CS 584 Project Write up Poster session for final Due on day of final HTML format Email a tar zipped file to me I will post Poster session for final 4 page poster

Graphs A graph G is a pair (V, E) Graphs can be directed or undirected V is a set of vertices E is a set of edges Graphs can be directed or undirected Terms path adjacent incident

Graph Algorithms We will consider the following algorithms Minimum Spanning Tree Single Source Shortest Paths All pairs shortest paths Transitive closure

Minimum Spanning Tree A spanning tree is a subgraph of G that is a tree containing all the nodes of G. A minimal spanning tree is a spanning tree of minimal weight If the graph G is not connected, it does not have a spanning tree. It has a spanning forest.

Prim’s MST Algorithm Procedure PRIM_MST(V, E, w, r) VT = {r}; d[r] = 0; for all v in (V - VT) do if E[r,v] exists set d[v] = w[r, v] else set d[v] = infinite end for while VT != V find a vertex u such that d[u] = min(d[v] | v in (V - VT)) VT = VT union {u} d[v] = min(d[v], w[u, v]) end while end Procedure

Parallelizing Prim’s Algorithm Since the value of d[v] for a vertex v may change every time a vertex is added, it is impossible to select more than one vertex at a time. So the iterations of the while loop cannot be done in parallel. What about parallelizing a single iteration?

Parallelizing Prim’s Algorithm Consider the calculation of the next node to add to the set. Calculates the min distance from any of the nodes already in the tree. Have all processors calculate a min of their nodes and then do a global min.

Data Decomposition | n/p | d[1..n] n

Analysis Computation ---> O(n2/p) Communication per iteration Global min ---> log2p Bcast min ---> log2p

Single Source Shortest Paths Find the shortest paths from a vertex to all other vertices. A shortest path is a minimum cost path Similar to Prim’s algorithm Note: Instead of storing distances, we store the min cost to a vertex from the vertices in the set.

Dijkstra’s Algorithm Procedure DIJKSTRA_SSP(V, E, w, s) VT = {s}; for all v in (V - VT) do if E[s,v] exists set L[v] = w[r, v] else set L[v] = infinite end for while VT != V find a vertex u such that L[u] = min(L[v] | v in (V - VT)) VT = VT union {u} L[v] = min(L[v], L[u] + w[u, v]) end while end Procedure

Parallelizing Dijkstra’s Algorithm Parallelized exactly the same way as Prim’s algorithm Exact same cost as Prim’s algorithm

All pairs shortest paths Find the shortest paths between all pairs of vertices. Three algorithms presented. Matrix Multiplication Dijkstra’s Floyd’s We will consider Dijkstra’s

Dijkstra’s Algorithm Two ways to parallelize source partitioned Partition the nodes Each processor computes Dijkstra’s sequential algorithm source parallel Run the parallel single source shortest path algorithm for all nodes Can subdivide the processors into sets and also divide the nodes into sets.

Analysis Source Partitioned Source Parallel No communication Each vertex requires O(n2) The algorithm can use at most n processors Source Parallel Communication is O(n log2 n) Each vertex requires O(n2/p) Can efficiently use more processors

Transitive Closure Determine if any two vertices are connected Computed by first computing all pairs shortest path if there is a shortest path, there is a path Parallelize the all pairs shortest path