Minimum Spanning Tree with Uncertain Random Weights

Transcription

1 Minimum Spanning Tree with Uncertain Random Weights Yuhong Sheng 1, Gang Shi 2 1. Department of Mathematical Sciences, Tsinghua University, Beijing 184, China College of Mathematical and System Sciences, Xinjiang University, Urumqi 8346, China sheng-yh12@mails.tsinghua.edu.cn 2. School of Information Science and Engineering, Xinjiang University, Urumqi 8346, China Department of Computer Sciences, Tsinghua University, Beijing 184, China shigang122@163.com Abstract This paper considers the minimum spanning tree problem with uncertain random weights in an uncertain random network. The concept of uncertain random minimum spanning tree is initiated for minimum spanning tree problem with uncertain random edge weights. A model is presented to formulate a specific minimum spanning tree problem with uncertain random edge weights involving a distance chance distribution model. The distance chance distribution model is one of minimum spanning tree of a deterministic network which can be solved by an optimization methods. Finally, a numerical example on an uncertain random network is put forward to illustrate the effectiveness of the proposed model and algroithm. Keywords: Minimum spanning tree, Uncertain random variable, chance distribution, model and algorithm 1 Introduction The minimum spanning tree problem is one of the most important network optimization problems, which has been widely applied to many fields such as transportation problems, communication network design, logistics problems, etc. This problem was first formulated by Borüvka [2] in 1926 as a method of constructing an efficient electricity network. A spanning tree of that network would be a subset of those edges that has no loops but still connects to every node. There might be several spanning trees possible in a network. The minimum spanning tree problem is to find a least-weight subgraph connecting all nodes. For a classical network, the edge weights of the network are deterministic constants, and some efficient algorithms have been developed by many researchers such as Kruskal s algorithm [1] and Prim s algorithm [19]. Corresponding author: shigang122@163.com. 1

2 In real applications, the weights of edge are not always deterministic constants. For example, the links in a communication network can be affected by collisions, congestions and interferences. In 1965, Frank and Hakimi [4] first investigated random network. In 1981, Ishii et al. [7] discussed the minimum spanning tree problem with random edge weights. Following that, Ishii and Matsutomi [8] presented a polynomial time algorithm to solve the problem stated in [7] when the parameters of probability distributions of the edge weights are unknown, and estimated the parameters by applying a confidence region from stochastic data. Bertsimas [1] and Frieze [3] studied a so-called probabilistic minimum spanning tree in which some points are present with certain probability and found a closed-form expression for the expected length of a given spanning tree. In 27, Liu [11] founded uncertainty theory, which provides an appropriate framework to describe another indeterministic phenomena such as human belief degrees, particularly those involving the linguistic ambiguity and subjective estimation. Liu [11] first presented uncertain measure as a set function satisfying four axioms. As a fundamental concept in uncertainty theory, the uncertain variable was presented by Liu [11]. Liu [11] proposed the concept of uncertainty distribution and inverse uncertainty distribution. After, Liu [14] presented the operational law of uncertain variables. In order to rank uncertain variables, Liu [11] proposed the concept of expected value of uncertain variables. Meanwhile, Liu [11] presented the concepts of variance, moments and entropy of uncertain variables. In 21, Liu [13] first explored uncertain network for modeling the project scheduling network with uncertain variables represent time. For the spanning tree problem of uncertain network, Zhang et al. [24] studied inverse the minimum spanning tree problem, in the meantime, Zhou et al. [26] introducted the quadratic minimum spanning tree problem of uncertain network. Recently, Zhou et al. [27] investigated path optimality conditions for the minimum spanning tree problem with uncertain edge weights. In practice, uncertainty and randomness together sometimes appear in a complex system. In order to describe this phenomenon, in 213, Liu [16] first proposed chance theory, which is a mathematical methodology for modeling complex systems with both uncertainty and randomness, including chance measure, uncertain random variable, chance distribution, operational law, expected value and so on. In 214, Liu [15] first discussed an uncertain random network in which some weights are random variables and others are uncertain variables. After that, Sheng et al. [21] gave an ideal chance distribution the minimum spanning tree of an uncertain random network and proposed a model and algorithm to find the minimum spanning tree of uncertain random network. In this paper, we will further study the minimum spanning tree problem for an uncertain random network in the framework chance theory. The remainder of this paper is organized as follows. Section 2 introduced some basic concepts and properties of uncertainty theory and chance theory. In Section 3, a model is presented the minimum spanning tree of an uncertain random network and an algorithm is given to find the minimum spanning tree. Section 4 provides a numerical example to illustrate its effectiveness. Section 5 gives a brief summary to this paper. 2

