Layout Problems on Lattice Graphs Josep D´ıaz† Mathew D. Penrose‡
Jordi Petit†
∗
Mar´ıa Serna†
Abstract This work deals with bounds on the cost of layout problems for lattice graphs and random lattice graphs. Our main result in this paper is a convergence theorem for the optimal cost of the Minimum Linear Arrangement problem and the Minimum Sum Cut problem, for the case where the underlying graph is obtained through a subcritical site percolation process. This result can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidian TSP. Finally we estimate empirically the value for the constant in the mentioned theorem.
∗
This research was partially supported by ESPRIT LTR Project no. 20244 — ALCOM-IT, CICYT Project TIC97-1475-CE, and CIRIT project 1997SGR-00366. † Departament de Llenguatges i Sistemes Inform` atics. Universitat Polit`ecnica de Catalunya. Campus Nord C6. c/ Jordi Girona 1-3. 08034 Barcelona (Spain). {diaz,jpetit,mjserna}@lsi.upc.es ‡ Department of Mathematical Sciences, University of Durham, South Road, Durham DH1 3LE, England.
[email protected]
1
1
Introduction
Several well-known optimization problems on graphs can be formulated as Layout Problems. Their goal is to find a layout (linear ordering) of the nodes of an input graph such that a certain function is minimized. For the problems we consider below, finding an optimal layout is NP-hard in general, and therefore it is natural to develop and analyze techniques to obtain tight bounds on restricted instances [1, 7, 4]. Graphs encoding circuits or grids are typical instances of linear arrangement problems. We consider these instances as sparse graphs that have clustering and geometric properties. For these classes of graphs, not much is known. In this paper, we are concerned with lattice graphs, and random instances of lattice graphs. For most of the layout problems it is an open problem to find exact or approximated polynomial time algorithms for lattice graphs which are non square [10, 9]. A graph is said to be a lattice graph if it is a node-induced subgraph of the infinite lattice, that is, its vertex set is a subset of Z2 and two vertices are connected whenever they are at distance 1. Percolation theory provides a framework to study lattice graphs in a probabilistic setting. We consider site percolation, where nodes from the infinite lattice are selected with e0 be the set of all open nodes some probability p (selected nodes are called “open”). Let C e0 can connected to the origin. A basic question in percolation theory is whether or not C e0 | = ∞, and set pc = inf{p : ϑ(p) > 0}, be infinite. Let ϑ(p) denote the probability that |C the critical value of p. It is well-known [8] that pc ∈ (0.5, 1). In this paper, we consider only subcritical limiting regimes p ∈ (0, pc ) in which all components are almost surely finite. Results for supercritical regimes are derived in [5, 13]. In order to deal with bounded graphs, we introduce the class of random lattice graphs with parameters m and p denoted by Lm,p that corresponds to the lattice graphs whose set of vertices are obtained through the random selection (independently and uniformly) of elements from {0, ..., m − 1}2 with probability p. Our layout problems are formally defined as follows. A layout ϕ on a graph G = (V, E) is a one-to-one function ϕ : V → {1, ..., n} with n = |V |. Given a graph G and a layout ϕ on G, let us define: L(i, ϕ, G) = {u ∈ V (G) : ϕ(u) ≤ i}
R(i, ϕ, G) = {u ∈ V (G) : ϕ(u) > i}
θ(i, ϕ, G) = {uv ∈ E(G) : u ∈ L(i, ϕ, G) ∧ v ∈ R(i, ϕ, G)} δ(i, ϕ, G) = {u ∈ L(i, ϕ, G) : ∃v ∈ R(i, ϕ, G) : uv ∈ E(G)} λ(uv, ϕ, G) = |ϕ(u) − ϕ(v)|
where uv ∈ E(G).
