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Chapter 15 Graphs Modified Part A - Terminology and Traversals

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Chapter Scope Directed and undirected graphs Weighted graphs (networks) Common graph algorithms Java Software Structures, 4th Edition, Lewis/Chase

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Terminology and Traversals Java Software Structures, 4th Edition, Lewis/Chase15 - 3

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Graphs Like trees, graphs are made up of nodes and the connections between those nodes. In graph terminology, we refer to the nodes as vertices and refer to the connections among them as edges Vertices are typically referenced by a name or label Edges are referenced by a pairing of the vertices (A, B) that they connect An undirected graph is a graph where the pairings representing the edges are unordered (order doesn’t matter) Java Software Structures, 4th Edition, Lewis/Chase

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Undirected Graphs An undirected graph: Java Software Structures, 4th Edition, Lewis/Chase

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Undirected Graphs An edge in an undirected graph can be traversed in either direction Two vertices are said to be adjacent if there is an edge connecting them Adjacent vertices are sometimes referred to as neighbors An edge of a graph that connects a vertex to itself is called a self-loop or sling An undirected graph is considered complete if it has the maximum possible number of edges connecting vertices. How many edges? Java Software Structures, 4th Edition, Lewis/Chase

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Undirected Graphs A path is a sequence of edges that connects two vertices in a graph The length of a path in is the number of edges in the path (or the number of vertices minus 1) An undirected graph is considered connected if for any two vertices in the graph there is a path between them Java Software Structures, 4th Edition, Lewis/Chase

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Undirected Graphs An example of an undirected graph that is not connected: Java Software Structures, 4th Edition, Lewis/Chase

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Cycles A cycle is a path in which the first and last vertices are the same and none of the edges are repeated A graph that has no cycles is called acyclic Java Software Structures, 4th Edition, Lewis/Chase

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Directed Graphs A directed graph, sometimes referred to as a digraph, is a graph where the edges are ordered pairs of vertices (edges have arrows) This means that the edges (A, B) and (B, A) are separate, directional edges in a directed graph Java Software Structures, 4th Edition, Lewis/Chase

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Directed Graphs A directed graph with – vertices A, B, C, D – edges (A, B), (A, D), (B, C), (B, D) and (D, C) Java Software Structures, 4th Edition, Lewis/Chase

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Directed Graphs Previous definitions change slightly for directed graphs – a path in a direct graph is a sequence of directed edges that connects two vertices in a graph – a directed graph is connected if for any two vertices in the graph there is a path between them if a directed graph has no cycles, it is possible to arrange the vertices in a sequence such that vertex A precedes vertex B in the sequence, if an edge exists from A to B Java Software Structures, 4th Edition, Lewis/Chase

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Directed Graphs A connected directed graph and an unconnected directed graph: Java Software Structures, 4th Edition, Lewis/Chase

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A tree as a graph A tree is an undirected acyclic graph. A rooted tree has one vertex designated as the root. A directed tree is a directed graph with edges directed away from the root, and which would be a tree if the directions on the edges were ignored

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Weighted Graphs A weighted graph, sometimes called a network, is a graph with weights (or costs) associated with each edge The weight of a path in a weighted graph is the sum of the weights of the edges in the path Weighted graphs may be either undirected or directed For weighted graphs, we represent each edge with a triple including the starting vertex, ending vertex, and the weight (Boston, New York, 120) Java Software Structures, 4th Edition, Lewis/Chase

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Weighted Graphs We could use an undirected network to represent flights between cities The weights are the cost of flight Java Software Structures, 4th Edition, Lewis/Chase

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Weighted Graphs A directed version of the graph could show different costs depending on the direction Java Software Structures, 4th Edition, Lewis/Chase

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Common Graph Algorithms There are a number of a common algorithms that may apply to undirected, directed, and/or weighted graphs These include – various traversal algorithms – algorithms for finding the shortest path length (edge count) – algorithms for finding the least costly path in a network – algorithms for answering simple questions (such as connectivity) Java Software Structures, 4th Edition, Lewis/Chase

