow problem, and the multicommodity flow problem. This will be a very efficient way of introducing the four problems. (Perhaps under 10 minutes of class time.) Breadth first search animation

Preliminary to Topological Sorting LEMMA. If each node has at least one arc going out, then the first inadmissible arc of a depth first search determines a directed cycle. COROLLARY 1. If G has no directed cycle, then there is a node in G with no arcs going. And there is at least one node in G with no arcs coming in. COROLLARY 2. If G has no directed cycle, then one can relabel the nodes so that for each arc (i,j), i < j. 1 4 6 7 3

Initialization 6 1 Determine the indegree of each node LIST is the set of nodes with indegree of 0. “Next” will be the label of nodes in the topological order. 5 2 8 3 7 4 next 1 2 3 4 5 6 7 8 Node Indegree LIST 7

Select a node from LIST 6 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 5 2 8 3 7 7 4 1 1 next 1 2 3 4 5 6 7 8 Node LIST Indegree 2 2 3 2 1 1 1 2 7 5

Select a node from LIST 6 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 8 3 7 7 4 1 1 2 next 1 2 3 4 5 6 7 8 Node LIST Indegree 2 2 1 3 2 1 1 1 2 4 5 7 6

Select a node from LIST 3 6 6 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 8 3 7 7 4 1 2 1 3 next 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 2 1 3 1 2 1 1 2 4 5 7 2 6

Select a node from LIST 3 6 6 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 4 3 2 1 next 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 2 1 3 2 1 1 1 2 4 7 5 6 2 1

Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 1 3 5 2 4 1 next 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 1 2 2 3 2 1 1 1 2 1 4 7 5 1 2 6

Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 2 5 5 2 2 8 3 4 7 7 4 4 1 6 6 3 5 4 2 1 next 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 2 1 3 2 1 1 2 1 1 2 1 2 1 4 6 7 5 8

Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 2 5 5 2 2 8 8 3 4 7 7 4 4 1 6 2 7 4 5 3 6 1 next 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 1 2 2 3 1 1 2 1 1 1 2 8 3

Select a node from LIST 5 3 6 6 1 1 next := next +1 order(i) := next; update indegrees update LIST Select a node from LIST and delete it. 7 8 2 5 5 2 2 8 8 3 3 4 7 7 4 4 1 6 2 5 8 6 7 3 1 4 next List is empty. The algorithm terminates with a topological order of the nodes 1 2 3 4 5 6 7 8 Node LIST Indegree 1 2 2 1 2 1 3 1 2 1 1 2 1 3