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Chapter 9: Graphs

Mark Allen Weiss: Data Structures and Algorithm Analysis in Java. Chapter 9: Graphs. Summary. Lydia Sinapova, Simpson College. Summary of Graph Algorithms. Basic Concepts Topological Sort Shortest Paths Spanning Trees Scheduling Networks Breadth-First and Depth First Search

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Chapter 9: Graphs

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  1. Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Chapter 9: Graphs Summary Lydia Sinapova, Simpson College

  2. Summary of Graph Algorithms • Basic Concepts • Topological Sort • Shortest Paths • Spanning Trees • Scheduling Networks • Breadth-First and Depth First Search • Connectivity

  3. Basic Concepts • Basic definitions: vertices and edges • More definitions: paths, simple paths, cycles, loops • Connected and disconnected graphs • Spanning trees • Complete graphs • Weighted graphs and networks • Graph representation • Adjacency matrix • Adjacency lists

  4. Topological Sort RULE:If there is a path from u to v, then v appears after u in the ordering. Graphs: directed,acyclic Degree of a vertex U: the number of outgoing edges Indegree of a vertex U: the number of incoming edges The algorithm for topological sort uses "indegrees" of vertices. Implementation: with a queue Complexity: Adjacency lists: O(|E| + |V|), Matrix representation: O(|V|2)

  5. Shortest Path in Unweighted Graphs Data Structures needed: Distance Table Queue Implementation: Breadth-First search using a queue Complexity: Matrix representation:O(|V|2) Adjacency lists - O(|E| + |V|)

  6. Algorithm • Store s in a queue, and initialize • distance = 0 in the Distance Table • 2.While there are vertices in the queue: • Read a vertex vfrom the queue • For all adjacent verticesw: • If distance = -1 (not computed) • Distance = (distance to v) + 1 • Parent = v • Append wto the queue

  7. Shortest Path in Weighted Graphs – Dijkstra’s Algorithm 1. Storesin a priority queue withdistance = 0 2. While there are vertices in the queue DeleteMina vertexvfrom the queue For all adjacent verticesw: Compute new distance Store in / update Distance table Insert/update in priority queue

  8. Complexity O(E logV + V logV) = O((E + V) log(V)) Each vertex is stored only once in the queue – O(V) DeleteMin operation is :O( V logV ) Updating the priority queue – search and inseart:O(log V) performed at most for each edge:O(E logV)

  9. Spanning Trees Spanning tree: a tree that contains all vertices in the graph.   Number of nodes: |V| Number of edges: |V|-1

  10. Spanning Trees of Unweighted Graphs Breadth-First Search using a queue Complexity:O(|E| + |V|) - we process all edges and all nodes

  11. Min Spanning Trees of Weighted Graphs – Prim’s Algorithm Find a spanning tree with the minimal sum of the weights. Similar to shortest paths in a weighted graphs. Difference: we record the weight of the current edge, not the length of the path. Implementation: with a Priority Queue Complexity:O(|E| log (|V|) )

  12. Min Spanning Trees of Weighted Graphs – Kruskal’s Algorithm The algorithm works with the set of edges, stored in Priority queue. The tree is built incrementally starting with a forest of nodes only and adding edges as they come out of the priority queue Complexity: O(|E| log (|V|))

  13. Scheduling Networks • Find the earliest occurrence time (EOT) of an event • Find the earliest completion time (ECT) of an activity • Find the slack of an activity: how much the activity can be delayed without delaying the project • Find the critical activities: must be completed on time in order not to delay the project Directed simple acyclic graph Nodes:events, Source, Sink Edges:activities (name of the task, duration)

  14. EOT and ECT Earliest occurrence time of an event EOT(source) = 0 EOT( j ) = max ( EOTs of all activities preceding the event) Earliest completion time of an activity ECT( j, k ) = EOT( j ) + duration( j, k) 1. Sort the nodes topologically 2. For each event j : initialize EOT(j) = 0 3. For each event i in topological order For each event j adjacent from i : ECT(i,j) = EOT(i) + duration (i,j) EOT(j) = max(EOT(j) , ECT(i,j))

  15. Slack of activities Compute latest occurrence time LOT(j) , slack(i, j) 1. Initialize LOT(sink) = EOT(sink) 2. For each non-sink event i in the reverse topological order: LOT(i) = min(LOT( j ) – duration( i, j)) 3. For each activity (i,j) slack( i, j ) = LOT( j ) – ECT( i, j) Critical activities: slack(j) = 0 Critical path in the project: all edges are activities with slack = 0 The critical path is found using breadth-first (or depth-first) search on the edges Complexity: O(V + E) with adjacency lists, O(V2) with adjacency matrix

  16. bfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from beginning of L if w not visited add (v,w) to T visit(w) BFS and DFS dfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from end of L if w not visited add (v,w) to T visit(w) Visit ( vertex v) mark v as visited for each edge (v,w) add edge (v,w) to end of L Complexity: O( V + E)

  17. Graph Connectivity • Connectivity • Biconnectivity – no cut vertices • Articulation Points and Bridges • Connectivity in Directed Graphs • Strongly connected • Unilaterally connected • Weakly connected

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