The problems we consider are: • Bandwidth (Bandwidth): Given a graph G = (V, E), find minbw(G) = minϕ bw(ϕ, G) where bw(ϕ, G) = maxuv∈E λ(uv, ϕ, G). 2
• Minimum Linear Arrangement (MinLA): Given a graph G = (V, E), find minla(G) = P minϕ la(ϕ, G), where la(ϕ, G) = uv∈E λ(uv, ϕ, G) P = ni=1 |θ(i, ϕ, G)|. • Minimum Cut Width (MinCut): Given a graph G = (V, E), find mincut(G) = minϕ cut(ϕ, G) where cut(ϕ, G) = maxni=1 |θ(i, ϕ, G)|. • Vertex Separation (VertSep): Given a graph G = (V, E), find minvs(G) = minϕ vs(ϕ, G) where vs(ϕ, G) = maxni=1 |δ(i, ϕ, G)|. • Minimum Sum Cut (MinSumCut): Given a graph G = (V, E), find P minsc(G) = minϕ sc(ϕ, G) where sc(ϕ, G) = ni=1 |δ(i, ϕ, G)|. • Bisection (Bisection): Given a graph G = (V, E), find minbis(G) = minϕ bis(ϕ, G) where bis(ϕ, G) = |θ(bn/2c, ϕ, G)|. Only a few results are known for those problems when restricted to lattice graphs. For the particular case of square or rectangular lattices, it is known that Bandwidth can be solved optimally using a diagonal mapping and that MinLA can also be solved with a nontrivial ordering [9, 10]. Results for the case of d-dimensional c-ary arrays (a generalization of square lattices) on the Bisection, MinCut and MinLA problems are presented in [11]. On the other hand, [12] presents a dynamic programming algorithm to solve Bisection on lattice graphs without holes. With regard to their complexity, Bandwidth, MinCut and VertSep remain NP-complete even when restricted to lattice graphs [5]. For the remaining layout problems, the complexity on lattice graphs is open. In the first part of this paper (Section 2), we deal with general lattice graphs. The results of this section are upper bounds for several layout problems on any lattice graph. In the second part of the paper (Section 3), we move to a randomized setting where we deal with random lattice graphs. The main result in that section is to prove that for p ∈ (0, pc ), there exist two nonzero finite constants βla (p) and βsc (p) such that minla(Lm,p )/m2 converges in probability to βla (p) and minsc(Lm,p )/m2 converges in probability to βsc (p), as m → ∞. This result can be viewed as an analogue of the celebrated Beardwood, Halton and Hammersley theorem on the cost for the traveling salesman problem (TSP) on random points distributed in [0, 1]d , BHH Theorem [2].
Let X = {Xi } be a sequence of independent and uniformly
distributed points in [0, 1]d . Let mintsp(n) denote the length of the optimal solution of the TSP among the first n points of X. Then, there exists a constant βtsp (d) such that mintsp(n)/n(d−1)/d converges to βtsp (d) almost surely as n → ∞.
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A key property to prove BHH-like results is geometric subadditivity [16, Chapter 3]. This property does not hold for our layout problems, therefore we take a completely different approach using percolation theory. Finally, in the third part of this paper (Section 4), we report results on computational experiments in order to get an empirical estimate of the value of βla (p).
