Makino, Kozo

Kyoto University Research Information Repository

Title2-Disjoint Path Covers in Mesh-Torus(New Trends in Theoryof Computation and Algorithm)Author(s)Makino, KozoCitation数理解析研究所講究録 (2006), 1489: 181-187Issue Date2006-05URL http://hdl.handle.net/2433/58210RightType Departmental Bulletin PaperTextversionpublisherKyoto University2-Disjoint Path Covers in Mesh-Torus(Kozo Makino)(makino2@is.titech.ac.jp)Dept. of Mathematical and Computing Sciences,Tokyo Inst. of Technology, JapanAbstractFor a graph $G$ , a $k$-disjoint path cover prvblem, is finding $k$ vertex-disjointpaths joining $k$ distinct source-sink pairs that cover all vertices in the graph.We call a set containing such $k$ disjoint paths k-DPC. For a reculanglargrid graph(we call it mesh), it is known that there exists a necessary andsufficient condition for the existence of 1-DPC in mesh of size $\geq 4\mathrm{x}4[3]$ . Inthis paper, we treat a mesh-torus $M$ , which is obtained from mesh by addingcolumn- and row- wraparound edges, of size even $\mathrm{x}$ even. More precisely,such $M$ is expressed by $M=(V=V_{1}\cup V_{2}, E)$ , since mesh-torus is bipartite.For any 2 distinct source-sink pairs in $M$ of size $\geq 4\mathrm{x}4$ : $(s,t),$ $(u,v)$ , iftwo of them are in $V_{1}$ , the others are in $V_{2}$ , we show the existence of 2-DPCjoining $s$ and $t,$ $u$ and $v$ .1 IntroductionOne of issues in various interconnection networks is finding node-disjoint paths. A node-disjointpath can be used as parallel paths for efficient data routing among vertices. An interconnectionnetworks is modeled as a graph, in which vertices and edges corresponded to nodes and links,respectively. In a graph, a node-disjoint path problem in interconnection networks is calledvertex-disjoint path problem(disjoint path problem, for short).There are three categories of disjoint paths, i.e., one-to-one, one-to-many, and many-to-many.One-to-one type has a single source $s$ and a single sink $t$ . The purpose is finding disjoint pathsjoining $s$ and $t.$ One-to-many type deals with the disjoint paths joining a single source 8 and $k$distinct sinks $t_{1},$ $t_{2},$ $\cdots,$ $t_{k}$ . $\mathrm{M}\mathrm{i}\mathrm{y}- \mathrm{t}\triangleright \mathrm{m}\mathrm{a}\mathrm{n}\mathrm{y}$ type deals with the disjoint paths joining $k$ distinctsources $s_{1},$ $s_{2},$ $\cdots,$ $s_{k}$ and $k$ distinct sinks $t_{1},t_{2},$ $\cdots,$ $t_{k}$ . The works in many-to-many type havea relative paucity because of its difficulty and some results can be found in [5, ?]. All ofthree types of disjoint paths do not care whether it covers all vertices in the graph or not. A k-disjoint path cover pmblem($k$-DPCP, for short) in a graph $G$ is finding disjoint paths containingall vertices in $G$ . We denote a set containing such $k$ disjoint paths is k-DPC. If necessary, wedenote $k- \mathrm{D}\mathrm{P}\mathrm{C}[G, (s_{1},t_{1}), (s_{2}, t_{2}), \cdots, (s_{k}, t_{k})]$ . J.H.Park et al. [8] proposed a certain graph $G$which has a $k$-DPC (more precisely, $f(\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{o}\mathrm{r}/\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s})- \mathrm{f}\mathrm{a}\mathrm{u}\mathrm{l}\mathrm{t}$ -free $k$-DPC) for any sourceand sink sets.The case of $k=1$ , the problem