3 2 Preliminaries This section reviews some basic preliminaries from uncertainty theory and chance theory. 2.1 Uncertainty Theory Let L be a σ-algebra on a nonempty set Γ. A set function M : L [, 1] is called an uncertain measure if it satisfies the following axioms: (i) (Normality) M{Γ} = 1 for the universal set Γ. (ii) (Duality) M{Λ} + M{Λ c } = 1 for any event Λ. (iii) (Subadditivity For every countable sequence of events Λ 1, Λ 2,, we have { } M Λ i M {Λ i }. axiom. i=1 i=1 Besides, the product uncertain measure on the product σ-algebra L is defined by the following product (iv) (Product Axiom) (Liu [12]) Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2, The product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } where Λ k are arbitrarily chosen events from L k for k = 1, 2,, respectively. i=1 k=1 The uncertainty distribution of a monotonous function of uncertain variables can be obtained by the following theorem. Theorem 1. (Liu [14]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f(ξ 1, ξ 2,, ξ n ) is strictly increasing with respect to ξ 1, ξ 2,, ξ m and strictly decreasing with respect to ξ m+1, ξ m+2,, ξ n, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with an inverse uncertainty distribution Φ 1 (α) = f(φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 n (1 α)). Theorem 2. (Liu [14]) Let ξ 1, ξ 2,, ξ n be independent uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n ) is an uncertain variable with uncertainty distribution Φ(x) = sup f(x 1,x 2,,x n)=x min Φ i(x i ). 1 i n 3

4 2.2 Chance Theory In 213, Liu [16] first proposed chance theory, which is a mathematical methodology for modeling complex systems with both uncertainty and randomness, including chance measure, uncertain random variable, chance distribution, operational law, expected value and so on. After that, Guo and Wang [5], Sheng and Yao [22] verified some results of variance of uncertain random variables. In application, Liu [17] founded uncertain random programming in 213. As extensions, Zhou et al. [25] proposed uncertain random multiobjective programming for optimizing multiple, incommensurable and conflicting objectives. After that, uncertain random programming was developed steadily and applied widely, Qin [2] proposed uncertain random goal programming in order to satisfy as many goals as possible in the order specified, Ke et al. [9] proposed uncertain random multilevel programming for studying decentralized decision systems. For dealing with uncertain random systems, Liu and Ralescu [18] invented the tool of uncertain random risk analysis. Meanwhile, Wen et al. [23] presented the tool of uncertain random reliability analysis for dealing with uncertain random systems. Let (Γ, L, M) be an uncertainty space and (Ω, A, Pr) be a probability space. Then the product (Γ, L, M) (Ω, A, Pr) is called a chance space. Definition 1. (Liu [16]) Let (Γ, L, M) (Ω, A, Pr) be a chance space, and let Θ L A be an uncertain random event. Then the chance measure of Θ is defined as Ch{Θ} = 1 Pr{ω Ω M{γ Γ (γ, ω) Θ} r}dr. Liu [16] proved that a chance measure satisfies normality, duality, and monotonicity properties, that is (i) Ch{Γ Ω} = 1; (ii) Ch{Θ} + Ch{Θ c } = 1 for and event Θ; (iii) Ch{Θ 1 } Ch{Θ 2 } for any real number set Θ 1 Θ 2. Besides, Hou [6] proved the subadditivity of chance measure, that is, { } Ch Θ i Ch{Θ i } i=1 i=1 for a sequence of events Θ 1, Θ 2, In order to describe uncertain random variables, Liu [16] presented a definition of chance distribution. Definition 2. (Liu [16]) Let ξ be an uncertain random variable. Then its chance distribution is defined by Φ(x) = Ch{ξ x} for any x R. The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain variable is just its uncertainty distribution. Theorem 3. (Liu [17]) Let η 1, η 2,, η m be independent random variables with probability distributions Ψ 1, Ψ 2,, Ψ m, respectively, and let τ 1, τ 2,, τ n be independent uncertain variables. Then the uncertain 4

