Mesh Based APSP algorithm
Sathish Vadhiyar
Introduction
 Mesh based Cartesian topology can
be used to solve APSP (all-pair
shortest path)
 First, leading up to APSP
 Prim’s MST
 Dijikstra’s SSSP
Minimal Spanning Tree – Prim’s
Algorithm
 Spanning tree of a graph, G (V,E) – tree
containing all vertices of G
 MST – spanning tree with minimum sum
of weights
 Vertices are added to a set Vt that
holds vertices of MST; Initially contains
an arbitrary vertex,r, as root vertex
Minimal Spanning Tree – Prim’s
Algorithm
 An array d such that d[v in (V-Vt)] holds
weight of the edge with least weight
between v and any vertex in Vt; Initially
d[v] = w[r,v]
 Find the vertex in d with minimum weight
and add to Vt
 Update d
 Time complexity – O(n2)
Parallelization
 Vertex V and d array partitioned across P
processors
 Each processor finds local minimum in d
 Then global minimum across all d performed
by reduction on a processor
 The processor finds the next vertex u, and
broadcasts to all processors
Parallelization
 All processors update d; The owning
processor of u marks u as belonging to Vt
 Process responsible for v must know w[u,v]
to update d[v]; 1-D block mapping of
adjacency matrix
 Complexity – O(n2/P) + (OnlogP) for
communication
Single Source Shortest Path –
Dijikistra’s Algorithm
 Finds shortest path from the source
vertex to all vertices
 Follows a similar structure as Prim’s
 Instead of d array, an array l that
maintains the shortest lengths are
maintained
 Follow similar parallelization scheme
All-Pairs Shortest Paths
 To find shortest paths between all pairs
of vertices
 Dijikstra’s algorithm for single-source
shortest path can be used for all
vertices
 Two approaches
All-Pairs Shortest Paths
 Source-partitioned formulation: Partition the
vertices across processors
 Works well if p<=n; No communication
 Can at best use only n processors
 Time complexity?
 Source-parallel formulation: Parallelize SSSP for a
vertex across a subset of processors
 Do for all vertices with different subsets of
processors
 Hierarchical formulation
 Exploits more parallelism
 Time complexity?
All-Pairs Shortest Paths
Floyd’s Algorithm
 Consider a subset S = {v1,v2,…,vk} of
vertices for some k <= n
 Consider finding shortest path between
vi and vj
 Consider all paths from vi to vj whose
intermediate vertices belong to the set
S; Let pi,j(k) be the minimum-weight path
among them with weight di,j(k)
All-Pairs Shortest Paths
Floyd’s Algorithm
 If vk is not in the shortest path, then
pi,j(k) = pi,j(k-1)
 If vk is in the shortest path, then the
path is broken into two parts – from vi to
vk, and from vk to vj
 So di,j(k) = min{di,j(k-1) , di,k(k-1) + dk,j(k-1) }
 The length of the shortest path from vi
to vj is given by di,j(n).
 In general, solution is a matrix D(n)
Parallel Formulation
2-D Block Mapping
 Processors laid in a 2D mesh
 During kth iteration, each process Pi,j
needs certain segments of the kth row
and kth column of the D(k-1) matrix
 For dl,r(k): following are needed
 dl,k(k-1) (from a process along the same
process row)
 dk,r(k-1) (from a process along the same
process column)
 Figure 10.8
Parallel Formulation
2D Block Mapping
 During kth iteration, each of the root(p)
processes containing part of the kth row
sends it to root(p)-1 in same column;
 Similarly for the same row
 Figure 10.8
 Time complexity?
Sources/References
 END