corresponds to hamiltonian path problem in a graph $G$ ,that is, the problem is to find a path joining given $s$ and $t$ covers all vertices in $G$ . For 1-DPCP(hamiltonian path problem), there are some definitions in previous works$[6, 8]$ . A graph1489 2006 181-187 181$G$ is hamiltonian(resp.hamiltonian-connected) if there exists a hamiltonian cycle (resp. if eachpair of vertices are joined by a hamiltonian path) in $G$ . Note that if a graph $G$ is hamltonian,then $2\leq\delta(G)$ where $\delta(G)$ is minimum degree of vertices of $G$ . If $G$ is hamiltonian-connected,then $3\leq\delta(G)$ . The bipartite graph $G=(V_{1}\cup V_{2},E)$ can not be hamilitonian-connectedbecause there is no hamiltonian path from $s$ to $t$ for $s,t\in V_{2}$ such that $|V_{1}|\geq|V_{2}|$ . A bipartitegraph $G$ with $|V_{1}|\geq|V_{2}|$ is bihamiltonian-conneced if one of following are satisfied:(1) if $|V_{1}|=|V_{2}|$ , there is a hamiltonian path from $s$ to $t$ for all $s\in V_{1}$ and $t\in V_{2}$ .(2) if $|V_{1}|=|V_{2}|+1$ , there is a hamiltonian path from 8 to $t$ for all $s,t\in V_{1}$ .From the definition above, bihamiltonian-connected is defined, only at $|V_{1}-V_{2}|\leq 1$ .For given a graph $G$ , it is difficult to determine whether $G$ is (bi)hamiltonian-connected or not.Of course, almost of classes of graphs is $\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{h}\mathrm{a}\dot{\mathrm{u}}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{a}\mathrm{n}$ -connected. But a rectangular gridgraph(we $\mathrm{c}\mathrm{a}!^{\iota}$ it mesh) $R(m,n)$ of size $\geq 4\mathrm{x}4$ is bihamiltonian-connected[3].In this paper, we discuss about the case of $k=2.$ For reviewing the problem, we define the2-disjoint path cover problem formally. $V(P_{1})$ means a set of vertices included in the path $P_{1}$ .2-Disjoint Path Cover Problem(2-DPCP)Instance: graph $G=(V,E)$ and $s,t,\mathrm{u},v\in V$ .Question: Find 2 paths $P_{1},P_{2}$ such that(i) $P_{1}$ connects 8 and $t,$ $P_{2}$ connects $\mathrm{u}$ and $v$ .(ii) $P_{1}$ and $P_{2}\cdot \mathrm{a}\mathrm{r}\mathrm{e}$ disjoint.$(\mathrm{i}\mathrm{i}\mathrm{i})V(P_{1})\cup V(P_{2})=V$ .Like a 1-DPCP, we define some new definitions for 2-DPCP. We call the given four ver-tices $(\epsilon,t, u,v)$ endpoint. Given $G$ and four endpoints, $[G, (\mathit{8},t), (u,v)]$ is solvable if there is a$2- \mathrm{D}\mathrm{P}\mathrm{C}[G, (\mathit{8},t), (\mathrm{u},v)]$ . A graph $G$ is coverable if $[G, (s,t), (u,v)]$ is solvable for any pairs ofvertices: $(\mathit{8}, t)$ , $(u,v)$ . Note that if a graph $G$ is coverable, then $3\leq\delta(G)$ . A bipartite graph $G$is $bi$-coveroble if one the followings conditions are satisfied:(A) if $|V_{1}|=|V_{2}|,$ $[G, (\mathit{8},t), (u,v)]$ is solvable for any $s,t,u,v$ such that two endpoints of$\{\epsilon,t,u,v\}$ are in $V_{1}$ , the rest are in $V_{2}$ .