2
Bounds for lattice graphs
In this section we give some deterministic upper bounds on the costs minbis, mincut and minvs on subsets of the square integer lattice Z2 . Each subset L of vertices in Z2 is identified with a lattice graph, namely the maximal subgraph of the 2-dimensional integer lattice with vertex set L. Lemma 1. For any lattice graph L with n vertices, and m ∈ {1, 2, . . . , n}, there is a layout √ ϕ on L such that |θ(m, ϕ, L)| ≤ 23/2 n + 1. Proof. We are looking for a subset S of L consisting of m vertices, such that there are at √ most 23/2 n + 1 edges between S and L\S. Let α > 0 be a constant, to be chosen later. For √ x ∈ Z let Sx = {y ∈ Z : (x, y) ∈ L}. Let V = {x ∈ Z : |Sx | ≥ α n}. For i ∈ Z, let Hi denote the half-space (−∞, i] × R. Set i0 = min{i ∈ Z : |L ∩ Hi | ≥ m}. Consider the case i0 ∈ / V . Then define S to be a set of the form S = L ∩ (Hi0 −1 ∪ ({i0 } × (−∞, j])) with j chosen so that S has precisely m elements. With this definition of S for i0 ∈ / V , the number of horizontal edges between S and √ L\S is at most |Si0 |, and hence is at most α n. There is at most one vertical edge between √ S and L\S, so the number of edges from S to L\S is at most α n + 1 when i0 ∈ / V. Now consider the other case i0 ∈ V . Let I = [i1 , i2 ] be the largest integer interval which √ includes i0 and is contained in V . Then i1 − 1 ∈ / V , and i2 + 1 ∈ / V . Also, as |V | ≤ α−1 n, √ i2 − i1 + 1 ≤ α−1 n. We have |L ∩ Hi1 −1 | < m ≤ |L ∩ Hi2 |. For j ∈ Z let Tj = [i1 , i2 ] × (−∞, j]. Choose j0 so that |L ∩ (Hi1 −1 ∪ Tj0 −1 )| < m ≤ |L ∩ (Hi1 −1 ∪ Tj0 )|,
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and let S be L ∩ (Hi1 −1 ∪ Tj0 −1 ∪ ([i1 , i3 ] × {j0 })), with i3 ∈ [i1 , i2 ] chosen so that S has precisely m elements. We estimate the number of edges between S and L\S for the case i0 ∈ V . Since i1 − 1 ∈ / V , and i2 + 1 ∈ / V , the number of horizontal edges between S and L\S is at most √ √ 2α n+1. Also, since i2 −i1 +1 ≤ α−1 n, the number of vertical edges between S and L\S is √ √ at most α−1 n. Combining these estimates we find that there are at most (2α + α−1 ) n + 1 edges between S and L\S, whether or not i0 ∈ V .
√ The minimum value of 2α + α−1 (achieved at α = 2−1/2 ) is 2 2. Setting α = 2−1/2 in
the above definition, we have the partition required. Using Lemma 1 and taking m = bn/2c we get the following result. √ Theorem 1. For any lattice graph L with n vertices minbis(L) ≤ 23/2 n + 1. For the MinCut problem the bound changes in the constant: √ Theorem 2. For any lattice graph L with n vertices, mincut(S) ≤ 14 n. Proof. First suppose we have n = 2m for an integer m. The proof is based on repeated bisection, with the size guaranteed by lemma 1. Partition L into complementary subsets S0 , S1 of size 2m−1 , with at most 23/2 2m/2 + 1 edges between them. Recursively for each i, i = 1, . . . , m − 1, partition a set Sj at level i into disjoint subsets Sj0 , Sj1 of size 2m−i−1 , with at most 23/2 2(m−i)/2 + 1 edges between them, for j = 0i , . . . , 1i (binary sequences of length i). The decomposition of L into subsets Sα generated in this way can be represented, in the obvious way, as vertices in a complete binary tree of depth m. At level m, one ends up with 2m singletons, each represented by a binary sequence of length m. Let ϕ be the ordering of the elements of L corresponding to the lexicographic ordering on the leaves of the tree. Given x ∈ S, let ϕ−1 (x) be the binary sequence such that {x} = Sϕ−1 (x) . Then {x} is a leaf of the binary tree, and has one ancestor set at each of levels 0, 1, 2, . . . , m − 1. Therefore |θ(ϕ−1 (x), ϕ, L)| is the sum of the numbers of edges in the bisection of the m ancestors of x in the tree. Thus, −1
|θ(ϕ
(x), ϕ, L)| ≤
m X
(23/2 2j/2 + 1) = 4
j=1
2m/2 − 1 + m. 21/2 − 1
Therefore we have, mincut(L) ≤
4 1/2 2 −1
5
(2m/2 − 1) + m,
We can drop the assumption that n = 2m , by taking m so that n ≤ 2m < 2n, and adding extra points until one has a set of size 2m . By monotonicity this process does not reduce the mincut cost.