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Graph Traversals There are two main types of graph traversal algorithms – breadth-first: behaves much like a level-order traversal of a tree – depth-first: behaves much like the preorder traversal of a tree One difference from trees: there is no root node present in a graph Graph traversals may start at any vertex in the graph Java Software Structures, 4th Edition, Lewis/Chase

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Graph Traversals Java Software Structures, 4th Edition, Lewis/Chase

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BFT starting from vertex v create a queue Q visit v, mark v as visited and put v into Q while Q is non-empty remove the head element u of Q visit, mark & enqueue all (unvisited) neighbors of u Java Software Structures, 4th Edition, Lewis/Chase

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DFT starting from vertex v create a stack S visit, mark v as visited and push v onto S while S is non-empty peek at the top u of S if u has an (unvisited) neighbor w, visit w, mark w and push it onto S else pop S Java Software Structures, 4th Edition, Lewis/Chase

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Chapter 15 Graphs Modified Part B – Data Structures for Graphs

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Basic Data Structures for Graphs Adjacency lists Adjacency matrices Java Software Structures, 4th Edition, Lewis/Chase

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For weighted graph - network Can add weight field to nodes in adjacency lists.

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Chapter 15 Graphs Modified Part C – Graph Algorithms 1

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Basic Graph Algorithms Un-weighted graphs/digraphs – Connectivity – Cycle detection – Shortest path – Spanning tree – Topological sort Java Software Structures, 4th Edition, Lewis/Chase

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Basic Graph Algorithms Weighted graphs/digraphs – Minimum spanning tree – Cheapest path – Traveling salesman problem Java Software Structures, 4th Edition, Lewis/Chase

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Dynamic graphs Adding and removing vertices Adding and removing edges Java Software Structures, 4th Edition, Lewis/Chase

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Connectivity For a graph (undirected) how can we determine if it is connected? Answer: Do a depth first traversal from any vertex and see if all of the vertices are reached. For a digraph, how do can we determine if it is connected? Answer: Do a depth first traversal from every vertex and see if all other vertices are reached. Java Software Structures, 4th Edition, Lewis/Chase

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Java Software Structures, 4th Edition, Lewis/Chase

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Cycle detection For a graph, how can we detect if it contains any cycles? Do a depth first traversal and look for “back edges”, edges from current node to a node already visited. For a digraph – Do the same, for every vertex as starting vertex. Java Software Structures, 4th Edition, Lewis/Chase

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Finding shortest path (un-weighted) For a graph, how do we find the shortest path (fewest edges) from vertex A to vertex B? Do a breadth first traversal starting at A and going until B is reached, recording the path length and predecessor at each node visited. Need more info stored at each node visited. For a digraph – Same Java Software Structures, 4th Edition, Lewis/Chase

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Shortest path form A to B create a queue Q visit A, mark A as visited, set path length=0, set pred=null, and put A into Q repeat until B is visited remove the head element u of Q for each w: (unvisited) neighbors of u visit w, mark w, set path length of w, set pred of w=u, and enqueue w Follow the preds backwards from B to A Java Software Structures, 4th Edition, Lewis/Chase

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Java Software Structures, 4th Edition, Lewis/Chase

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Chapter 15 Graphs Modified Part D – Graph Algorithms 2

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Basic Graph Algorithms Un-weighted graphs/digraphs – Spanning tree – Topological sort Java Software Structures, 4th Edition, Lewis/Chase

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Constructing a spanning tree Assume an undirected graph A spanning tree is a subgraph of the original graph which includes all of the vertices but only some of the edges. If the graph has N vertices, the spanning tree will have N-1 edges. It will have no cycles; it is a tree It uses the minimum number of edges to connect all of the vertices together -- Unique? Lots of practical applications If a network contains any cycles, an edge can be removed and the graph is still connected. Java Software Structures, 4th Edition, Lewis/Chase