(B) if $|V_{1}|=|V_{2}|+1,$ $[G, (s,t), (u,v)]$ is solvable for any $\epsilon,t,u,v$ such that three endpoints of$\{\mathit{8},t,u,v\}$ are in $V_{1}$ , the rest is in $V_{2}$ .In this paper, we consider a given graph $G$ is a mesh-torus $M(m,n)$ , which is obtained from$R(m,n)$ by adding column- and row-wraparound edges. Note that $M(m,n)$ is not bipartite if$m$ or $n$ is odd. Therefore, when $m$ and $n$ is even, we can discuss about whether mesh-torus is$\mathrm{b}\mathrm{i}$-coverable or not. In addition, $|V_{1}|=|V_{2}|$ is true since $m$ and $n$ is even. So if a graph $G$ ismesh-torus, the only case we consider is the condition (A) of $\mathrm{b}\mathrm{i}$-coverable. And our result isfollowing:Theorem 1.1. If $n$ and $m$ are $even\geq 4,$ $M(n,m)$ is bi-coverable.For showing the Theorem 1.1, we introduce a certain operation (roughly speaking, inserttwo columns or rows to mesh-torus) for creating any instance of 2-DPCP from some “$p\dot{n}m\epsilon$”instance. We show that the $\mathrm{e}\mathrm{x}\mathrm{i}_{8}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}$ of 2-DPC is retained after this operation. Then, ourclaim is proved if all “prime” instance is all bi-coverable.This paper is organized as follows. In section 2, we explain the definitions and essential factsin previous papers. Next section, we provide the operation and its properties. Then, Theorem1.1 is showed in section 4. In the final section, we conclude our result again. This paper, someproofs are omitted because of page constraint.1822 PreliminariesHere we prepare necessary notions and notations, and review some important facts on l-disjointpath cover problem.A mesh $R(m,n)$ , more simply $R$ if $m$ and $n$ are fixed, is a graph of $m\mathrm{x}n$ vertices suchthat (i) each vertex $v$ corresponds to a grid point $(x,y),$ $x\in\{1,2, \ldots,m\}$ and $y\in\{1,2, \ldots,n\}$ ,and (ii) each edge corresponds to an edge between a pair of adjacent grid points. We considerthat a mesh is identical to its corresponding rectangular subgrid on $\mathrm{R}^{2}$ ; vertices and edges areregarded as corresponding grid points and edges. For each vertex $v$ in $R(m,n)$ , we use $v_{x}$ and$v_{y}$ to denote its x- and $y$-coordinate respectively. A vertex $v$ is called even if $(v_{x}+v_{y})$ is even,and odd otherwise. For our explanation, we assume that all vertices are colored either black orwhite in the $\mathrm{f}\mathrm{o}\mathrm{u}\dot{\mathrm{o}}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$: color every even vertex by white, and odd vertex by black. We denote$V_{1}$ is a set of white vertices, $V_{2}$ is $V\backslash V_{1}$ .A column-toru8 $T_{\mathrm{c}}(m,n)$ is $R(m,n)$ with $m$ column-wraparound edges; $((i,n),$ $(i, 1))(1\leq$$i\leq m)$ . $\mathrm{A}\cdot me\mathit{8}h- tom\mathit{8}M(n,m)$ is obtained $\mathrm{k}\mathrm{o}\mathrm{m}T_{\mathrm{c}}(m,n)$ by adding $n$ row-wraparound edges;$((m,i),$ $(1,l)’)(1\leq i\leq n)$ . Note that $M(m,n)$ is bipartite graph ff $m$ and $n$ is even, $T_{\mathrm{c}}(m,n)$is bipartite iff $n$ is even. We assume that all vertices in column- and mesh-torus are colored bysame way at mesh. In the follwing, four endpoints are satified that two endpoints are in $V_{1}$ ,the others are in $V_{2}$ .Fact 1. $Ifm,n\geq 4$ , or $m=3$ and $n$ is odd, $R(m,n)\dot{\mathrm{u}}bihamiltonian- connected[SJ$.Fact 2. $R(2,n)$ is bihamiltonian-connected exoept that two endpoints are in same non-boundary$mw[\mathit{3}J$ .Lemma 2.1. Given a $T_{\mathrm{c}}(1,n)(n\geq 4)$ and a source $\epsilon$ , there are two sinks $t\mathit{8}uch$ that there tsa $hamilton|an^{-}$’ $path$ joining $s$ and $t$ .Lemma 2.2. $T_{\mathrm{C}}(2,n)$ is