5 random variable ξ = f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) has a chance distribution Φ(x) = R F (x, y 1,, y m )dψ 1 (y 1 ) dψ m (y m ) m where F (x, y 1,, y m ) is the uncertainty distribution of uncertain variable f(y 1, y 2,, y m, τ 1, τ 2,, τ n ) for any real numbers y 1, y 2,, y m. 2.3 Uncertain Random Network In this section, we introduce some contents of uncertain random network. The term network is a synonym for a weighted graph, where the weights may be understood as cost, distance or time and so on. Assume that in a network some weights are random variables and others are uncertain variables. In order to model this type of network, Liu [15] presented a concept of uncertain random network. Definition 3. (Liu [15]) Assume N is the collection of nodes, U is the collection of uncertain edges, R is the collection of random edges, and W is the collection of uncertain and random edge weights. Then the quartette (N, U, R, W) is said to be an uncertain random network. The uncertain random network degenerates to a random network (Frank and Hakimi [4]) if all weights are random variables; and degenerates to an uncertain network (Liu [14]) if all weights are uncertain variables. Figure 1 shows an uncertain random network (N, U, R, W) of order 4 in which N = {1, 2, 3, 4}, U = {(1, 2), (3, 4)}, R = {(1, 4), (2, 3)}, W = {w 12, w 23, w 14, w 34 }. 1 ξ 1 4 τ 1 2 ξ 2 τ 2 3 Figure 1: An Uncertain Random Network 5

6 Theorem 4. (Ideal chance distribution minimum spanning tree, Sheng et al.[21]) Let (N, U, R, W) be an uncertain random network. Assume the uncertain weights ξ ij have ragular uncertainty distributions Υ ij for (i, j) U, and the random weights ξ ij have probability distributions Ψ ij for (i, j) R, respectively. Then the shortest path from a source node to a destination node has a chance distribution Φ(x) = F (x; y ij, (i, j) R) dψ ij (y ij ) where F (x; y ij, (i, j) R) is the uncertainty distribution of uncertain variable f(y ij, (i, j) R, ξ ij, (i, j) U), and it is determined by its inverse uncertainty distribution F 1 (α; y ij, (i, j) R) = f(υ 1 ij (α), (i, j) U, y ij, (i, j) R), and f may be calculated by the Kruskal s algorithm. Theorem 5. ( Sheng et al. [21]) Let T be a spanning tree of an uncertain random network (N, U, R, W) and let Φ T (x) be the chance distribution of the weight of the spanning tree T. Then we always have Φ T (x) Φ(x) where Φ(x) is the ideal chance distribution of the minimum spanning tree. 3 Model of Minimum Spanning Tree First, some assumptions are listed as follows. (1) There is a connected and undirected network. (2) The weight of each edge (i, j) is a positive finite; (3) The weight of each edge (i, j) is either an uncertain variable or random variable. (4) All the uncertain variables and the random variables are independent. In a deterministic network (N, E, W), where E is the collection of edges and W is the collection of deterministic weights, the weight of the minimum spanning tree can be easily found by Kruskal s algorithm [1] and Prim s algorithm [19]. Obviously, the weight of the minimum spanning tree is a function with respect to w ij, where w ij denote the weights of the (i, j) edges, i.e., f({w ij (i, j) E}), is an increasing function with respect to weight w ij. Actually, it is easy to verify that f(w 1 ) f(w 2 ), where w 1 = {w ij (i, j) E}, w 2 = {wij (i, j) E}, and w ij wij, for all (i, j) E. In a real-life network optimization situation, the minimum spanning tree problem often requires satisfying additional constraints. For an uncertain random network, the weight of minimum spanning tree is an uncertain random variable, we can obtain the ideal chance distribution of the weights of the minimum spanning tree by Theorem 4. In that paper, the minimum spanning tree of an uncertain random 6

