Uncertain Distribution-Minimum Spanning Tree Problem

Transcription

1 International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 24, No. 4 (2016) c World Scientific Publishing Company DOI: /S Uncertain Distribution-Minimum Spanning ree Problem Jian Zhou, Xiajie Yi, Ke Wang and Jing Liu School of Management, Shanghai University, Shanghai , China zhou jian@shu.edu.cn yixiajie88@shu.edu.cn ke@shu.edu.cn jing liu@shu.edu.cn Received 12 February 2015 Revised 1 November 2015 his paper studies the minimum spanning tree problem on a graph with uncertain edge weights, which are formulated as uncertain variables. he concept of ideal uncertain minimum spanning tree (ideal UMS) is initiated by extending the definition of the uncertain α-minimum spanning tree to reflect the overall properties of the α-minimum spanning tree weights at any confidence level α [0, 1]. On the basis of this new concept, the definition of uncertain distribution-minimum spanning tree is proposed in three ways. Particularly, by considering the tail value at risk from the perspective of risk management, the notion of uncertain -distribution-minimum spanning tree (-distribution- UMS) is suggested. It is shown that the -distribution-ums is just the uncertain expected minimum spanning tree when = 0. For any [0, 1], this problem can be effectively solved via the proposed deterministic graph transformation-based approach with the aid of the -distribution-path optimality condition. Furthermore, the proposed definitions and solutions are illustrated by some numerical examples. Keywords: Minimum spanning tree; uncertain distribution-minimum spanning tree; ideal UMS; -distribution-ums; tail value at risk. 1. Introduction Minimum spanning tree (MS) problem 1 is of high importance in network optimization and also of great significance to applications, which can be seen as a realistic representation of many practical problems in the real world such as communication network, 2 complex brain networks, 3 cluster analysis, 4 and image segmentation. 5 he classical MS problem has usually been solved under the deterministic circumstance, which means that the edge weights associated to a graph are completely known and assumed to be crisp numbers. For this case, some well-known algorithms such as the Kruskal algorithm 6 and the Prim algorithm 7 can be employed to solve them in polynomial time. However, the edge weights are not always deterministic Corresponding author. 537

2 538 J. Zhou et al. (nor crisp) in fact. Considering nondeterministic factors may be encountered in the MS problems, two main streams of research have been developed in the past few decades, namely the stochastic MS problem (e.g. Refs. 8 11) and the fuzzy MS problem (e.g. Refs ), which model the nondeterministic phenomena as randomness and fuzziness, respectively. he stochastic MS problem and fuzzy MS problem provide some appropriate frameworks to model the MS problem and also work well in some practice while dealing with the nondeterministic factors which can be interpreted as frequency and subjective indeterminacy, respectively. In spite of these successes, both randomness and fuzziness have been shown to be improper to describe some nondeterministic phenomena, particularly those arisen from the shortage of statistical data and thus involving some linguistic ambiguity and subjective estimation. 17 As for the MS problem, due to some economic reasons or technical difficulties, we often lack observed data to get an accurate estimation on the edge weights of a graph (or a network) on which the problem is concerned. Consequently, some domain experts are invited to make an assessment and give some belief degrees about the uncertain edges weights. However, it has been illustrated by many studies (e.g. Ref. 18) that human beings usually overweight unlikely events, and thus the belief degrees given by these experts may have much larger variance than the real frequency. his makes the probability theory and the fuzzy set theory inappropriate to deal with the MS problem in this situation since both of them may lead to counterintuitive results. 17 Fortunately, the uncertainty theory, proposed by Liu in and redefined in as a powerful tool to rationally deal with human beings belief degrees, provides an alternative appropriate framework to handle it. Since the uncertainty theory provides a new powerful mathematical approach to handling nondeterministic phenomena involving subjective estimation, it has been applied to a great deal of uncertain network optimization problems (e.g. Refs ). Particularly, on the basis of the uncertainty theory, Peng and Li 24 considered the uncertain minimum spanning tree (UMS) problem where the edge weights are assumed to be uncertain variables. Following that, Peng et al. 25 reviewed the recent advances in uncertain network optimization, and introduced some general uncertain network optimization models based on uncertain programming. Recently, some other variants of the MS problem have also been studied within the framework of Liu s uncertainty theory such as the uncertain inverse MS problem 26 and the uncertain quadratic MS problem. 27 As for the UMS problem, after it was proposed by Peng and Li, 24 Zhou et al. 28 further discussed its path optimality conditions as well as effective solving methods. Since the edge weights are uncertain variables in the UMS problem, the weights of spanning trees are uncertain as well. Consequently, the definition of MS for the classical MS problem becomes useless for the UMS problem. herefore, three typical types of uncertain minimum spanning trees, namely expected UMS, 24 α- UMS, 24 and most UMS, 28 were proposed according to different decision criteria