bihamiltonian-connected.Remark. Given $T_{\mathrm{c}}(2,n)(n\geq 4)$ and a source 8, the number of candidates of sink $t$ such thatthere exists a hamiltonian path joining $\epsilon$ and $t$ and $t_{x}=i(i=1,2)$ is at least 2.Lemma 2.3. Given a $T_{\mathrm{c}}(1,n)(n\geq 4)$ and two $\mathit{8}ou|oe\mathit{8}S=(1,\mathit{8}_{y}),u=(1,u_{y})(u_{y}>s_{y})$ , thereare two pair8 $(t_{1},v_{1}),$ $(t_{2},v_{2})$ such that $2- DPC\beta_{\mathrm{c}}’(1,n),$ $(s,t_{1}’),$ $(u,v_{1})Jex|\epsilon t\mathit{8}(i=1,2)$ .3 Unit Insertion and Its PropertiesFor proving our theorem, we introduce an operation for expanding a mesh-torus of a given2-DPCP instance. A unit $|n\mathit{8}e\hslash ion$ is inserting two columns or rows to a mesh-torus $M$ . Then,the following two lemmas are essential for our theorem..Lemma 3.1. Let $\mathrm{Y}$ be an $imtan\alpha$ of 2-DPCP obtaind fivm some $X=[M(m,n), (s,t), (u,v)]$($m,n$ are $even\geq 4$) by any unit $ime\hslash ion$ to M. If $Xi\mathit{8}$ solvable, so $\dot{u}$ Y.In our argument, we define some notions for two column insertions; two row insertions aretreated in the same way. For any unit insertion, a inserted line –a betweenness of two adjacentcolumns– is called chink, a inserted 2 $\mathrm{x}n$ component is called river. Now, $X$ has a 2-DPC.For any unit insertion and a 2-DPC, an edge, is a part of 2-DPC, which moves across the chink183is called bridge. Let $b$ be the number of bridges. And the lemma is proved by simple analysisabout $b,$ $\mathrm{i}.\mathrm{e}.,(1)b=0$ and (2) $b\neq 0$ . For each case, we can construct a 2-DPC of $\mathrm{Y}$ from2-DPC of $X$ , easily. So the proof of this lemma is omitted.The empty column is a column which has no endpoint.Lemma 3.2. Let $X$ be any instance of 2-DPCP $[M(m,n), (\mathit{8},t), (u,v)]$ ($m,n$ are $even\geq 4$).The $Xi\mathit{8}$ obtained flom one of the folloutng $in\mathit{8}tanoes$ of 2-DPCP by some unit insertion;$(A)[M(4,4), (\mathit{8}’,t’), (u’,v’)]$$(B)[M(6,l), (\mathit{8}’,t), (u’,v’)](l=4,6,8)$ and each $\mathit{8}ide$ of any empty column are non-empty.$(C)[M(8,l), (\mathit{8}’,t), (u’,v’)]\beta=4,6,8)$ and each side of any empty column are non-empty.This proof is easily by simple case analysis of distances of endpoints. So the proof isomitted. An instance of 2-DPCP $X$ is prime, if $X$ is one of (A), (B), (C). The Figure 1expresses a ambiguous shapes (i.e., almost rows are omitted) of (B) and (C). In this figure, acolumn with circle(s) means that the column has endPoint(s).In tihe next s..ection, we prove the primes’ solvality.4 Prime InstancesIn this section, we show the existence of -DPC for all prime instances. Let $S_{1}$ be a set ofall instances is categorized in (B) and (C) described in the Lemma 3.2, and $S_{2}$ be a set of allinstances of (A). We denote $[M(m,n), (c_{1}- c_{2} -... - c_{m})](\Sigma_{1\epsilon[1,m1^{\mathrm{q}}}=4, c_{i}\geq 0(\forall 1\leq i\leq m))$is a set of instances of 2-DPCP such that column $i$ has a $c_{i}$ endpoint(s). For example, the$(\mathrm{B})-(1)$ in the Figure 1 is an element of $[M(6,$ $l)$ , (1-0-1-0-2-0)$]$ . In the follwing, we often say$[M(m,n), (c_{1}- c_{2} -... -c_{m})]$ is $\mathrm{b}\mathrm{i}$-coverable if for any four endpoints satisfying such conditiondescribed as above, it is solvable.Lemma 