7 network was been finded by a model and algorithm. This paper will propose two new models and an algorithm to obtain the minimum spanning tree. Given a connected and undirected uncertain random network (N, U, R, W), where N = {1, 2,, n}, W = {ξ ij, (i, j) U R}. A spanning tree of that uncertain random network is a subnetwork T = (N, U, R, W ) such that its weight total is the least and U U, R R, W = {ξ ij, (i, j) U R } which is a tree and connects all the nodes together. A minimum spanning tree is then a spanning tree with weight less than or equal to the weight of every other spanning tree. The minimum spanning tree contains a subset of the edges that forms a tree and includes every node. Define ξ = {ξ ij, (i, j) U R}, the weigh of minimum spanning tree is f(ξ). We use following representation to denote a spanning tree {x ij, (i, j) U R} where x ij = represent edge (i, j) / T and x ij = 1 represent edge (i, j) T. Then the weight of the spanning tree {x ij, (i, j) U R} is x ij ξ ij It is clear that x ij ξ ij is also an uncertain random variable. Typically, {x ij, (i, j) U R} is a spanning tree if and only if x ij = n 1 (S) x ij S 1, 2 S n (1) x ij = {, 1}, (i, j) U R where U R(S) denotes the collection of edges of induced subgraph from S N. 3.1 Distance Chance Distribution Model In practice, the decision-makers want to find the minimum spanning tree with the minimum distances between ideal chance distribution and chance distribution of spanning tree. In this paper, we give the following a definition of the minimum spanning tree about distance chance distribution: Definition 4. Let (N, U, R, W) be an uncertain random network. A spanning tree T is called the minimum spanning tree about distance chance distribution if max (Φ T (x) Φ(x)) x<+ is minimum for any real number x, where Φ T (x) is the chance distribution of weight of T and Φ(x) is the ideal chance distribution of the minimum spanning tree. Let Φ(x) be ideal chance distribution of the minimum spanning tree, and let T be any spanning tree and Ψ(x x ij, (i, j) U R) be its chance distribution. In order to find the minimum spanning 7

8 tree, according to Definition 4, we minimize distances between the ideal chance distribution of minimum spanning tree and chance distribution of the spanning tree T, i.e., the objection functions are denoted max (Φ(x) Ψ(x x ij, (i, j) U R)) x<+ and the minimum spanning tree can be obtained the following model: min {x ij,} max (Φ(x) Ψ(x x ij, (i, j) U R)) x<+ s.t. x ij = n 1 x ij S 1, S N, 2 S n (2) (S) x ij = {, 1}, (i, j) U R where Φ(x) is the ideal chance distribution of the minimum spanning tree and Ψ(x x ij, (i, j) U R) = Ch x ij ξ ij x is the chance distribution of for the spanning tree {x ij, (i, j) U R} of uncertain random network. Considering the definition of chance distribution, the following theorem reformulates model (2). Theorem 6. The model (2) can be reformulated as the following model: min max Φ(x) Υ x x ij y ij dψ(y) {x ij,} x<+ s.t. x ij = n 1 x ij S 1, S N, 2 S n (S) (3) x ij = {, 1}, (i, j) U R where Φ(x), Ψ(y) and Υ(r) can be obtained by Φ(x) = F (z; y ij, (i, j) R) Ψ(y) = x ij y ij y dψ ij (y ij ), dψ ij (x ij y ij ), Υ(r) = sup min Υ ij(x ij r ij ). (i,j) U x ij r ij = r (i,j) U 8

