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The set-to-set disjoint paths problem is to find
$n$
vertex-disjoint paths
${ \boldsymbol s}_{i}\leadsto { \boldsymbol t}_{j_{i}}$
(
$1\le i\le n$
,
$\{j_{1},j_{2},\ldots,j_{n}\}=\{1,2,\ldots,n\}$
) between two sets of vertices
$S=\{ { \boldsymbol s}_{1}, { \boldsymbol s}_{2},\ldots, { \boldsymbol s}_{n}\}$
and
$T=\{ { \boldsymbol t}_{1},...
Citations
... The problems of building disjoint paths in a graph have received significant attention in literature. Refer to, for example, [4]- [8] for details. It is often important to find disjoint paths that collectively pass through all vertices. ...
One of the key problems in parallel processing is finding disjoint paths in the underlying graph of an interconnection network. The
disjoint path cover
of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. Given disjoint source and sink sets,
S
= {
s
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>
,...,
s<sub>k</sub>
} and
T
= {
t
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>
,...,
t<sub>k</sub>
}, in graph
G
, an
unpaired many-to-many k-disjoint path cover
joining
S
and
T
is a disjoint path cover {
P
<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>
,...,
P<sub>k</sub>
}, in which each path
P<sub>i</sub>
runs from source
s<sub>i</sub>
to some sink
t<sub>j</sub>
. In this paper, we reveal that a nonbipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties of the unpaired type, retains such a good property. As a result, an
m
-dimensional nonbipartite torus,
m
≥ 2, with at most
f
vertex and/or edge faults has an unpaired many-to-many
k
-disjoint path cover joining arbitrary disjoint sets
S
and
T
of size
k
each, subject to
k
≥ 2 and
f
+
k
≤ 2
m
– 2. The bound of 2
m
– 2 on
f
+
k
is nearly optimal.
The distributed computing or parallel computing system uses an interconnection network as a topology structure to connect a large number of processors. The disjoint paths of interconnection networks are related to parallel computing and the fault tolerance. l-path cover of graph \(G=(V(G), E(G))\) consists of (internally) disjoint paths \(P_k\)s (\(1 \le k \le l\)), where \(\cup _{k=1}^lV(P_k)=V(G)\). The bubble-sort star graph is bipartite and has favorable reliability and fault tolerance which are critical for multiprocessor systems. We focus on the one-to-one 1-path cover, one-to-one \((2n-3)\)-path cover, and many-to-many 2-path cover of the bubble-sort star graph \(BS_n\). More specifically, let \(V(BS_n)=V_e \cup V_o\) with \(V_e \cap V_o=\emptyset\), for \(\{u, x\} \subset V_e\) and \(\{v, y\} \subset V_o\), we prove that (1) \(BS_n\) contains a 1-path cover, i.e., Hamiltonian path \(P_{uv}\), (2) \(BS_n\) contains one-to-one \((2n-3)\)-path cover \(P_k\)s (\(1 \le k \le 2n-3\)) between u and v, and (3) \(BS_n\) contains many-to-many 2-path cover \(P_{uv}\) and \(P_{xy}\), where \(n \ge 3\). Since \(BS_n\) is \((2n-3)\)-regular graph, the one-to-one \((2n-3)\)-path cover is the maximal one-to-one path cover.
The bijective connection graph encompasses a family of cube-based topologies, and
$n$
-dimensional bijective connection graphs include the hypercube and almost all of its variants with the order
$2^{n}$
and the degree
$n$
. Hence, it is important to design and implement algorithms that work in bijective connection graphs. The set-to-set disjoint paths problem is as follows: given a set of source nodes
$S=\{ \boldsymbol s _{1}, \boldsymbol s _{2},\ldots, \boldsymbol s _{p}\}$
and a set of destination nodes
$D=\{ \boldsymbol d _{1}, \boldsymbol d _{2},\ldots, \boldsymbol d _{p}\}$
in a
$k$
-connected graph
$G=(V,E)$
with
$p\le k$
, construct
$p$
paths
$P_{i}$
:
$\boldsymbol s _{i}\leadsto \boldsymbol d _{j_{i}}$
(
$1\le i\le p$
) such that
$\{j_{1},j_{2},\ldots,j_{p}\}=\{1,2,\ldots,p\}$
and the paths
$P_{i}$
are node-disjoint. Finding a solution to this problem is an important issue in parallel and distributed computation as well as the node-to-node disjoint paths problem and the node-to-set disjoint paths problem. In this paper we propose an algorithm that constructs
$p~(\le n)$
disjoint paths between any pair of node sets in
$n$
-dimensional bijective connection graphs in polynomial-order time of
$n$
. We give a proof of correctness of the algorithm as well as the estimates of the time complexity
$O(n^{3}p^{4})$
and the maximum path length
$n+p-1$
. According to a computer experiment in a locally twisted cube as an example of a bijective connection graph to construct
$n$
disjoint paths, the average time complexity of the algorithm is
$O(n^{2})$
, and the average maximum path is
$0.6333n-0.266$
.