4.1. For any imtanoe of $S_{1}$ , it $become\mathit{8}$ to a $\mathit{8}om\epsilon$ element of [$T_{\mathrm{c}}(3,l),$ a-0-3)].It is easily to show by using Facts and Lemmas in section 2. So it is omitted.Lemma 4.2. $A[T_{c}(3,l), (1- 0- 3)]$ ($l$ is even) is bi-coverable.The proof of the Lemma, is simple case analysis, is omitted.Remark. If $l=4$ , there is only two case of the position of $r;r_{\nu}=2$ or 4. For each assignmentof endpoints, it is easy to show that there is a 2-DPC contains the edge $((3,3),(3,4))$ .Lemma 4.3. A $M(4,4)$ is $bi$-covera$bl\epsilon$ .proof. We prove the claim by case analysis. We divide all instance of 2-DPCP by the orderof a number of endpoints. That is, our cases are (1) $[M(4,4),(4- 0- 0- 0)],$ (2) $[M(4,4),(3- 1-0-$$0)],$ (3) $[M(4,4),(3- 0- 1- 0)],$ (4) $[M(4,4),(2- 2- 0- 0)],$ (5) $[M(4,4),(2- 0- 2- 0)],$ (6) $[M(4,4),(2- 1- 1- 0)]$ ,(7) $[M(4,4),(2- 1- 0- 1)]$ , and (8) $[M(4,4),(1- 1- 1- 1)]$ .Case 1 There are only two patterns of the combination of pairs. A 2-DPC for thesepatterns are in the Figure 2.184Case 2 We can deliver the three endpoints in the column 1 to the column 4 by coveringthe all vertices in the column 1. Then, we obtain the instance of $[T_{c}(3,4),(1- 0- 3)]$ . ByLemma 4.2, the all instance of this case is solvable.Case 3 We reset a $\mathrm{y}$-coordinate of some white endpoint in the column 1 is 1. By the Lemma4.2 (Remark), the left 3 $\mathrm{x}4$ column-torus has 2-DPC, and it contains a $((3,3)(3,4))$ .Then, we cut the connection of that edge, and connect $(3, 3)$ to $(4, 3)$ , $(3, 4)$ to $(4, 4)$ . Next,we connect $(3, 4)$ to $(4, 4)$ by a path crossing column-wraparound edge. In consequence,we obtain a 2-DPC for $[M(4,4),(3- 0- 1- 0)]$ .Case 4 We can deliver one endpoint in the column 2 to the column 1 and the other tothe column 3 by $\mathrm{c}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\dot{\mathrm{i}}\mathrm{g}$ all vertices in the column 2. Then, we obtain a instance of$[T_{\mathrm{c}}(3,4),(1- 0- 3)]$ . By the same argument of the case 2, all instance of this case is solvable.Case 5 Without loss of generality, we can set the vertices in column 1 are (a) [$(1,1)$ and$(1, 3)]$ , and $(\mathrm{b})$ [ $(1,1)$ and (1, 2)]. The case of (a), there are two patteas of the combinationof pairs. A 2-DPC for these patteas are in the Figure 3.The case of (b), there are three further cases of location of the rest two endpoints.We express such three cases in the Figure 4(upper side). Furthermore, we consider anassignment of endpoints for all cases in the figure. A -DPC for these patterns are in theFigure 4(under side). So, in the case of 4, all instance of this case is solvable.Case 6 and 7 We can obtain an instance of $[T_{c}(3,4),(1- 0- 3)]$ from all instance of these casesby sune argument in the case 2 or 4 (the delivered endpoints are in column 2 to column1 in case 6, are in column 1 to column 2 in case 7).Case 8 The rest cases which we have to consider is only one; i.e., each column and rowhave just one endpoint. And there are only two patterns of combination of pairs. TheFigure 5 expresses a 2-DPC for these two patterns. $\mathrm{O}$Therefore, we prove our main theorem.Theorem 1.1. If $n$, and $m$ are even) 4, $M(n, m)i\mathit{8}$ bi-coverable.5 Conclusion