9 Proof: According to Theorem 4, the ideal chance distribution of the minimum weight can be written as Φ(x) = According to Theorem 3, Ch F (z; y ij, (i, j) R) x ij ξ ij x is calculated by Υ x x ij y ij dψ(y), dψ ij (y ij ). where The theorem is completed. Ψ(y) = x ij y ij y dψ ij (x ij y ij ), Υ(r) = sup min Υ ij(x ij r ij ). (i,j) U x ij r ij = r (i,j) U 3.2 Solution Algorithm Generally speaking, uncertain random programming models are difficult to solve by traditional methods. This paper proposes models can be denote the minimum spanning tree of the classical network. So, in order to solve above models, we delelop a algorithm to find the minimum spanning tree of an uncertain random network. In order to find the minimum spanning tree in an uncertain random network, we develop the following algorithm. Algorithm 1. Step 1. Calculate and save the value of ideal chance distribution of the minimum spanning tree of an uncertain random network. Step 2. Using the enumeration method, obtain all spanning trees in the uncertain random network. Step 3. Calculate the distance between the ideal chance distribution and chance distribution of each spanning trees obtained in Step 2. Step 4. Compare the distance, and choose the minimum value, whose corresponding spanning tree is the minimum spanning tree. 4 Numerical Example In this section, we will give an example to illustrate the conclusions presented above. The uncertain random network (N, U, R, W) is shown in Figure 3. There is an uncertain random network with 6 nodes 9

10 and 9 edges. Assume the uncertain weights τ 12, τ 23, τ 34, τ 45, τ 56 have regular uncertainty distributions Υ 12, Υ 23, Υ 34, Υ 45, Υ 56 and the random weights ξ 13, ξ 24, ξ 35, ξ 46 have probability distributions Ψ 13, Ψ 24, Ψ 35, Ψ 46, respectively. Then by Theorem 4 we can obtain an ideal chance distribution of the minimum spanning tree of the network (N, U, R, W), that is Φ(x) = F (x; y 13, y 24, y 35, y 46 )dψ 13 (y 13 )dψ 24 (y 24 )dψ 35 (y 35 )dψ 46 (y 46 ) where F (x; y 13, y 24, y 35, y 46 ) is determined by its inverse uncertainty distribution F 1 (α; y 13, y 24, y 35, y 46 ) = f ( Υ 1 12 (α), Υ 1 23 (α), Υ 1 34 (α), Υ 1 45 (α), Υ 1 56 (α), y ) 13, y 24, y 35, y 46 and f may be calculated by the Kruskal s algorithm or Prim s algorithm for each given α. ξ τ 12 ξ 46 1 τ 23 τ 34 6 τ 45 ξ τ 56 ξ 35 Figure 3: Uncertain Random Network (N, U, R, W) The weight information of each edge is listed in Table 1, in which ξ ij are random variables, τ ij are uncertain variables. In Table 1, U represents uniformly probability distribution, L represents linear uncertainty distribution and Z represents zigzag uncertainty distribution. Table 1: Edges of probability distribution and uncertainty distribution edge(i, j) τ ij edge(i, j) ξ ij (1,2) L(1, 12) (1,3) U(18, 2) (2,3) L(14, 16) (2,4) U(11, 13) (3,4) L(15, 18) (3,5) U(14, 16) (4,5) Z(11, 13, 16) (4,6) U(15, 18) (5,6) L(12, 14) By Algorithm 1, we can obtain the minimum spanning tree of model (2) in Figure 4. 1