3 Uncertain Distribution-Minimum Spanning ree Problem 539 by Peng and Li, 24 and Zhou et al. 28 he expected UMS compares the expected values of the weights of all possible spanning trees to determine the minimum one, whereas the α-ums considers the fractile of the uncertainty distributions of the spanning tree weights regarding a predetermined confidence level α, and the most UMS is an equivalent counterpart of the α-ums. 28 It should be noted that for a given graph, the found UMS may be different under different decision criteria, and all of these three types of UMS have their own limitations. For the purpose of discussing the performance of possible UMSs under different decision criteria, and putting forward a more persuasive alternative decision criterion, the notion of ideal UMS is initiated in this paper. Ideal UMS, which can be seen as an extension of the α-ums, reflects the overall properties of the α- minimum spanning tree weights at any confidence level α [0, 1]. Usually, such an ideal UMS may not really exist and it is a suppositional concept, but it can provide a new natural and persuasive perspective to define a UMS. A UMS defined by referring to the ideal UMS is called the uncertain distribution-minimum spanning tree (distribution-ums). Specifically, we propose the definition of distribution- UMS in three different ways and thus three types of distribution-ums are defined accordingly, i.e., max-distribution-ums, overall-distribution-ums, and -distribution-ums. In particular, the -distribution-ums is suggested by considering the tail value at risk from the perspective of risk management. Some discussions concerning the path optimality condition as well as related solutions are also presented for the -distribution-ums problem. he rest of this paper is organized as follows. Section 2 introduces some basic concepts concerning the uncertain variable as well as the classical MS problem. Section 3 proposes the notion of ideal UMS. In Sec. 4, three types of distribution- UMS are defined, and some properties of them are discussed as well. Following that, path optimality condition and solutions of the -distribution-ums problem are presented in Sec. 5. Finally, numerical examples are given in Sec. 6 for illustration. 2. Preliminaries In this section, some fundamental concepts concerning the uncertain variable as well as the classical MS problem are introduced briefly. For more details, the reader may refer to Refs. 19, Uncertain variable Definition 1. (Liu 19 ) Let L be a σ-algebra on a nonempty set Γ. A set function M : L [0, 1] is called an uncertain measure if it satisfies the following axioms: Axiom 1. (Normality Axiom) M{Γ} = 1 for the universal set Γ; Axiom 2. (Duality Axiom) M{Λ} + M{Λ c } = 1 for any event Λ;

4 540 J. Zhou et al. Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ 1, Λ 2, Λ 3,..., we have M { i=1 Λ i} i=1 M{Λ i}. Besides, the triplet (Γ, L, M) is called an uncertainty space. Definition 2. (Liu 19 ) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} = {γ Γ ξ(γ) B} is an event. Definition 3. (Liu 19 ) Let ξ be an uncertain variable. Its uncertainty distribution is defined by Φ(x) = M{ξ x} for any real number x. Definition 4. (Liu 19 ) An uncertain variable ξ is called linear if it has a linear uncertainty distribution 0, if x a, Φ(x) = (x a)/(b a), if a x b, (1) 1, if x b, denoted by L(a, b), where a and b are real numbers with a < b. Definition 5. (Liu 19 ) An uncertainty distribution Φ is said to be regular if its inverse function (α) exists and is unique for each α (0, 1). Let ξ be an uncertain variable with regular uncertainty distribution Φ. hen the inverse function is called the inverse uncertainty distribution of ξ. Note that the inverse uncertainty distribution is well defined on the open interval (0, 1). If required, we may extend the domain to [0, 1] via (0) = lim α 0 + (α) and (1) = lim α 1 (α). It is clear that the distribution of a linear uncertain variable ξ L(a, b) is regular, and its inverse uncertainty distribution is (α) = (1 α)a + αb. (2) Definition 6. (Liu 29 ) he uncertain variables ξ 1, ξ 2,..., ξ n are said to be independent if M { n i=1 {ξ i B i }} = n i=1 M{ξ i B i } for any Borel sets B 1, B 2,..., B n of real numbers. heorem 1. (Liu 19 ) Let ξ 1, ξ 2,..., ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,..., Φ n, respectively, and f : R n R a continuous and strictly increasing function. hen the uncertain variable ξ = f(ξ 1, ξ 2,..., ξ n ) has an inverse uncertainty distribution Ψ 1 (α) = f( 1 (α), Φ 1(α),..., Φ 1 n (α)). (3) Definition 7. (Liu 19 ) Let ξ be an uncertain variable. hen the expected value of ξ is defined by + 0 E[ξ] = M{ξ r}dr M{ξ r}dr (4) 0 provided that at least one of the two integrals is finite. 2

5 Uncertain Distribution-Minimum Spanning ree Problem Classical minimum spanning tree problem Following many earlier studies on the MS problem (e.g. Refs. 28,30), we formulate the classical MS as follows. Let G = (V, E, w) denote a connected graph, where V = {v 1, v 2,..., v n } is the vertex set, E = {e 1, e 2,..., e m } is the edge set, and w = (w 1, w 2,..., w m ) is the edge weight vector. A spanning tree = (V, S) of G is a connected acyclic subgraph containing all vertices, where S is the set of edges contained in. For the sake of brevity, a spanning tree is denoted by its edge set S in this paper. hen the classical MS problem is to find a spanning tree with the minimum total edge weight among all spanning trees of G. Definition 8. (Minimum Spanning ree) Given a connected graph G = (V, E, w), a spanning tree 0 is said to be a minimum spanning tree if e i 0 w i e j w j holds for any spanning tree. Furthermore, with regard to a spanning tree, the edges in the tree are called tree edges, whereas the edges not in the tree are called non-tree edges. For any nontree edge e j, it is known that there must exist a unique path between the vertices of e j such that it is composed of tree edges. his unique path is called the tree path of edge e j and denoted by P j. On the basis of these concepts, the following path optimality condition of MS was proposed by Ahuja et al. 30 heorem 2. (Ahuja et al. 30 ) Given a connected graph G = (V, E, w), a spanning tree 0 is a minimum spanning tree if and only if w i w j 0, e j E\ 0, e i P j (5) where E\ 0 is the set of non-tree edges, and P j is the tree path of edge e j. 3. Ideal Uncertain Minimum Spanning ree In this section, we first briefly review the uncertain minimum spanning tree problem, particularly three typical types of UMS. hen, the concept of ideal UMS is initiated. Some theorems concerning the weight distribution of the ideal UMS are presented as well hree typical types of UMS In many real-world applications, since the edge weights cannot be precisely known due to the lack of observed data, the MS problem should be considered on a graph with uncertain edge weights. Consequently, the uncertain version of the classical MS problem, called the uncertain minimum spanning tree problem, which is formulated as follows, was investigated by Peng and Li, 24 and Zhou et al. 28 Let G = (V, E, ξ) denote an uncertain graph consisting of the vertex set V = {v 1, v 2,..., v n }, the edge set E = {e 1, e 2,..., e m }, and the edge weight vector

