CS 584

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

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

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

4. 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.

5. 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} for all v in (V - VT) do d[v] = min(d[v], w[u, v]) end while end Procedure

7. 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?

8. 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.

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

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

11. 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.

12. 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} for all v in (V - VT) do L[v] = min(L[v], L[u] + w[u, v]) end while end Procedure

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

14. 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

15. 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.

16. Analysis • Source Partitioned • 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

17. 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