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Finding a spanning tree Do a depth first traversal and record all edges traversed to get to a “new” edge. Java Software Structures, 4th Edition, Lewis/Chase

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Find Spanning tree – DFT starting from vertex v create a stack S Initialize spanning tree edge list to empty mark v as visited and push v onto S while S is non-empty peek at the top u of S if u has an (unvisited) neighbor w, add edge from u to w to spanning tree, mark w and push it onto S else pop S Java Software Structures, 4th Edition, Lewis/Chase

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Java Software Structures, 4th Edition, Lewis/Chase

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Topological sort A diagraph is acyclic if it has no cycles. Such a graph is often referred to as a directed acyclic graph, or DAG, for short. DAGs are used in many applications to indicate precedence among events. A directed graph G is acyclic if and only if a depth-first search of G yields no back edges. Topological sort of a DAG: a linear ordering (sequence) of vertices such that if (u, v) is an edge of the DAG, then u appears somewhere before v in that ordering. Java Software Structures, 4th Edition, Lewis/Chase

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Topological sort Do a DFT of the DAG and put vertices onto the front of a linked list as they are finished. https://www.cs.usfca.edu/~galles/visualization/T opoSortDFS.html https://www.cs.usfca.edu/~galles/visualization/T opoSortDFS.html Java Software Structures, 4th Edition, Lewis/Chase

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Alternate algorithm for top. sort of DAG 1.Compute the indegrees of all vertices 2.Find a vertex U with indegree 0 and append it to the ordering 3.If there is no such vertex then there is a cycle and the vertices cannot be ordered. Stop. 4.Remove U and all its edges (U,V) from the graph. 5.Update the indegrees of the remaining vertices. 6.Repeat steps 2 through 4 while there are vertices to be processed. Java Software Structures, 4th Edition, Lewis/Chase

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Chapter 15 Graphs Modified Part E – Graph Algorithms 3

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Basic Graph Algorithms Weighted graphs/digraphs – Minimum spanning tree – Cheapest path – Traveling salesman problem Java Software Structures, 4th Edition, Lewis/Chase

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MST for weighted graph We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree. A minimum spanning tree (MST) or minimum weight spanning tree is a spanning tree with weight less than or equal to the weight of every other spanning tree. We will assume positive weights only. Java Software Structures, 4th Edition, Lewis/Chase

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Greedy algorithms A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. Java Software Structures, 4th Edition, Lewis/Chase

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Prim’s algorithm 1.Initialize a tree with a single vertex, chosen arbitrarily from the graph. 2.Grow the tree by one edge: Of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree. 3.Repeat step 2 (until all vertices are in the tree). Java Software Structures, 4th Edition, Lewis/Chase https://www.cs.usfca.edu/~galles/visualization/Algorithms.html

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Kruskal's algorithm create a forest F (a set of trees), where each vertex in the graph is a separate tree create a set S containing all the edges in the graph while S is nonempty and F is not yet spanning – remove an edge with minimum weight from S – if that edge connects two different trees, then add it to the forest, combining two trees into a single tree At the termination of the algorithm, the forest forms a minimum spanning forest of the graph. If the graph is connected, the forest has a single component and forms a minimum spanning tree. Java Software Structures, 4th Edition, Lewis/Chase

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Shortest (cheapest) path in a weighted graph Dijkstra’s shortest path algorithm computes the shortest paths to all vertices from a starting vertex. It is a greedy algorithm It works by growing a tree, adding the “closest” node to the current tree to get the next version of the tree. The node added is the one that is adjacent to a node already in the tree and closest to the start node using only edges already included in the tree and one frontier edge connecting it that tree. After adding a node, the distances of the nodes already in the tree must be updated because the new node may provide a shorter path to some of those nodes. Java Software Structures, 4th Edition, Lewis/Chase

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Let the node at which we are starting be called the initial node. Let the distance of node Y be its distance from the initial node. Dijkstra's algorithm will assign some initial distance values and will try to improve them step by step. Java Software Structures, 4th Edition, Lewis/Chase Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes. 2.Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisited nodes called the unvisited set consisting of all the nodes. 3.For the current node, consider all of its unvisited neighbors and calculate their tentative distances. Compare the newly calculated tentative distance to the current assigned value and assign the smaller one. For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B (through A) will be = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, keep the current value.