! √ 25/2 4 mincut(L) ≤ n + (log2 (n) + 1) − 1/2 21/2 − 1 2 −1 √ ≤ 13.657 n + log2 (n) − 8.
√ But notice that for any x > 0 we have (log2 (x) − 8)/ n < 0.067, therefore the above bound √ for mincut(L) is at most 14 n for all n. As a consequence of the previous theorem, and the fact that for any graph G, minla(G) ≤ n mincut(G) and minsc(G) ≤ n minvs(G), we can extend the previous result to the remaining problems. √ Corollary 1. For any lattice graph L with n vertices, minvs(L) ≤ 14 n, √ √ minla(L) ≤ 14n n and minsc(L) ≤ 14n n. Comparing with the known results on square lattice graphs, we can see the previous bounds are equal up to a constant.
3
Stochastic results
Let us describe some basic concepts of site percolation for the lattice Lm with vertex set Vm = ([0, m) ∩ Z)2 . Given p ∈ (0, 1), site percolation with parameter p on Lm is obtained by taking a random set of open vertices of Vm with each vertex being open with probability p independently of the others. Let Lm,p be the subgraph of Lm obtained by taking all edges between open vertices. We say that Lm,p is a random lattice graph. Denote by Prp and Ep the probability and expectation with respect to the described process of site percolation with parameter p. By a cluster we mean the set of vertices in any connected component of Lm,p . Let C0 denote the cluster in Lm,p that includes (0, 0) (possibly the empty set), and write |C0 | for the cardinality of C0 . A similar site percolation process can be generated analogously on the infinite lattice with vertex set Z2 and edges between nearest neighbors. In the same way we can extend Prp e0 the cluster including the origin for site and Ep to this infinite process. Let us denote by C e0 | denote its cardinality. It may be the case that C e0 is empty. percolation on Z2 and let |C Notice that we can view the random lattice graph as generated by a site percolation process on Z2 and taking the open vertices in Vm . In this section we consider random lattice graphs generated by subcritical limiting regimes (p < pc ), in which all clusters are almost surely finite. We begin by giving bounds 6
for the MinCut and VertSep problems on the subcritical percolation process on the lattice Lm . Theorem 3. Assume 0 < p < pc ; then there exists constants 0 < c1 < c2 such that minvs(Lm,p ) mincut(Lm,p ) √ √ lim Pr c1 ≤ ≤ ≤ c2 = 1. m→∞ log m log m Proof. Recall that for any graph G, minvs(G) ≤ mincut(G). As the mincut of a disconnected graph is the maximum of the mincuts of its connected components, then mincut(Lm,p ) = maxx∈Vm mincut(Cx ). Hence by Theorem 2, for any positive constant c2 , i h i h p p Pr mincut(Lm,p ) ≥ c2 log m = Pr ∪x∈Vm {mincut(Cx ) ≥ c2 log m} . h i e0 | ≥ n ≤ By the site percolation version of Theorem 5.75 of [8], there exists α > 0 such that Pr |C e−αn , therefore by Theorem 2 i h p Pr mincut(Lm,p ) ≥ c2 log m ≤Pr ∪x∈Vm {|Cx | ≥ (c2 /14)2 log m} ≤m2 exp(−α(c2 /14)2 log m). √ √ Choosing c2 > 56/ α we get Pr mincut(Lm,p ) ≥ c2 log m → 0. To get a lower bound for minvs(Lm,p ), let δ > 0 and let T1 , . . . , Tj(m) be disjoint lattice subsquares of Lm , each of side d(δ log m)1/2 e, where j(m) = bm/d(δ log m)1/2 ec2 . Set γ = log(1/p) so that p = e−γ . Let Aj be the event that all sites in Tj are open. Then Pr [Aj ] = exp(−γd(δ log m)1/2 e2 ) ≥ m−γδ .
h i j(m) Hence, Pr ∩j=1 Acj ≤ (1 − m−γδ )j(m) ≤ exp(−m−γδ j(m)) which tends to zero provided δ is chosen so that γδ < 2. But as minvs(Lm ) ≥ m, then j(m)
∪j=1 Aj ⊂ {minvs(Lm,p ) ≥ (δ log m)1/2 }, and taking c1 =
√
δ we obtain the lower bound.
