Ellen Walker CPSC 201 Data Structures Hiram College Graphs Ellen Walker CPSC 201 Data Structures Hiram College

Basic Graph Terminology Edge (link) cycle C A C A No cycle D D B cycle B Vertex (node) Path Note: {A,C} in unordered is like (A,C) + (C A) in ordered Physical layout isn’t relevant, only the connections. In the directed graph, not every node is reachable from every other node, even though the underlying undirected graph is connected E E Directed Graph Undirected Graph V = {A,B,C,D,E} E = {(A,B), (A,E), (A,D), (D,E), (A,C).(C,A)} V = {A,B,C,D,E} E = {{A,B}, {A,C}, {A,D}, {A,E}, {D, E}}

More Terminology C A is adjacent to B, C, D and E (degree is 4) A D is adjacent to A and E ( degree of D is 2) There are 2 connected components (ABCDE and FG) Path BAE is a simple path ; Path BABAC is not. Path ADEA is a cycle A D B F E G

Graph Application Example: Prerequisites CPSC 171 CPSC 240 CPSC 172 CPSC 152 CPSC 386 CPSC 201 CPSC 331 CPSC 361 CPSC 400 CPSC 401

Graph Application Example: Travel Planning 2.5 CLE 1 DEN 1 2 2 ORD PHL 4 4 1.5 SFO 1.5 4 CVG 2 Calculating time cost of connecting flights from CLE to SFO - time in hrs flying time 4 CLT SJC 4

Graphs and Trees Every tree is a graph It is connected It has no cycles Either all links are directed “downward” away from the root, or it is undirected Every connected, no-cycle, undirected graph can be a tree Any node can be chosen as the root

Graph ADT A set of methods Other options might be available Create a new graph (specify # vertices) Iterate through vertices of graph Iterate through neighbors of a vertex Is there an edge between 2 vertices? What is the weight of an edge between 2 vertices? Insert an edge into the graph Other options might be available E.g. add vertices, connect a vertex to another, etc. No built-in graph class in Java

Representing Vertices and Edges Vertex represented as integer Doesn’t allow named vertices Index into an array of names if you need them Edge represented by a class Source vertex, destination vertex, weight (default 1.0) For undirected graph, add each edge in both directions

Graph Representations Adjacency List Each vertex is associated with a list of edges Looks a lot like a hash table; linked lists hanging off an array Adjacency Matrix 2D matrix: M[R][C] = weight of edge from R to C

Adjacency List Example (Directed) 2 1 2 3 4 1 2 3 4 3 4 1 4

Adjacency List Example (Undirected) 2 1 2 3 4 1 2 3 4 3 1 4 3 4

Adjacency Matrix Example 10 2 1 2 3 4 5 8 7 x 10 8 7 5 3 1 7 5 4 Value at row,col = weight from row to col The value “x” means infinity (Double.POSITIVE_INFINITY in Java) Unweighted graph can use boolean: (edge = true)

Graph Traversals vs. Tree Traversals Tree has no cycles; graph can have cycles Mark visited nodes, and don’t repeat them! Tree is connected; graph does not have to be After your traversal is “finished” check to see if there are leftover (not visited) nodes A graph traversal induces a tree structure on the graph The edges that are used to “find” nodes are part of the tree The root of the tree is the node that started the traversal

Breadth-First vs. Depth-First Visit all my immediate neighbors before visiting their neighbors In tree - all children before any grandchildren (level-order traversal) Depth-first Visit first child, first grandchild, … as deep as possible before looking at second child

Breadth First Algorithm For each vertex If the vertex is unseen, offer it to the queue and mark it identified While the queue is not empty Current_vertex = q.poll(); For each neighbor of the current vertex If the neighbor is not yet visited, offer it to the queue and mark it identified Mark current vertex as visited End for End While

Three Kinds of Nodes Unseen (white in your textbook) Nodes that have not yet been seen at all Identified (pale blue in your textbook) Nodes that are currently in the queue Must have at least one visited neighbor Visited (dark blue in your textbook) Nodes that have been completed All neighbors are visited or identified

Breadth-First Search example

Depth First Search Algorithm For each vertex If the vertex is unseen, push it on the stack and mark it identified While the stack is not empty Current_vertex = q.pop(); For each neighbor of the current vertex If the neighbor is not yet visited, push it on the stack and mark it identified Mark current vertex as visited End for End While

Depth First Search Example

Recursive Depth First Search Rec_dfs(current-node){ mark current-node identified for each neighbor of current_node if neighbor is not identified (set parent of neighbor to current node) rec_dfs(neighbor) mark current-node visited }

Comments on Recursive DFS Recursive DFS visits the same nodes, in the same order as Stack DFS Recursive DFS actually uses a stack (Why?) Why is there no recursive BFS?

DFS / BFS analysis Each search visits every node, and considers every neighbor of every node The number of nodes visited is O(V) The number of neigbors considered is O(E) Every vertex is a neighbor for one of its edges Time proportional to V+E But, since E >> V (there cannot be fewer than O(V) edges, but there can be O(V2 )), overall time is O(E) in both cases

Common Graph Algorithms Dijkstra’s algorithm Shortest path from one vertex to all others in a weighted graph If the graph isn’t weighted use BFS. The path you take to the node is by definition the shortest. Prim’s algorithm Finding the spanning tree with the minimum total weight A spanning tree is a subset of (V-1) edges that minimally connects the graph

Dijkstra’s Algorithm The main idea: If the (best possible known) cost from point A to point B is x And there is a connection from B to C that costs y, Then the (best possible known) path from point A to point C through B costs x+y

Keeping Track of Paths We will find all possible distances from a starting node, A Create a table of all other nodes (e.g. B,C,D) Initially, distances are all infinite (unreachable) Now, we’ll visit one node at a time (starting with A) and adding all its links to the table…

Visiting a node Assume you are looking at node A Suppose there is an edge from A to B costing k Let d(A) and d(B) be the distances (in the table) from the starting node to A and B respectively If d(A)+k < d(B), change d(B) to d(A)+k and mark B’s predecessor as A Do the above for all edges of A

Putting it all together Initialize the table Visit the starting node While (not all nodes have been visited) Visit the node that is closest to the starting point When done, the table will contain all shortest distances