6 542 J. Zhou et al. ξ = (ξ 1, ξ 2,..., ξ m ), where ξ i, i = 1, 2,..., m, are independent uncertain variables with regular uncertainty distributions. hen, the weight of a spanning tree of the graph G can be denoted as W (, ξ) = e i ξ i. (6) Since ξ i, i = 1, 2,..., m, are uncertain variables, W (, ξ) is an uncertain variable as well. Consequently, Definition 8 for the classical MS problem becomes useless for the uncertain version of the problem. In this case, the following three typical types of UMS, including expected UMS, α-ums, and most UMS, were proposed according to different decision criteria. Definition 9. (Peng and Li, 24 Expected UMS) Given a connected uncertain graph G = (V, E, ξ), a spanning tree 0 is called an uncertain expected minimum spanning tree if E[W ( 0, ξ)] E[W (, ξ)] (7) holds for any spanning tree, where E[W ( 0, ξ)] and E[W (, ξ)] are the expected values of weights of spanning trees 0 and, respectively. Definition 10. (Peng and Li, 24 α-ums) Given a connected uncertain graph G = (V, E, ξ) and a predetermined confidence level α (0, 1], a spanning tree 0 is called an uncertain α-minimum spanning tree if min{ω M{W ( 0, ξ) ω} α} min{ω M{W (, ξ) ω} α} (8) holds for any spanning tree, and min{ω M{W ( 0, ξ) ω} α} is called the α-minimum spanning tree weight, denoted by W α ( 0, ξ). Definition 11. (Zhou et al., 28 Most UMS) Given a connected uncertain graph G = (V, E, ξ) and a predetermined weight supremum x, a spanning tree 0 is called an uncertain most minimum spanning tree if Φ 0(x ) Φ (x ) (9) holds for any spanning tree, where Φ 0 and Φ are the uncertainty distributions of tree weights W ( 0, ξ) and W (, ξ), respectively. Obviously, the expected-ums takes the expected value of the tree weights into consideration while determining the minimum spanning tree, whereas the α-ums minimizes the critical value ω at a given confidence level α such that M{W ( 0, ξ) ω} α. Similarly to the α-ums, the most-ums maximizes the chance that the tree weight does not exceed a predetermined supremum x. he equivalent relationship between the α-ums and the most-ums was revealed by Zhou et al. 28 as well.

7 Uncertain Distribution-Minimum Spanning ree Problem Ideal UMS: Definition and properties As mentioned above, the expected-ums only considers the mean or expected value of the tree weights. As a result, more characters concerning the distributions of the tree weights are not reflected under this decision criterion. For example, two trees with the same expected weight but different variances are treated as the same according to this criterion. Meanwhile, as for the α-ums (or the most UMS, which is with equivalent relationship to the α-ums), it only reflects the local properties of the distributions at a given confidence level (or a predetermined supremum). Consequently, the minimum spanning tree found under this decision criterion may vary with the predetermined confidence level (or the predetermined supremum). aking the examples given by Zhou et al. 28 (see Examples 3 and 4 in Ref. 28) for instance, Fig. 1 shows the α-minimum spanning tree weights (denoted by the solid line) of a given graph with respect to different confidence levels α. he weight distributions of the related spanning trees are illustrated in the figure as well, which are denoted by dashed lines. It is shown that the cases 0 < α 4, 4 < α 0.78 and 0.78 < α 1 are associated with different α-umss, respectively. In short, all the three typical types of UMS have their own limitations. herefore, an alternative decision criterion is suggested in this paper to result in a socalled ideal uncertain minimum spanning tree, which can be seen as a natural α (135,1) (127,1) (138,1) (114.56,0.78) (116.88,0.78) 0.6 (92.29,4) (96.84,4) W α ( 0, ξ) Fig. 1. α-minimum spanning tree weights with respect to different confidence levels.