Notice that the above theorem only gives an order of magnitude result for the minimal cost and we do not have a convergence result. The order of magnitude is Θ(log m), which contrast with the supercritical case p > pc , for which minvs(Lm,p ) and mincut(Lm,p ) are Θ(m) [13]. In the next lemma we prove that for a subcritical site percolation with parameter p, the e0 ) and |C e0 | is finite. We also give a similar result expected ratio of the value of the minla(C e0 = ∅, we use the convention 0/0 = 0 for the MinSumCut problem. To cover the case C throughout the remainder of the paper. 7
Lemma 2. For any p ∈ (0, pc ), # " e0 ) minla(C ∈ (0, ∞) Ep e0 | |C
" and
Ep
# e0 ) minsc(C ∈ (0, ∞). e0 | |C
e0 ⊂ [−n, n]2 }; then by considering the lexicographic ordering Proof. Let R0 = min{n : C of vertices one sees that minsc(Lm ) ≤ m3 and minla(Lm ) ≤ m3 , which together with e0 ) ≤ (2R0 + 1)3 and minla(C e0 ) ≤ (2R0 + 1)3 . The monotonicity gives us that minsc(C statement of the lemma follows from the fact that Prp [R0 > n] decays exponentially in n [8, Chapter 3]. We use this lemma to state one of our main results, namely that the value of minla on random lattices, divided by m2 , converges in probability to a constant. In section 4 we give some empirical bounds for the value of the constant for minla. The theorem also includes a similar result for minsc. Recall [3], that if {Xn } is a sequence of random variables and let Pr
X be a random variable, Xn converges in probability to X (Xn −→ X) if, for every > 0 we have limn→∞ Pr [|Xn − X| > ] = 0, and Xn converges to X almost surely (a.s) if as n → ∞, Pr [lim sup Xn = X = lim inf Xn ] = 1 Theorem 4. Assume 0 < p < pc , then as m → ∞, " " # # e0 ) e0 ) minla(Lm,p ) Pr minla(C minsc(C minsc(Lm,p ) Pr −→ Ep and −→ Ep . e0 | e0 | m2 m2 |C |C Proof. Recall Cx is the cluster including x for Lm,p . Think of Lm,p as being embedded in a site percolation process on the infinite lattice Z2 , with clusters in this latter process denoted ex . C X minla(Cx ) minla(Lm,p ) = m−2 2 m |Cx | x∈Vm
= m−2
X minla(C X ex ) + m−2 ex | |C
x∈Vm
x∈Vm
ex ) minla(Cx ) minla(C − ex | |Cx | |C
! .
Using theorem VII.6.9 from [6] and the Kolmogorov zero-one law, " # X minla(C ex ) Pr e0 ) minla(C −2 m −→ Ep . ex | e0 | | C | C x∈Vm Writing ∂Vm for the set of x ∈ Vm with lattice neighbors in Z2 \Vm , we get ! X minla(Cx ) minla(C X ex ) ex ) minla(C m−2 − ≤ 2m−2 ex | |Cx | |Cx | |C x∈Vm x∈Vm ,Cx 6=Cx X ey ). ≤ 2m−2 minla(C y∈∂Vm
8
(1)
ey )] is finite and does not depend on y. Hence the mean By the proof of Lemma 2, Ep [minla(C of the above expression tends to zero. The result for minla then follows from (2), and the proof for minsc is just the same.