11 Figure 4: Minimum spanning tree of modle (2) 5 Conclusions Indeterministic factors often appear in network problems. In the past, probability theory and uncertainty theory have been employed to deal with these indeterministic factors. Chance theory provides a new approach to deal with indeterministic factors in a complex indeterministic network. In this paper, we investigated the minimum spanning tree problem of an uncertain random network. Under the framework of chance theory, the model of distance chance distribution was derived to formulate the minimum spanning tree. To solve those model, an algorithm was designed. At last, a numerical example was given for the minimum spanning tree of an uncertain random network. Acknowledgements This work was supported by National Natural Science Foundation of China Grants Nos , and References [1] Bertsimas D.J., The probabilistic minimum spanning tree problem, Networks, Vol. 2, No. 3, , 199. [2] Borüvka O., O jistém problému minimálním, Práce Mor. Přírodovéd. Spol. v Brně, Vol.3, 37-58, [3] Frieze A.M., On the value of a random minimum spanning tree problem, Discret Applied Mathematics, Vol.1, No.1, 47-56, [4] Frank H., Hakimi S.L., Probabilistic flows through a communication network, IEEE Transactions on Circuit Theory, Vol.12, , [5] Guo H.Y., Wang X.S., Variance of uncertain random variables, Journal of Uncertainty Analysis and Applications, Vol.2, Article 6, 214. [6] Hou Y.C., Subadditivity of Chance Measure, Journal of Uncertainty Analysis and Applications, Vol.2, Article 14,

12 [7] Ishii H., Shiode S., Nishida T., Namasuya Y., Stochastic spanning tree problem, Discrete Applied Mathematics, Vol.3, No.4, , [8] Ishii H., Matsutomi T., Confidence regional method of stochastic spanning tree problem, Mathematical and Computer Modelling, Vol.22, No.1, 77-82, [9] Ke H., Su T.Y., Ni Y.D., Uncertain random multilevel programming with application to product control problem, Soft Computing to be published. [1] Krauskal J.B., On the shortest spanning tree subtree of a graph and the travling saleman problem, Proceedings of the American Mathematical Society, Vol. 7, No. 1, 48-5, [11] Liu B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 27. [12] Liu B., Some research problems in uncertainty theory, Journal of Uncertain Systems Vol.3, No.1, 3-1, 29. [13] Liu B., Theory and practice of uncertain programming, 2nd ed., Springer-Verlag, Berlin, 29. [14] Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 21. [15] Liu B., Uncertain random graph and uncertain random network, Journal of Uncertain Systems, Vol.8, No.1, 3-12, 214. [16] Liu Y.H., Uncertain random variables: A mixture of uncertainty and randomness, Soft Computing, Vol.17, No.4, , 213. [17] Liu Y.H., Uncertain random programming with applications, Fuzzy Optimization and Decision Making, Vol.12, No.2, , 213. [18] Liu Y.H., Ralescu D.A., Risk index in uncertain random risk analysis, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.22, No.4, , 214. [19] Prim R.C., Shortest conection networks and some generalizations, Bell System Technical Journal, Vol. 36, , [2] Qin Z.F., Uncertain random goal programming, [21] Sheng Y.H., Qin Z.F., Shi G., Minimum spanning tree problem of uncertain random network, Journal of Intelligent Manufacturing, to be published. [22] Sheng Y.H., Yao K., Some formulas of variance of uncertain random variable, Journal of Uncertainty Analysis and Applications, Vol.2, Article 12, 214. [23] Wen M.L, Kang R, Reliability analysis in uncertain random system, [24] Zhang X., Wang Q.N., Zhou J., Two uncertaint programming models for inverse minimum spanning tree problem, Industrial Engineering & Management Systems, Vol.12, No. 1, 9-15, 213. [25] Zhou J., Yang F., Wang K., Multi-objective optimization in uncertain random environments, Fuzzy Optimization and Decision Making, to be published. [26] Zhou J., He X., Wang K., Uncertain quadratic minimum spanning tree problem, Journal of Communications, Vol.9, No.5, , 214. [27] Zhou J., Chen L., Wang K., Path optimality conditions for minimum spanning tree problem with uncertain edge weights, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, to be published. 12