13th International Conference on Membrane Computing - MTA Sztaki

Fast distributed DFS solutions for edge-disjoint paths in digraphs
accepts this token, becoming the new frontier. The visited notification housekeeping
is performed in parallel with the main search. A frontier cell, which does
not have any (more) unvisited arc, sends back a backtrack token to its search
path predecessor, to return the frontier.
There are two cases of backtrack: (1) because Algorithm C avoids revisiting
the already visited cells, if σ i has any temporarily visited child, σ k , we consider
this is a backtrack from σ k (as in the classical DFS); (2) σ i receives a backtrack
token from σ k . In both cases, σ i computes its reach-number as the minimum of
its depth (line 4.7) and its st-successors’ reach-numbers (lines 4.13–15, 4.18–19).
If σ i decreases its reach-number, it records and sends its new reach-number to
all its neighbours, which is performed in parallel with the main search. Furthermore,
if σ k ’s reach-number is greater than σ i ’s depth, then σ i sends a discarding
notification to σ k (line 4.20).
On receiving a discarding notification, cell σ i sets itself as permanently visited
and sets its reach-number as infinite (lines 5.4, 5.6). Also, it sends a permanently
visited notification and an update to all its temporarily visited parents (lines 5.7–
5.12), σ j ’s. On receiving an update from σ i , cell σ j computes its reach-number
(lines 6.4–11). If σ j increases its reach-number, it records and sends an update
to all its temporarily visited parents (line 6.12). If σ i ’s reach-number is greater
than σ j ’s depth, then σ j sends a discarding notification to σ i (line 6.14).
Once cell σ i is discarded, it also notifies all its st-successors to discard themselves
(lines 5.13–15). This pruning propagation is performed in parallel with the
main search. However, this propagation travels along search path traces, which
is not the shortest possible path. If the pruning propagation does not finish when
the round ends, we let the delayed pruning propagation discard cells in the next
round.
We now consider two race conditions: (1) a cell can be visited before it is
discarded by the current pruning propagation, which has not arrived yet; (2)
a cell can be visited before it is discarded by the delayed pruning propagation,
which has not arrived yet. To solve these problems, we use the same strategy.
Once discarded, the cell immediately backtracks (line 4.5) and sends an update
to all its temporarily visited parents (lines 5.7–12).
If the search path reaches the target cell, σ t (line 4.4), then round #r succeeds:
an augmenting path, α r , is found and σ t sends a path confirmation back
to σ s . A successful round #r increments the set of edge-disjoint paths: while
moving towards σ s , the confirmation reshapes the existing edge-disjoint paths
and the newly found augmenting path, α r , building a larger set of edge-disjoint
paths, and a new residual digraph. Thus, lines 3.17–19 of Pseudocode 3 are actually
done within Pseudocode 4, during the return from a successful search.
After receiving the path confirmation, σ s initiates a global reset, which changes
all temporarily visited cells and arcs to unvisited (line 3.20). This end-of-round
reset runs two steps ahead and in parallel with the next round, without affecting
it.
If the search path cannot reach σ t , the source, σ s , receives a backtrack token
from σ sr and the round fails (line 3.12). Then σ s sends a discarding notification
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