1. CS 584 Project Write up Poster session for final Due on day of final
HTML format
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) 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

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}
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}
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 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

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