8 544 J. Zhou et al. extension of the α-ums. o present the ideal UMS, we first define the ideal distribution of UMS as follows. Definition 12. (Ideal Distribution of UMS) Given a connected uncertain graph G = (V, E, ξ), for the set of all the spanning trees { 1, 2,..., k } of G, the function Φ (x) = max Φ j (x) (10) 1 j k is called the ideal distribution of the uncertain minimum spanning tree of G, where Φ j are the uncertainty distributions of the weights of j for j = 1, 2,..., k, respectively. It is clear that the ideal distribution of UMS is nothing but the envelope curve of the distributions of all the spanning trees of a given graph. aking Fig. 1 for example, the solid line is just the ideal distribution, which also reflects the α-minimum spanning tree weights with respect to different confidence levels. Definition 13. (Ideal UMS) Given a connected uncertain graph G = (V, E, ξ), a spanning tree (may not actually exist) with the ideal distribution of the uncertain minimum spanning tree Φ (x) is called the ideal uncertain minimum spanning tree of G, and denoted by. Concerning the distribution of the weight of an ideal UMS (i.e., the ideal distribution of UMS), we have the following theorems. heorem 3. Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions, its ideal distribution of the uncertain minimum spanning tree Φ (x) is a continuous and strictly increasing function with respect to x on {x 0 < Φ (x) < 1}. Proof. Following from the definition of ideal distribution of UMS, we just need to prove that Φ (x) = max 1 j k Φ j (x) is continuous and strictly increasing at each point x with 0 < Φ (x) < 1, where { j 1 j k} denotes the set of all the spanning trees of G. For the given graph G, there are at most C n 1 m spanning trees. hat is, the number of possible spanning trees is a limited and fixed value. hus, we can denote the set of all the spanning trees by listing them as { 1, 2,..., k }. It follows from heorem 1 that for any spanning tree j, the tree weight W ( j, ξ) is also an uncertain variable with a regular uncertainty distribution, denoted as Φ j (x). hen Φ j (x) and (α) are both continuous and strictly j increasing on the domains {x 0 < Φ j (x) < 1} and {α 0 α 1}, respectively. We define that x L = min 1 j k Φ 1 (0) = min j 1 j k lim α 0 + Φ 1 j (α) (11)

9 Uncertain Distribution-Minimum Spanning ree Problem 545 and x U = min 1 j k Φ 1 (1) = min j 1 j k lim α 1 Φ 1 j (α). (12) It is easy to verify that for arbitrary x (x L, x U ) (i.e., 0 < max 1 j k Φ j (x) < 1), Φ (x) = max 1 j k Φ j (x) exists and is unique, which means Φ (x) is a function map of x. Now let us prove Φ (x) is continuous and strictly increasing. According to the arithmetic rule of limitation of compound function, we have lim x x Φ (x) = lim max Φ x x j (x) = max 1 j k = max Φ 1 j k j (x ) = Φ (x ). hat means Φ (x) is continuous. For arbitrary x L < x 1 < x 2 < x U, we get 1 j k lim Φ x x j (x) (13) Φ (x 1 ) = max 1 j k Φ j (x 1) and Φ (x 2 ) = max 1 j k Φ j (x 2). (14) For each spanning tree j, Φ j (x) is continuous and strictly increasing on the domain {x 0 < Φ j (x) < 1}, and Φ j (x) = 0 if x lim α 0 + (α) and Φ j j (x) = 1 if x lim α 1 (α). Since x L < x j 1 < x 2 < x U, there must exist at least one spanning tree 0 such that lim α 0 + (α) < x 0 1 < x 2 < lim α 1 (α). hen 0 we have Φ 0(x 1 ) < Φ 0(x 2 ). For any other spanning tree j 0, Φ j (x 1 ) Φ j (x 2 ) holds. herefore, we have max Φ 1 j k j (x 1) < max Φ 1 j k j (x 2). (15) hat is, Φ (x 1 ) < Φ (x 2 ). Hence Φ (x) is strictly increasing with respect to x (x L, x U ), i.e., 0 < Φ (x) < 1. According to heorem 3, for a graph with uncertain edge weights which have regular distributions, the distribution of its ideal UMS is continuous and strictly increasing, and thus its inverse function exists. By utilizing its inverse uncertainty distribution, an equivalent formulation of the ideal distribution of UMS can be obtained as follows. heorem 4. (Equivalent Formulation of Ideal Distribution) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions Φ i, i = 1, 2,..., m, respectively, the set of all the spanning trees in G is denoted as { 1, 2,..., k }. For the inverse function of the ideal distribution of the uncertain minimum spanning tree of G, called the inverse ideal distribution of UMS, we have (Φ ) 1 (α) = min 1 j k Φ 1 j (α), 0 α 1, (16) where j denote the inverse distributions of the weights of j for j = 1, 2,..., k, respectively.

10 546 J. Zhou et al. Proof. Since ξ i, i = 1, 2,..., m, are independent uncertain variables with regular uncertainty distributions, the distribution Φ j (x) and inverse distribution (α) j are both continuous and strictly increasing on the domains {x 0 < Φ j (x) < 1} and {α 0 α 1}, respectively. Furthermore, it follows from heorem 3 that (Φ ) 1 (α) exists and is unique on the domain 0 < α < 1, i.e., x (x L, x U ). For arbitrary x 0 (x L, x U ), according to the definition of ideal distribution, we have hen denote Φ (x 0 ) = max 1 j k Φ j (x0 ). (17) α 0 = Φ (x 0 ) and α j = Φ j (x 0 ). (18) We have and α 0 = max 1 j k α j (19) (Φ ) 1 (α 0 ) = j (α j ) = x 0. (20) Since j is strictly increasing, we obtain that for any spanning tree j, hat is j (α 0 ) j (α j ) = (Φ ) 1 (α 0 ). (21) (Φ ) 1 (α 0 ) = min 1 j k Φ 1 j (α 0 ), 0 < α 0 < 1, (22) which means Eq. (16) holds if 0 < α < 1. By extending the domain of α to [0, 1] via and j (0) = lim α 0 + Φ 1 j (α), j (1) = lim α 1 Φ 1 j (α), (23) (Φ ) 1 (0) = lim α 0 +(Φ ) 1 (α), Eq. (16) can be similarly verified. (Φ ) 1 (1) = lim α 1 (Φ ) 1 (α), (24) Following from the definitions of ideal distribution and ideal UMS and heorem 4, it is easy to find that the ideal UMS is a spanning tree such that it is always the most UMS for any predetermined weight supremum, and also the α-ums for any given confidence level. Consequently, such an ideal UMS may not really exist for most cases, and it is actually a suppositional concept. However, it provides a new perspective to define an uncertain minimum spanning tree. On the basis of this novel concept, some new decision criteria to find a UMS are discussed in the remainder of this paper.