and RemarksIn this paper, we show a $\mathrm{b}\mathrm{i}$-coverability of even $\mathrm{x}$ even mesh-torus larger than 4 $\mathrm{x}4$. However,there is no idea for another size cases; even $\mathrm{x}$ odd, odd $\mathrm{x}$ odd, since such mesh-torus is notbipartite. But, some instance expressed by $X=$ [$M$ (odd, even),(s, $t),$ $(u,v)$] has 2-DPC. Sonext problem is whether $M$(odd, even) is coverable or not. Furthermore, we are interested inthe class or property of graphs which is (bi-)coverability. By the way, in this paper, we treatonly 2-DPCP in mesh-torus. A mesh-torus is -regular bipartite graph with some restriction.For $k$-DPCP, what is a minimum number $l$ such that a $l$-regular bipartite graph with somerestrict(as a mesh-torus) is (bi-)coverable (expanding the definition of coverability of 2-DPCPto $k$-DPCP)? For example, a mesh-torus is not coverable for $3\mathrm{D}\mathrm{P}\mathrm{C}\mathrm{P}$ since $5\leq\delta$ is necessarycondition where 6 is minimun degree of the graph.185AcknowledgementThe author would like to thank Professor Osamu Watanabe for supervising this work, ProfessorHiro Ito for hosting an important research meeting about combinatorial games and puzzles atSep. 2005, and Professor Gisaku Nakamura for helpful advices. Thanks also go to ProfessorAkinori Kawachi and member of Watanabe’s group. for significant discussions.References[1] F.Luccio and $\mathrm{C}.\mathrm{M}\mathrm{u}\mathrm{g}\mathrm{n}\dot{\mathrm{a}}$ , “Hamiltonian Paths on a Rectangular Chessboard”, 16th Annual AllertonConference (1978) pp.73-78.[2] Y.Perl and Y.Shiloach, “Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graphs”,Joumal of the $A\epsilon\epsilon o\mathrm{c}\mathrm{i}\mathrm{a}uon$ for Computing Machinery 25(1) (l978) pp.1-9.[3] $\mathrm{A}.\mathrm{I}\mathrm{t}\dot{\mathrm{u}}$ , C.H.Papadimitriou and J.L.Szwarcfiters, “Hamilton Paths in Grid Graphs”, SEAM Journalon Computing 11 (4) (1982) pp.676-686.[4] H.Everett, “Finding Hamilton Paths in Non-rectangular Grid Graphs”, Congressus Numerantium53 (1986) PP.185-192.[5] S. Madhavapeddy, I.H. Sudborough “A Topological Property of hypercubes: Node Disjoint Paths”,in Proc. of $\Re b$ IEEE Symposium on Parallel and Distributed Processing (1990) pp. 532-539.[6] H.C.Kim and J.H.Park, t‘Fault Hamiltonicity of Two-Dimensional ‘Ibrus Networks”, Workshop onAlgorithms and Computation (WAAC 2000) pp. 110-117.[7] S.D.Chen, H.Shen, and R.Tbpor, “An efficient algorithm for constructing Hamiltonian path8 inmeshes”, Parallel (lOmputing 28 (2002) pp. 1293-1305.[8] J.H.Park, H.C.Kim, and H.S.Lim, $u_{\mathrm{M}\mathrm{a}\mathrm{n}\mathrm{y}- \mathrm{t}\mathrm{o}-}$-Many Disjoint Path Covers in a Graph with FaultyElements”, Proc. of the International Symposium on Algoritbms and Computation (ISAAC 2004)pp. 742-753.rm(C)Figure 1: shapes of prime186Figure 2: 2-DPCs for the case 1Figure 3: 2-DPC8 for the case 5-(a)\dagger \dagger $\downarrow$Figure 4: 2-DPCs for the case 5-(b)Figure 5: 2-DPCs for the case 8187

2-Disjoint Path Covers in Mesh-Torus(New Trends in Theory of Computation and Algorithm)

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2-Disjoint Path Covers in Mesh-Torus(New Trends in Theory of Computation and Algorithm)

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