Parallelized Critical Path Search in Electrical Circuit Designs

In practice, for both the aforementioned generalizations,
there exist more algorithms than for running a single-pair
shortest path algorithm on all relevant pairs of vertices.
2.1. Bidirectional Dijkstra Search
If a point-to-point shortest path search is performed, then
the start point as well as the target point are known. The
original unidirectional shortest path search algorithm introduced
by Dijkstra [5], starts from the start point and performs
a shortest path search in a depth first manner until
the target point is reached. The runtime for this search is
O(n log n).
Luby and Ragde [15] presented a bidirectional version
of the Dijkstra algorithm with an expected run time of
O( √ n log n). Such a bidirectional search consists of two
phases. In the first phase, two unidirectional Dijkstra search
runs start from the start point as well as from the target.
Both runs span a tree by alternating between start and target,
and expanding the next level of reachable nodes. From
these trees, the minimum distance to the start with respect
to the target is known. As long as there are no visible edges
in both runs, alternate nodes can be expanded and added to
the tree. Thus, the shortest path between the start and target
nodes is within the two expanded search trees. In the second
phase, the shortest path out of the two trees needs to be
collected.
According to Fu et al. [9], the time complexity of the
bidirectional Dijkstra algorithm is 1
8 O(n2 ) on a single-core
or uniprocessor systems. It is obvious that this approach
can optimally be distributed over two processes running on
a dual-core or multi-processor machine.
3. General Approach to the Shortest Path
Problem
A more general approach to speed-up the point-to-point
shortest path problem is to divide the graph into separate
sub-graphs. Each of this sub-graphs can be processed on its
own in a separate process.
The main challenge is to divide the circuit represented
by the graph into meaningful units. Figure 1(a) shows a
graph with 16 nodes. The only meaningful criteria by which
the graph could be divided are partition size and number
of interlinking nodes. Usually, the created sub-graphs are
balanced, i.e. each partition should have the same amount
of nodes, as shown in Figure 1(b).
The graph partitioning of our approach uses arithmetic
and logic modules information, as depicted in Figure 2. As
a result, this heuristic can be used to speed-up the partitioning
process. In general, there are two methods of graph partitions:
static and dynamic. These methods are explained
next.
(a) A graph with 16 nodes.
(b) A graph with three balanced partitions.
Figure 1. Graph divided by minimizing the
number of interlink nodes.
Figure 2. Two different ways to partition a circuit
depending on the electrical modules.