4
Empirical estimate of the limiting constant
Let us focus now on the MinLA problem. Our BHH-like Theorem 4 shows that for graphs in Lm,p with p ∈ (0, pc ), there exists a nonzero finite constant βla (p) that as m goes to infinity, minla(Lm,p )/m2 converges in probability to βla (p), where βla (p) = Ep [minla(C˜0 )/|C˜0 |]. This result is of importance in order to predict the expected value of a random lattice graph L sampled from Lm,p , because for big values of m, we will have minla(L) ∼ βla (p)m2 . The purpose of this section is to estimate the value of βla (p) (for some values of p) through a computational experiment. The setting of the basic experiment for a given value of p is obtained through the following procedure: 1. Generate at random a connected component C˜0 centered at the origin where the probability of a node to be open is p. 2. If |C˜0 | is small enough, (a) Compute minla(C˜0 ). 3. Else (a) Compute a lower bound of minla(C˜0 ). (b) Compute an upper bound of minla(C˜0 ). e0 To perform step 1, we have applied the following algorithm. With probability 1 − p, C e0 component, we start by including is empty. In order to generate at random a non empty C e0 , marking it as “alive” and marking the rest of nodes as “waiting”. Then, for node (0, 0) in C each “alive” node, we process each of its “waiting” neighbors. Tossing a coin, with probability e0 and its mark changes to “alive”; with probability 1 − p p, a “waiting” node is included in C its mark changes to “dead”. This procedure is iterated until no “alive” nodes exist. Remark that this procedure correctly generates random connected grid graphs, because for each node only one coin is tossed. Moreover, the procedure ends, because we are dealing with the subcritical phase. When |C˜0 | ≤ 15, we are able to compute the exact value of minla(C˜0 ) (step 2.1) in reasonable time. Otherwise we use the Degree method to obtain a lower bound, and Randomized Successive Augmentation algorithms to compute an upper bound. Both techniques are described and analyzed in [15]. 9
Method mesh33
Upper bound Succ. Augm. SS+SA 35456 34515
Lower bound Degree 3135
Optimal 31680
Table 1: Results for a 33 × 33 mesh This basic experiment (generation, lower bound and upper bound computation) has been repeated 10000 times, for each p = 0, 025i, 4 ≤ i ≤ 18. In this way, for each value of p, e0 )/|C e0 |]. we have obtained an average lower bound and an average upper bound of Ep [minla(C The obtained results are shown in Figure 1. As it can be seen, the estimation of βla (p) degrades as p increases, but for p = 0.45, the factor is less than 3. However for the highest values of p the standard deviation measured is quite big, so the obtained results may not be close to the expected value. We conjecture that the actual value of βla (p) for high values of p is closer to the upper bound than to the lower bound. There are two reasons for that. On one hand, the upper bounds we have obtained using another upper bounding technique (Spectral Sequencing with Simulated Annealing, SS+SA [14]) are only slightly lower than the ones obtained by multiple runs of Successive Augmentation algorithms. SS+SA is considered to be a powerful heuristic to approximate the MinLA problem on geometric instances, but requires very long runs. On the other hand, our experience is that even if the Degree method is the method that delivers the higher lower bounds, these are usually far from the optimal values. Table 1 reproduces a result of [15] that gives strength to this conjecture, by showing upper bounds and lower bounds obtained with all these methods for a 33 × 33 mesh.
5
Conclusions
For the sake of clarity we have presented our results in section 3 for Z2 , but they can be generalized easily to random lattice graphs in higher dimensions, subsets of Zd for d > 2. Moreover the convergence in theorem 4 holds almost surely by a straightforward (but messy) extension of our argument given here. It remains as an open problem to find better lower bounding techniques for connected lattice graphs, this would allow us a more accurate estimation of the constant in Theorem 4.
10
beta_LA(p) 14 Upper bound (Average) Lower bound (Average) 12
10
beta
8
6
4
2
0 0.1
0.15
0.2
0.25
0.3
0.35
0.4
p
Figure 1: Lower and upper bounds for the value of βla (p).
11
0.45
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