11 Uncertain Distribution-Minimum Spanning ree Problem Uncertain Distribution-Minimum Spanning ree In this section, we propose the definition of uncertain distribution-minimum spanning tree based on the notion of ideal UMS in three ways, which imply three different decision criteria for determining the UMS, respectively. Accordingly, three types of distribution-ums are obtained, namely max-distribution-ums, overalldistribution-ums, and -distribution-ums. Some properties of the proposed distribution-umss are discussed as well Max-distribution-UMS aking the ideal UMS as a reference to define a UMS, a natural way is to find a spanning tree such that it is the closest one to the ideal UMS among all spanning trees. Particularly, to measure the degree of closeness between a given spanning tree and the ideal UMS by the maximum distance between their weight distributions, a so-called max-distribution-ums is suggested as follows. Definition 14. (Max-Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ), a spanning tree 0 is called an uncertain max-distribution-minimum spanning tree if max x (Φ (x) Φ 0(x)) max x (Φ (x) Φ (x)) (25) holds for any spanning tree, where Φ 0, Φ and Φ are the weight distributions of spanning trees 0, and the ideal UMS, respectively. In this definition, the degree of closeness between the spanning tree and the ideal UMS is measured by the maximum distance between their weight distributions Φ (x) and Φ (x) with respect to an arbitrary x in the whole domain as shown in Fig. 2. α 1 Φ (x ) Φ (x ) distance Φ (x) Φ (x) 0 x x Fig. 2. he degree of closeness measured by the maximum distance. It is easy to see that this way is followed from the idea of the worst-case analysis, and the result may be influenced by some individual extreme points. o overcome this limitation, a natural extension of this definition is to consider the whole area surrounded by the weight distributions Φ (x) and Φ (x) as shown in Fig. 3.

12 548 J. Zhou et al. Fig. 3. he degree of closeness measured by area Overall-distribution-UMS By measuring the degree of closeness between a spanning tree and the ideal UMS by the area surrounded by Φ (x) and Φ (x), a corresponding new type of distribution-ums, called the overall-distribution-ums, is proposed as follows. Definition 15. (Overall-Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ), a spanning tree 0 is called an uncertain overall-distributionminimum spanning tree if + 0 (Φ (x) Φ 0(x))dx + 0 (Φ (x) Φ (x))dx (26) holds for any spanning tree, where Φ 0, Φ and Φ are the weight distributions of 0, and the ideal UMS, respectively. Remark 1. It is easy to verify that inequality (26) can be formulated in an equivalent form as follows, + 0 (1 Φ 0(x))dx + 0 (1 Φ (x))dx. (27) Since the weight of a spanning tree W (, ξ) cannot take negative values (i.e., Φ (x) = 0 for any x 0), the left and right items in inequality (27) are just the expected values of the weights of spanning tree 0 and, respectively, according to the definition of expected value of an uncertain variable (see Definition 7). hat is, + (1 Φ 0 0(x))dx = E[W ( 0, ξ)] and + (1 Φ 0 (x))dx = E[W (, ξ)]. herefore, we have the following theorem. heorem 5. (Equivalent Definition I of Overall-Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ), a spanning tree 0 is an uncertain overalldistribution-minimum spanning tree if and only if it is an uncertain expected minimum spanning tree. Proof. It follows from Remark 1 and the definition of expected UMS (Definition 9) immediately.

13 Uncertain Distribution-Minimum Spanning ree Problem 549 According to heorem 4, if the edge weights of the graph G are all independent uncertain variables with regular uncertainty distributions, its ideal distribution of UMS can be formulated in an equivalent form by using its inverse distribution. Consequently, an equivalent definition of the overall-distribution-ums can be obtained for the case of G with uncertain edge weights having regular uncertainty distributions as follows. heorem 6. (Equivalent Definition II of Overall-Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions, a spanning tree 0 is an uncertain overall-distribution-minimum spanning tree if and only if 0 ( 0 (α) (Φ ) 1 (α))dα 0 ( (α) (Φ ) 1 (α))dα (28) holds for any spanning tree, where, 0 and (Φ ) 1 are the inverse distributions of the weights of 0, and the ideal UMS, respectively. Proof. Since all the edge weights ξ i are independent uncertain variables with regular uncertainty distributions, Φ (x) and (α) are both continuous and strictly increasing on the domains {x 0 < Φ (x) < 1} and {α 0 α 1}, respectively. It follows from heorems 3 and 4 that both Φ (x) and (Φ ) 1 (α) are also continuous and strictly increasing on the domains {x 0 < Φ (x) < 1} and {α 0 α 1}, respectively. herefore, an equivalent form of inequality (26) can be immediately obtained as inequality (28), which calculates the area surrounded by Φ (x) and Φ (x) by integration with respect to α (see Fig. 3). It is clear that the overall-distribution-ums takes overall properties of the weight distributions into consideration. In other words, it considers all of the possible confidence levels α [0, 1] to measure the degree of closeness between (α) and (Φ ) 1 (α) according to heorem 6. However, in reality, we scarcely care about the cases where the confidence level α is relatively low. Conversely, we always prefer to concern the situation that the confidence level α is above a certain level. From the perspective of risk management, the tail value at risk 31 (also known as conditional value at risk) is usually being paid more concerns. herefore, we further extend the definition of overall-distribution- UMS to a more general form by introducing a predetermined confidence level in the following distribution-ums For a given confidence level, if the degree of closeness between a spanning tree and the ideal UMS is measured by the area surrounded by (α) and