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Java Software Structures, 4th Edition, Lewis/Chase When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again. 5.If all nodes have been marked visited or if the smallest tentative distance among the nodes in the unvisited set is infinity (occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished. 6.Select the unvisited node with the smallest tentative distance, and set it as the new "current node" then go back to step 3.

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CN5E by Tanenbaum & Wetherall, © Pearson Education-Prentice Hall and D. Wetherall, 2011 First 5 steps in computing the shortest paths from A.

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Dijkstra's algorithm picks the unvisited vertex with the lowest-distance. It then calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Marked visited (set to red) when done with neighbors. Java Software Structures, 4th Edition, Lewis/Chase

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Animated demos Java Software Structures, 4th Edition, Lewis/Chase https://www.cs.usfca.edu/~galles/visualization/Algorithms.html

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Traveling salesman problem Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? Java Software Structures, 4th Edition, Lewis/Chase

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Brute Force Solutions of TSP Java Software Structures, 4th Edition, Lewis/Chase The most direct solution would be to try all permutations (ordered combinations) and see which one is cheapest (using brute force search). The running time for this is O(N!) where N is the number of cities, so this solution becomes impractical somewhere around 20 cities.

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Java Software Structures, 4th Edition, Lewis/Chase An exact solution for 15,112 German towns was found in 2001 using the cutting-plane method proposed by George Dantzig, Ray Fulkerson, and Selmer M. Johnson in 1954, based on linear programming. The computations were performed on a network of 110 processors located at Rice University and Princeton University. The total computation time was equivalent to 22.6 years on a single 500 MHz Alpha processor.

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Java Software Structures, 4th Edition, Lewis/Chase In March 2005, the travelling salesman problem of visiting all 33,810 points in a circuit board was solved using Concorde TSP Solver: a tour of length 66,048,945 units was found and it was proven that no shorter tour exists. The computation took approximately 15.7 CPU-years. In April 2006 an instance with 85,900 points was solved using Concorde TSP Solver, taking over 136 CPU-years.

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Java Software Structures, 4th Edition, Lewis/Chase TSP is an NP-hard problem - Comp 314/Comp 332

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Chapter 15 Graphs Modified Part F – Graph Algorithms 4

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Dynamic graphs Adding and removing vertices Adding and removing edges Java Software Structures, 4th Edition, Lewis/Chase

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Dynamic structures of graphs So far we have worked with static data structures for graphs. We didn’t add vertices or edges, and we didn’t delete vertices or edges. Now let’s consider how to handle those cases Java Software Structures, 4th Edition, Lewis/Chase

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With adjacency matrices Adding a vertex means adding a row and column to a matrix Adding an edge (assuming the vertices are already there) means filling in an entry (or two) in the matrix Deleting a vertex means removing a row and a column from the matrix, which deletes the edges incident with that vertex Deleting an edge means zeroing out an entry in the matrix. Arrays are not the most flexible structures for adding and deleting. Java Software Structures, 4th Edition, Lewis/Chase

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With adjacency lists Adding a vertex means adding a new list (with any additional vertex info you are keeping). Adding an edge means adding node(s) to all list(s) for vertices that edge is incident with. Deleting a vertex means removing a list; this will delete the edges from that vertex, and other lists may have to be updated as well Deleting an edge means removing the node(s) for it in the appropriate list(s). This structure is more flexible, especially if you use an ArrayList, or any list for the vertex info and links. Java Software Structures, 4th Edition, Lewis/Chase

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