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Basic Graph Algorithms

Basic Graph Algorithms. Based on slides provided by Prof. Jennifer Welch. Broadcast Over a Rooted Spanning Tree. Suppose processors already have information about a rooted spanning tree of the communication topology tree: connected graph with no cycles

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Basic Graph Algorithms

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  1. Basic Graph Algorithms Based on slides provided by Prof. Jennifer Welch

  2. Broadcast Over a Rooted Spanning Tree • Suppose processors already have information about a rooted spanning tree of the communication topology • tree: connected graph with no cycles • spanning tree: contains all processors • rooted: there is a unique root node • Implemented via parent and children local variables at each processor • indicate which incident channels lead to parent and children in the rooted spanning tree

  3. Broadcast Over a Rooted S.T. • root initially sends M to its children • when a processor receives M from its parent • sends M to its children • terminates (sets a local boolean to true) Exercise to transform this pseudocode into a state machine style description

  4. Complexity Analysis • Synchronous model: • time is depth of the spanning tree, which is at most n - 1 • number of messages is n - 1, since one message is sent over each spanning tree edge • Asynchronous model: • same time and messages

  5. Convergecast • Again, suppose a rooted spanning tree has already been computed by the processors • parent and children variables at each processor • Do the opposite of broadcast: • leaves send messages to their parents • non-leaves wait to get message from each child, then send combined info to parent

  6. f a b c d e c,f,h b,d f,h d e,g g h g h Convergecast solid arrows: parent-child relationships dotted lines: non-tree edges

  7. Finding a Spanning Tree Given a Root • Having a spanning tree is very convenient. • How do you get one? • Suppose a distinguished processor is known, to serve as the root. • Modify the flooding algorithm…

  8. Finding a Spanning Tree Given a Root • root sends M to all its neighbors • when non-root first gets M • set the sender as its parent • send "parent" msg to sender • send M to all other neighbors (if no other neighbors, then terminate) • when get M otherwise • send "reject" msg to sender • use "parent" and "reject" msgs to set children variables and know when to terminate (after hearing from all neighbors)

  9. c b b c a a d f f d e e g h g h Execution of Spanning Tree Alg. Both models: O(m) messages O(diam) time m = number of edges Asynchronous: not necessarily BFS tree Synchronous: always gives breadth-first search (BFS) tree

  10. Big O Notation • Consider f(n) and g(n) such thatf(n) and g(n) are non-negative for all n • f(n) is O(g(n) if there exists N, anda positive constant k such that,for all n > N f(n) ≤ k g(n)

  11. Spanning Tree Without a Specified Root • Read section 2.5

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