13th International Conference on Membrane Computing - MTA Sztaki

Fast distributed DFS solutions for edge-disjoint paths in digraphs
Our algorithms use a synchronous version of Cidon’s distributed DFS [2, 16],
which avoids case (2) above. When a node is first visited, it immediately marks all
incoming arcs as visited, by sending visited notifications to its digraph parents.
These notifications run in parallel with the main search, without delaying it. All
parents are thus timely notified and, if they become visited, will not send the
visiting token to this already visited node. For example, in Figure 2 (a), search
path σ 0 .σ 1 .σ 2 .σ 3 .σ 4 .σ 6 does not revisit cell σ 3 . This is a powerful optimisation,
which reduces the DFS complexity from O(m) to O(n); we use it, but this is not
intrinsically related to our novel proposal.
When τ cannot explore further, it backtracks. The search is successful when
the search path reaches the target. A successful search path becomes a new
augmenting path and is used to increase the number of edge-disjoint paths:
while conceptually a distinct operation, the new edge-disjoint paths are typically
formed while the successful search path returns on its steps, back to the
source (this successful return is distinct from the backtrack).
Given a current search path arc, (σ i , σ j ), σ i is a search path predecessor (sppredecessor)
of σ j , and σ j is a search path successor (sp-successor) of σ i . Given a
previous search path arc, (σ i , σ j ), σ i is a search tree predecessor (st-predecessor)
of σ j , and σ j is a search tree successor (st-successor) of σ i . Until the end of
current search round, these arcs are considered temporary visited. At the end of
the round, for the next search round, these arcs may become permanently visited
or revert to unvisited.
In this paper, we propose a novel procedure to detect “dead” nodes, based
on two numerical search-specific attributes: the (known) node depth and a new
attribute which we call reach-number. A node’s depth, σ i .depth, is the number
of hops from the source to itself in the search tree. A node’s reach-number is
the minimum of its depth and all its st-successors’ reach-numbers: σ i .reach =
min({σ i .depth} ∪ {σ j .reach | (σ i , σ j ) ∈ E ′ }), where E ′ is the current residual
arcs set and assuming that unvisited nodes and discarded nodes have infinite
reach-numbers.
As algorithmically determined, depths and reach-numbers start as infinite
and are further dynamically adjusted during the search process:
1. When the search path first visits a node:
(a) both its depth and reach-number are set to the current hop count.
2. When the search path backtracks to a node:
(a) its reach-number can decrease;
(b) its st-successors, which have finite reach-numbers greater than this node’s
depth, can be discarded.
3. When a node is discarded:
(a) its own reach-number becomes infinite and
(b) its parents’ reach-numbers can increase.
4. When a node’s reach-number is increased, without being discarded (because
one or more of its st-successors have increased their own reach-numbers):
(a) its parents’ reach-numbers can also increase and
175