14 550 J. Zhou et al. (Φ ) 1 (α) on condition that α, a so-called -distribution-ums is suggested as follows. Definition 16. (-Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions, and a predetermined confidence level [0, 1], a spanning tree 0 is an uncertain -distributionminimum spanning tree if ( 0 (α) (Φ ) 1 (α))dα ( (α) (Φ ) 1 (α))dα (29) holds for any spanning tree, where, 0 and (Φ ) 1 are the inverse distributions of the weights of 0, and the ideal UMS, respectively. he geometric interpretation of the definition of -distribution-ums are illustrated in Fig. 4. Comparing it with Fig. 3, it is easy to see that the overalldistribution-ums utilizes the whole area surrounded by the weight distribution of and the ideal distribution to measure the degree of closeness between a spanning tree and the ideal UMS, whereas the -distribution-ums only considers the area satisfying α. Fig. 4. Uncertain -distribution minimum spanning tree. Remark 2. he definition of -distribution-ums is a natural extension of the overall-distribution-ums since the latter is just a special case of the former. If = 0, the -distribution-ums is an overall-distribution-ums, and also an expected UMS according to heorem 5. Remark 3. It is easy to verify that inequality (29) in the definition of - distribution-ums can be formulated by the following equivalent form, For convenience, tree. Φ 1 0 (α)dα (α)dα. (30) (α)dα is called the -distribution weight of the spanning

15 Uncertain Distribution-Minimum Spanning ree Problem 551 In order to interpret the idea behind the notion of -distribution-ums more clearly from the perspective of risk management, the concept of tail value at risk under uncertainty proposed by Peng 31 is introduced as follows. Definition 17. (Peng 31 ) Let ξ be an uncertain variable, and (0, 1) the risk confidence level. hen the tail value at risk of ξ is defined by ξ VaR () = 1 1 inf{x M{ξ x} α}dα. (31) Based on the concept of tail value at risk, an equivalent definition of the - distribution-ums is obtained as follows. heorem 7. (Equivalent Definition I of -Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions, and a predetermined confidence level (0, 1), a spanning tree 0 is an uncertain -distribution-minimum spanning tree if and only if W ( 0, ξ) VaR () W (, ξ) VaR () (32) holds for any spanning tree, where W ( 0, ξ) VaR () and W (, ξ) VaR () are the tail values at risk of the weights of spanning trees 0 and, respectively. Proof. Since all the edge weights ξ i are independent uncertain variables with regular uncertainty distributions, Φ (x) and (α) are both continuous and strictly increasing on the domains {x 0 < Φ (x) < 1} and {α 0 α 1}, respectively. herefore, we have W ( 0, ξ) VaR () W (, ξ) VaR () inf{x M{W ( 0, ξ) x} α}dα 0 (α)dα inf{x M{W (, ξ) x} α}dα (α)dα. According to heorem 7, the definition of -Distribution-UMS equals to taking the tail value at risk of the weight distribution of a spanning tree as the decision criterion to determine the minimum spanning tree. he solutions of the -distribution- UMS are discussed in the following sections. 5. -Distribution-Path Optimality Condition he path optimality condition formulated by heorem 2, which is a necessary and sufficient condition of the classical minimum spanning tree, provides a useful property for solving the classical MS problem. Following from this idea, in

16 552 J. Zhou et al. this section, some equivalent definitions of the -distribution-ums as well as the -distribution-path optimality condition are presented. Subsequently, an effective method to find the -distribution-ums is given on the basis of the proposed -distribution-path optimality condition. First of all, concerning the weight of a spanning tree, we have the following result. heorem 8. Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions Φ i, i = 1, 2,..., m, respectively, and a predetermined confidence level [0, 1], for any spanning tree, we have (α)dα = e i where is the inverse distribution of the weight of. i (α)dα, (33) Proof. Since W (, ξ) = e i ξ i and ξ i are independent uncertain variables, according to heorem 1, we have hen Eq. (33) can be immediately obtained. (α) = i (α), α [0, 1]. (34) e i Based on heorem 8, we can further get an equivalent definition of the - Distribution-UMS by utilizing the integration of the inverse uncertainty distributions of the edge weights. heorem 9. (Equivalent Definition II of -Distribution-UMS) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions Φ i, i = 1, 2,..., m, respectively, and a predetermined confidence level [0, 1], a spanning tree 0 is an uncertain -distribution minimum spanning tree if and only if e i 0 holds for any spanning tree. i (α)dα e j Proof. It follows immediately from Remark 3 and heorem 8. j (α)dα (35) Following from the path optimality condition for classical MS, we extend it to the -distribution-ums and get a similar path optimality condition, called the -distribution-path optimality condition. It is also an equivalent definition of the -distribution-ums. heorem 10. (-Distribution-Path Optimality Condition) Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions Φ i,

17 Uncertain Distribution-Minimum Spanning ree Problem 553 i = 1, 2,..., m, respectively, and a predetermined confidence level [0, 1], a spanning tree 0 is an uncertain -distribution-minimum spanning tree if and only if Φ i (α)dα Φ j (α)dα 0, e j E\ 0, e i P j (36) where E\ 0 is the set of non-tree edges, and P j is the tree path of non-tree edge e j. Proof. Given a connected uncertain graph G = (V, E, ξ), we can construct a corresponding deterministic graph Ḡ = (V, E, (α)dα) consisting of the same Φ 1 ξ vertex set V and edge set E with uncertain graph G, as well as the deterministic edge weight vector Φ 1 ξ (α)dα = ( Φ 1 1 (α)dα, Φ 1 2 (α)dα,..., Φ 1 m (α)dα). If 0 is the -distribution-ums of G, according to heorem 9, e i 0 Φ 1 i Φ 1 j (α)dα e j (α)dα holds for any spanning tree. hen 0 is also the MS of Ḡ. Following from heorem 2, inequality (36) is obtained. On the other hand, if 0 satisfies (36), according to heorem 2, 0 is the MS of Ḡ. hen we have e i 0 Φ 1 i (α)dα e j Φ 1 j (α)dα holds for any spanning tree. It follows from heorem 9 that 0 is the -distribution-ums of G. According to heorem 10, the -distribution-ums problem can be transformed to a classical MS problem following from the -distribution-path optimality condition as follows. heorem 11. Given a connected uncertain graph G = (V, E, ξ) where ξ i are with regular uncertainty distributions Φ i, i = 1, 2,..., m, respectively, and a predetermined confidence level [0, 1], a spanning tree 0 is the uncertain -distribution minimum spanning tree of G if and only if 0 is the minimum spanning tree of a deterministic graph Ḡ = (V, E, Φ 1 ξ (α)dα), where the deterministic tree weight vector is ξ (α)dα = ( 1 (α)dα, 2 (α)dα,..., m (α)dα). (37) Proof. It follows immediately from the proving process of heorem 10. heorem 11 describes the relationship between the -distribution-ums of an uncertain graph and its counterpart of a deterministic graph, which also provides an effective method to solve the -distribution-ums problem. In terms of this theorem, if we intend to find the -distribution-ums of a given uncertain graph G = (V, E, ξ) with a predetermined confidence level [0, 1], where ξ i, i = 1, 2,..., m, are uncertain variables with regular uncertainty distributions Φ i, respectively, we can turn to find the MS of its corresponding deterministic graph by employing an existing algorithm for the classical MS problem (e.g., the Kruskal algorithm). he solving process is summarized as follows.

18 554 J. Zhou et al. Solution to the -distribution-ums problem via deterministic graph transformation Step 1. Calculate Φ 1 i (α)dα, i = 1, 2,..., m, respectively, and then obtain a deterministic graph Ḡ = (V, E, Φ 1 ξ (α)dα). Step 2. Use the Kruskal algorithm to find the minimum spanning tree of Ḡ, denoted by 0. Step 3. Return 0 as the uncertain -distribution-minimum spanning tree of G. 6. Numerical Examples In this section, we give some numerical examples of the uncertain distributionminimum spanning tree problems to illustrate the conclusions presented above. A connected graph G consisting of 6 vertices and 9 edges is shown in Fig. 5. he edge weights are independent uncertain variables, denoted as ξ i, i = 1, 2,..., 9. B ξ 4 D ξ 1 ξ 8 A ξ 3 ξ 5 ξ 7 F ξ 2 ξ 9 C ξ 6 E Fig. 5. Uncertain graph for numerical examples. Suppose that ξ i, i = 1, 2,..., 9, are all linear uncertain variables as listed in able 1. It is known that the expected value of a linear uncertain variable ξ L(a, b) is E[ξ] = (a + b)/2. (38) In addition, the inverse uncertainty distributions of linear uncertain variables can be calculated on the basis of Eq. (2). As a result, we obtain i (α) for a given confidence level α as well as Φ 1 i (α)dα for a given confidence level, i = 1, 2,..., 9. For instance, letting α = 0.8 and =, the values of i (0.8) and Φ 1 i (α)dα as well as the expected values E[ξ i ] are shown in able 1. Example 1. In order to find the overall-distribution-ums of the graph G in Fig. 5, we can solve the problem via the solution process proposed for the - distribution-ums problem by supposing that = 0 since the overall-distribution UMS is a special case of the -distribution-ums according to Remark 2. On the

19 Uncertain Distribution-Minimum Spanning ree Problem 555 able 1. Parameter values in Fig. 5. Edge i ξ i E[ξ i ] i (0.8) Φ 1 i (α)dα 1 L(10, 12) L(7, 11) L(4, 14) L(7, 10) L(9, 13) L(6, 12) L(3, 13) L(7, 13) L(3, 13) other hand, we can also find the overall-distribution-ums by solving the expected UMS problem (see Ref. 28 for details) since the expected UMS is equivalent to the overall-distribution-ums in terms of heorem 5. he two ways yield the same result as shown in Fig. 6. here are two spanning trees, 0 1 = {e 2, e 3, e 4, e 7, e 9 } and 0 2 = {e 2, e 4, e 6, e 7, e 9 }, both of which are the overall-distribution UMS (and also the expected UMS) with the same expected minimum weight E[W ( 0, ξ)] = E[ξ 2 ] + E[ξ 3 ] + E[ξ 4 ] + E[ξ 7 ] + E[ξ 9 ] = E[ξ 2 ] + E[ξ 4 ] + E[ξ 6 ] + E[ξ 7 ] + E[ξ 9 ] = = (39) B 8.5 D B 8.5 D A F A F C 9 E C 9 E (a) 0 1 Fig. 6. (b) 0 2 Overall-distribution-UMS (expected UMS). Example 2. Considering the α-ums problem, it is clear that the found UMS may be different with respect to different confidence levels α. For example, given a confidence level α = 0.8, the values of i (0.8) are listed in able 1. Via the approach of deterministic graph transformation (see Ref. 28 for details), we can find

20 556 J. Zhou et al. able 2. α-umss with respect to different confidence levels. α 0 W α( 0, ξ) 1 0 = {e 2, e 3, e 4, e 7, e 9 }; 2 0 = {e 2, e 4, e 6, e 7, e 9 } = {e 2, e 4, e 6, e 7, e 9 } = {e 2, e 4, e 6, e 7, e 9 } = {e 2, e 4, e 6, e 7, e 9 } = {e 1, e 2, e 4, e 6, e 9 } 55.5 that the 0.8-UMS consists of edges e 2, e 4, e 6, e 7, and e 9, namely 2 0 in Example 2. However, if the confidence level α changes into 0.9, another spanning tree 3 0, consisting of edges e 1, e 2, e 4, e 6, and e 9, becomes the new solution. he results of the α-ums problem of the graph G in Fig. 5 with different confidence levels are listed in able 2, where W α ( 0, ξ) is the α-minimum spanning tree weight. Figure 7 presents the weight distributions of the spanning trees 1 0, 2 0 and 3 0 in able 2 (to make the comparison more clearly, the case of α < is not shown in the figure). It can be seen that if α 0.875, 2 0 is the α-ums, and if α 1, 3 0 is the α-ums. In other words, the α-minimum spanning tree weight W α ( 0, ξ) is determined by different spanning trees with the varying of confidence level α. herefore, it is obvious that the ideal UMS of the graph G in Fig. 5 does not actually exist. Fig. 7. Weight distributions of the spanning trees 0 1, 0 2, and 0 3.

21 Uncertain Distribution-Minimum Spanning ree Problem 557 B D B ξ 4 D ξ 1 ξ 8 A F A ξ 3 ξ 5 ξ 7 F ξ 2 ξ 9 C E C 5.25 Deterministic graph with edge vector Φ 1 (α)dα -distribution-ums ξ ξ 6 E Fig. 8. he -distribution-ums problem with =. Example 3. Following from the proposed solution to the -distribution-ums problem, we can find the -distribution-ums of the graph G in Fig. 5 via the deterministic graph transformation method. aking a given confidence level = for instance, the values of Φ 1 i (α)dα and the corresponding deterministic graph with edge vector Φ 1 ξ (α)dα are shown in able 1 and Fig. 8(a), respectively. From heorem 10 we know that the MS of Fig. 8(a) (as shown in Fig. 8(b)) is also the -distribution-ums of Fig. 5. he -distribution-ums consists of edges e 2, e 4, e 6, e 7, and e 9 (denoted by 2 0 in Examples 2 and 3) with the -distribution weight e i 0 2 i (α)dα = + 2 (α)dα + 7 (α)dα + 4 (α)dα + 9 (α)dα = = (α)dα (40) As for the spanning tree 0 1, we have e i 0 1 i (α)dα = + 2 (α)dα + 7 (α)dα + 3 (α)dα + 9 (α)dα = = , 4 (α)dα (41) which implies that 0 1 is a worse solution than 0 2 to the -distribution-ums problem. In other words, considering the tail value at risk from the perspective of risk management, 0 2 is more appropriate to be treated as a UMS.

22 558 J. Zhou et al. From the results in Examples 2 and 3, we know that both 1 0 = {e 2, e 3, e 4, e 7, e 9 } and 2 0 = {e 2, e 4, e 6, e 7, e 9 } are the expected UMS and -UMS of the graph G in Fig. 5. hat is, according to the decision criteria with respect to expected UMS and α-ums with α =, 1 0 and 2 0 have the same performance and thus both of them are treated as the UMS. However, based on the notion of -distribution- UMS, it can be found that only 2 0 is treated as the UMS with a confidence level =, since the -distribution weight of 2 0 is less than that of 1 0. In summary, by comparing the results in Examples 2, 3 and 4, it can be seen that the proposed notion of distribution-ums provides an alternative effective way to address the UMS problem. 7. Conclusion In this paper, we discussed the uncertain minimum spanning tree problem where edge weights are uncertain variables, and proposed three types of uncertain distribution-minimum spanning trees, i.e., max-distribution-ums, overalldistribution-ums and -distribution-ums, by referring to the ideal UMS. It was shown that the overall-distribution-ums is a special case of the - distribution-ums, and also equivalent to the expected UMS. For the general case of the -distribution-ums, some equivalent definitions as well as the path optimality condition are proposed. Based on the path optimality condition, which shows that the notion of the -distribution-ums can be characterized by a set of constraints on non-tree edges and their corresponding tree paths, it was proved that the -distribution-ums problem has an equivalent counterpart in its corresponding deterministic graph, and thus it can be transformed to a classical MS problem and then be solved efficiently by some well-known algorithms. Acknowledgments his work was supported in part by grants from the National Social Science Foundation of China (No. 13CGL057), the Ministry of Education Funded Project for Humanities and Social Sciences Research (No. 14YJC630124), and the Shanghai Philosophy and Social Science Planning Project (No. 2014EGL002). References 1. L. G. Ronald and H. Pavol, On the history of the minimum spanning tree problem, Ann. Hist. Ccomput. 7 (1985) C. Chiang, C. H. Liu and Y. M. Huang, A near-optimal multicast scheme for mobile ad hoc networks using a hybrid genetic algorithm, Expert Syst. Appl. 33 (2007) C. J. Stam, P. ewarie, E. C. W. Van Straaten, A. Hillebrand and P. Van Mieghem, he trees and the forest: Characterization of complex brain networks with minimum spanning trees, Int. J. Psychophysiol. 92 (2014) J. C. Gower and G. J. S. Ross, Minimum spanning trees and single linkage cluster analysis, J. Roy. Stat. Soc. C-App. 18 (1969)

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