CS200: Graphs. Rosen Ch , 9.6, Walls and Mirrors Ch. 14

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1 CS200: Graphs Rosen Ch , 9.6, Walls and Mirrors Ch. 14

2 Trees as Graphs Tree: an undirected connected graph that has no cycles. A B C D E F G H I J K L M N O P

3 Rooted Trees A rooted tree is a tree in which one vertex has been designated as the root and every edge is directed away from the root

4 Trees as Graphs Tree: an undirected connected graph that has no cycles. An undirected graph is a tree iff there is a unique simple path (no repeated vertices) between any two vertices.

5 When is a graph a Tree? Can explicitly check that the graph is connected and has no cycles. (How?) Want an alternative characterization

6 When is a graph a Tree? A connected undirected graph with n vertices must have at least n-1 edges (proof: by induction on the number of vertices) A connected undirected graph that has n vertices and exactly n-1 edges cannot contain a cycle (proof: by contradiction with previous statement) A connected undirected graph that has n vertices and more than n-1 edges must contain a cycle.

7 When is a graph a Tree? Conclusion: A connected graph with n vertices and n-1 edges is a tree. In order to check if a graph is a tree need to check that it is connected and count the number of edges and vertices.

8 Spanning Trees Spanning tree: A subgraph of a connected undirected graph G that contains all of G s vertices and enough of its edges to form a tree. How to get a spanning tree: Remove edges until you get a tree. Add edges until you have a spanning tree

9 Spanning Trees - DFS algorithm dfstree(in v:vertex) Mark v as visited for (each unvisited vertex u adjacent to v) Mark the edge from u to v dfstree(u)

10 Spanning Tree Depth First Search Example A B C D E F G H I J K L M N O P

11 Minimum Spanning Tree Minimum spanning tree spanning tree minimizing the sum of edge weights Example use Connecting each house in the neighborhood to cable graph where each house is a vertex. Need the graph to be connected, and minimize the cost of laying the cables.

12 Prim s Algorithm Idea: incrementally build spanning tree by adding the least-cost edge to the tree

13 MST Example Prim s Algorithm (Start at d) a 8 7 b c h 11 i 4 14 d g f 1 2 e

14 Prim s Algorithm prims(in: G=(V,E):Graph) //V T current vertices in spanning tree //E T edges belonging to the spanning tree V T = {w} // w is an arbitrarily chosen vertex E T = ϕ //spanning tree contains no vertices initially for i = 1 to V - 1 do find a minimum-weight edge e=(u,v) among edges that connects a vertex in V T with a vertex in V V T add v to V T add e to E T return E T

15 Implementing Prim s Algorithm Each node not in the tree is associated with an attaching cost the weight of the smallest edge that connects it to the forming tree (infinity if no such edge exists). At each iteration we retrieve the node with the smallest attaching cost and update the attaching cost of its neighbors. Can use a priority queue! (need to add a method for updating priorities).

16 Shortest Path Algorithms (Dijkstra s Algorithm) Graph G(V,E) with non-negative weights ( distances ) Compute shortest distances from vertex s to every other vertex in the graph

17 Dijkstra s Algorithm u 1 v s x 2 y

18 Dijkstra s Algorithm (Initialize) u 1 v s x 2 y

19 Dijkstra s Algorithm (Step 1) u v 10 1 s x y

20 Dijkstra s Algorithm (Step 2) u v s x y

21 Dijkstra s Algorithm (Step 3) u v s x y

22 Dijkstra s Algorithm (Step 4) u v s x y

23 Dijkstra s Algorithm (Done) u v s x y

24 Shortest Path Algorithms (Dijkstra s Algorithm) Algorithm Maintain array d (minimum distance estimates) Init: d[s]=0, d[v]= v V-s Priority queue of vertices not yet visited select minimum distance vertex, visit v, update neighbors

25 Dijkstra s Algorithm Dijkstra(G: graph with vertices v 0 v n-1 and weights w[u][v]) // computes shortest distance of vertex 0 to every other vertex create a set vertexset that contains only vertex 0 d[0] = 0 for (v = 1 through n-1) d[v] = infinity for (step = 2 through n) find the smallest d[v] such that v is not in vertexset add v to vertexset for (all vertices u not in vertexset) if (d[u] > d[v] + w[v][u]) d[u] = d[v] + w[v][u]

26 Dijkstra s Algorithm Using a priority queue u v S0: S1: [5,x], [10,u] s S2: [7,y], [8,u], [14,v] S3: [8,u],[13,v] S4: [9,v] 5 7 S5: 5 x 2 7 y

27 Dijkstra s Algorithm How to obtain the shortest paths themselves? At each vertex maintain a pointer that tells you from which vertex you arrived.

28 MST as an optimization problem We can formulate the MST problem as follows: Given a spanning tree of a weighted graph G=(V,E) with weights w(e) assigned to edges in E define a cost function for a spanning tree T as: Objective: find a tree T* that minimizes S(T)

29 Greedy Algorithms MST and shortest path are examples of optimization problems. In principle need to consider all possible solutions Not feasible in most cases. We used a greedy approach instead: at each step we modified the solution in a way which was best given the limited information we had. In some cases the greedy approach leads to an optimal solution (Prim s, Dijkstra s)

Depth-first Search (DFS)

Depth-first Search (DFS) DFS Strategy: First follow one path all the way to its end, before we step back to follow the next path. (u.d and u.f are start/finish time for vertex processing) CH08-320201: