On the Location of a Constrained Tree Facility in a Tree Network with Unreliable Edges
Given a tree network with vertices where each edge has an independent operational probability, we are interested in finding the optimal location of a reliable service provider facility in a shape of subtree with exactly leaves and with a diameter of at most which maximizes the expected number of nodes that are reachable from the selected subtree by operational paths. Demand requests for service originate at perfectly reliable nodes. So, the major concern of this paper is to find a location of a reliable tree-shaped facility on the network in order to provide a maximum access to network services by ensuring the highest level of network connectivity between the demand nodes and the facility. An efficient algorithm for finding a reliable – tree core of is developed. The time complexity of the proposed algorithm is Examples are provided to illustrate the performance of the proposed algorithm.
The classical location theory is concerned with the location of a service facility on a network. This facility can be a single point or a specified number of points, located at either a vertex or along an edge of the network. The location of the facility depends on the distances between the demand vertices (customers) and their respective server. The location of special types of subgraphs such as paths, trees, or forests is considered as an extensive facility location problem. These facilities formed of connected structures have many applications in the fields such as transportation, communication, and computer sciences. The three criteria for optimality which are extensively studied in the literature are the following:(1)The median criterion or the minisum criterion in which the sum of the distances from all the vertices of the network to the facility is minimized.(2)The center criterion or the minimax criterion in which the maximum distance from the demand vertex to the facility is minimized.(3)The centdian criterion in which the convex combination of the weighted average distance and the maximum weighted distance from the facility to the demand points is minimized.
In recent years, there has been a growing interest in studying the location of facilities formed of connected structures which are also called extensive facilities on a tree network. The location of a path-shaped and tree-shaped facilities under the above three criteria has been studied by a number of authors. In this paper, we are studying the location of a reliable tree-shaped facility on a tree network with unreliable edges. A core of a tree is defined to be a path that is optimal with respect to the property of minimizing the sum of the distances from each vertex in the tree to the path (criterion 1). The generalization of the core of the tree is a tree core which is a subtree containing exactly leaves that minimizes the sum of the distances from each vertex in the tree network to the selected tree core facility. The tree core is a subtree with diameter at most having leaves which minimizes the sum of the distances or the weighted distances from all vertices in the tree network to this subtree facility.
Finding the core of a tree network has been considered by a several authors. Becker et al.  presented two efficient algorithms for finding the core of a tree. For unweighted trees they developed an time algorithm, while for weighted trees they provided a procedure with time complexity of . Peng and Lo  presented a recursive time algorithm for finding a core of specified length, that is, a path with length equal to a specified value , in unweighted trees. Morgan and Slater  presented a linear time algorithm for finding the core of a tree network. Minieka and Patel  explored some properties of a core of a specified length of a tree network. A core of a tree is a set of mutually disjoint paths in that minimizes the sum of the distances of all vertices in from any of the two paths. Wang  developed an time algorithm for the core problem, where is the number of vertices in . Becker et al.  considered the problem of finding an optimal location of a path-shaped facility on a tree network using combinations of the center and the median criteria. They studied two problems: finding a path which minimizes the sum of the distances such that the maximum distance from the vertices of the tree to the path is bounded by a fixed constant and such that the length of the path is not greater than a fixed value; finding a path which minimizes the maximum distance with the sum of the distances being not greater than a fixed value and with bounded length. They gave divide-and-conquer algorithms.
Wang et al.  developed two algorithms for finding a tree core of a tree . The first algorithm has time complexity for the weighted tree (each edge has an arbitrary length). The second algorithm has time complexity for the unweighted tree (the lengths of all edges are 1). Also, Peng et al.  presented two algorithms to find a tree core of a tree with vertices. The time complexities of these two algorithms are and Becker et al.  provided two algorithms; the first one for unweighted trees has time complexity of , whereas the second one for weighted trees has time complexity of . Shioura and Uno  proposed a linear time algorithm for finding a tree core of a tree network. Minieka  described methods for finding an optimal location for a path-shaped or tree-shaped facility of a specified size in a tree network. Four optimization criteria were examined: minimizing distancesum, minimizing eccentricity, maximizing distancesum, and maximizing eccentricity. Solution algorithms were presented. Kim et al.  studied the problem of locating a subtree facility on a tree network. They considered the cost of establishing the subtree facility in addition to the transportation cost associated with the travel of customers to the subtree facility. The objective function is to select a tree-shaped facility that will minimize the sum of the setup cost and the total transportation cost. Tamir er al.  developed an algorithm of time complexity for finding the optimal location of a tree-shaped facility of a specified size in a tree network with nodes, using the centdian criterion: a convex combination of the weighted average distance and the maximum weighted distance from the facility to the demand points (nodes of the tree). Tamir et al.  studied the location of a subtree facility on a tree network, both discrete and continuous under the condition that existing facilities are already located. They used the center and the median criteria.
The problems of finding a single point, a path, or a subtree on tree network with distance edges have been studied extensively in literature, a single facility location problem on a network with unreliable edges has been also considered by a number of authors. Melachrinoudis and Helander  addressed the reliable 1-median (relisum) location problem of a single facility on an undirected tree with unreliable edges. The objective of the problem is to maximize the expected number of nodes reachable by operational paths. They developed two polynomial algorithms; an algorithm which is a modification of the Floyd-Warshall algorithm for finding all pairs of the shortest paths in a graph and an algorithm which set up a depth-first node traversal and the decomposition nature of an operational path. Xue  presented a linear time algorithm for the same problem which was displayed by Melachrinoudis and Helander . Helander and Melachrinoudis  introduced path reliability measures for the expected number of accidents over a given planning horizon. Reliability refers to the probability of a hazmat transport vehicle completing a journey from an origin to a destination. They presented two different locations of the modeling frameworks: the reliable 1-median and a general framework for considering multiple routes. Santiváñez et al.  considered the problem of location a single facility on an undirected network with unreliable edges under the center criterion. The objective function is formally stated as either minimizing the maximum expected number of unsuccessful responses to demand requests over all nodes, called the reli-minmax problem, or maximizing the minimum expected number of successful responses to demand requests over all nodes, called the reli-maxmin problem. The problem is termed the reliable 1-center problem and finds applications in telecommunication and computer networks. As for subproblems of the most general reliable 1-center problem, Eiselt et al.  presented the problem of locating facilities on a network so that the total expected demand disconnected from the facilities is minimized. It is supposed that every edge has a probability of failure and that failures can never occur on two edges simultaneously. Eiselt et al.  showed how to optimally locate facilities on a network with unreliable node or edge in order to minimize the expected demand disconnected from the facilities. Ding and Xue  studied the problem of locating a node which maximizes the expected number of nodes that are reachable from it. Such a node is called a most reliable source (MRS) of the network. They presented a linear time algorithm for computing a most reliable source on an unreliable tree network where all links are immune to failures and each node has an independent transmitting probability and an independent receiving probability. Puerto et al.  considered the problem of locating two path-shaped facilities minimizing the expected service cost in the long run, assuming that paths may become unavailable and their failure probabilities are known in advance. They provided a polynomial time algorithm that solves the unreliable path location problem on tree networks in time, where is the number of vertices. Ding et al.  studied the problem of finding the 2-most reliable source (2-MRS) in an unreliable tree network. The 2-MRS problem aims to find a node pair from which the expected number of reachable nodes or the minimum reachability is maximized (i.e., Sum-Max 2-MRS and Min-Max 2-MRS).
The reliable tree core problem is stated as finding the optimal location of a tree-shaped facility (service provider) in an unreliable tree network with a diameter of at most and having leaves which maximizes the expected number of nodes that are reachable from it. This problem considers only edge failures while the nodes and the service provider facility are considered perfectly reliable. So, the model we present in this paper is stochastic in the sense that edges fail randomly which may cause a disconnection between demand nodes and service provider facility. The major concern of this work is to find a location of a service provider facility in order to provide a maximum access to network services, but in the context of connectivity, i.e., the probability that an operational path exists between two points of the network, and therefore the connectivity of the network is measured using path reliability.
For any two vertices , the vertex is called reachable from (or is called reachable from ) if there exists an operational path between them, i.e., every edge in this path is in an operational state. The objective is to identify a subtree that has exactly leaves, diameter of at most and the cumulative reliability from to , is maximized. This subtree is the reliable tree core facility of . The reliable tree core problem is motivated from a distributed database application in computer science; see Wang et al.  and Peng et al. .
The remainder of this paper is organized into five sections. In Section 2, we present notation, definitions, and some preliminary results. Section 3 is devoted to the formulation and a full description for the proposed problem. In Section 4, we present an efficient algorithm for finding a reliable tree core of . In Section 5, a numerical example is provided to illustrate the efficiency of the proposed algorithm. Finally, in Section 6, we give a concluding remarks and future research.
2. Definitions and Preliminaries
Let be a tree network, where is the node set and is the edge set. Let be a unique path in tree from node to node and let be its length which is defined in this study to be the number of edges in the path. Let and denote the set of nodes and edges on the path , respectively. Let denote the probability that edge in operational state such that and is the probability that the edge is in a failed state. All vertices are assumed to be perfectly reliable and any vertex is called a leaf vertex if the number of edges incident to it is equal to . The number of edges that can be in a failed state has no limitations, we assume that failures occur independently. Let be the tree obtained by making rooted at a vertex The diameter of is while the height of is , where is called the root of tree . Let be an a vertex in , and we denote by a subtree of rooted at a vertex Let denote the set of children of and denote the parent of . If the vertex is adjacent to , is called a subtree of
For each edge we call the endpoint of closest to the root , and we call the endpoint of farthest to the root. Each edge divides into two disjoint subtrees denoted by and . The subtree is a subtree of rooted at a vertex as defined above and the subtree is a subtree of rooted also at but induced by the set of vertices , where is the set of vertices of the subtree . If and are two vertices in an unreliable communication network, we use to denote the reliability that a message can be transmitted correctly from to Let be a subtree of and be the set of nodes of The expected number of nodes reachable by operational paths from the nodes of subtree of is called the reachability or connectivity of the subtree
A reliable , −tree core is a subtree with leaves and with a diameter of at most which maximizes the expected number of nodes that are reachable by operational paths from the nodes of . In other words, we seek for a subtree of with diameter of at most , having leaves, which maximizes the expected number of successful responses to demand requests over all nodes. A demand request is originating at a node which requires that some entity moves from the service provider located at a subtree to node along some path in the tree network.
Definition 1. Under the assumption that edges fail independently, the reliability of a path is defined as the product of the reliability of arcs in the path , i.e.,and the probability that at least one edge in the path has failed deeming the path unusable is
Note that when consists of one edge and when Also, note that and as long as all vertices have the same weight. The path is operational if and only if all edges of are operational simultaneously. Since for any there exists a unique path in connecting and , then it always holds that in
Definition 2. For a rooted tree , the total connectivity of any vertex or the expected number of nodes reachable from is defined bywhere
For a vertex and a path of , the reliability from to is
Also, for a vertex and a subtree of , the reliability from to is
Definition 3. For a rooted tree , the total connectivity of any path is the sum of the reliabilities from all the vertices of not in to the path where is the maximum reliability from to a vertex in
Definition 4. For a rooted tree , the total connectivity of any subtree is the sum of the reliabilities from all the vertices of not in to the subtree where is the maximum reliability from to a vertex in
Following the same notation developed by Ding and Xue , let be a rooted tree at any vertex. For any vertex in , let be the subtree of rooted at vertex . Let be the set of vertices of and let be the subtree induced by the set of vertices such that and If is a leaf, then and If , then and The quantities corresponding to are often referred as below quantities, while the ones corresponding to are referred as upper quantities. The two subtrees and are connected by the edge , see Figure 1.
Based on the previous decomposition of at node , Theorem 5 presents two formulas for computing the connectivity of a node in the subtrees and We will use to denote the expected number of nodes in which can be reached from node and use to denote the expected number of nodes in which can be reached from node
Proof. See Ding and Xue .
Based on the formulas in Theorem 5 for computing the expected number of nodes in the two subtrees and that can be reached from , the next lemma gives the combination of and to find expected number of nodes in that can be reached from which is also called the total connectivity of
Lemma 6. For a rooted tree , the total connectivity of any vertex or the expected number of nodes reachable from is given by
Proof. Ding and Xue .
Theorem 5 and Lemma 6 give the connectivity or the reachability of a node in an unreliable tree network. The next six lemmas give formulas for computing the cumulative reliability of an extensive facility in the form of single edge path-shaped facility, path-shaped facility, and a tree-shaped facility. These formulas generalize the results of Theorem 5 and Lemma 6 to present the connectivity of a path or a subtree instead of a node, also, using the concept of reliability saving which is very useful for computing the cumulative reliability from to a given path or subtree facility.
Lemma 7. For each edge , where and , the expected number of nodes reachable by operational paths from the service facility located as a single edge path-shaped facility which is called the relisum of this edge is given by
The relisum of the path-shaped facility with only one edge can be evaluated also by
We now introduce the definition of the reliability saving of a path which is the most important measure for guiding the optimal selection of a path with maximum connectivity or reachability starting from any vertex of the tree.
Definition 8. Given a path and a path with edges disjoint from , the reliability saving of with respect to is the increment of the reliable sum obtained by adding to , that is,
If the first path consists of only one vertex , we simply write . The reliability saving of a path in the subtree is defined by
The reliability saving of a given path in the subtree is defined recursively as follows:
where gives the reliability saving of a single edge path facility in the subtree
The following lemma gives two important formulas for calculating the reliability sum and the reliability saving of a given path in the subtree .
Lemma 9. Let , be a path in in the form , then we haveand
Proof. The reliability sum of a given path in the subtree is the sum of the reliability sums of the vertices in the path excluding the edges of this path. which can be written in the form Hence,This proves formula (18)
Since and by using formula (18), we get
The following lemma shows the using of the reliability saving concept in computing the reliability sum from to a given path.
Lemma 10. For all , the reliability sum of the path is
Proof. The proof follows immediately from the previous lemma.
The reliability saving can be calculated recursively by using a bottom up procedure as stated by the following lemma.
Lemma 11. Let be a path in a tree The reliability saving of the path can be defined recursively bywhere
For a subtree and a vertex , let be the subtree in induced by the vertex set . If we regard as a tree rooted at , can be seen as a subtree rooted at . If the subtree becomes larger, the reliability sum increases strictly. The following lemma shows that the reliability sum of the subtree increases by adding a path to it.
Lemma 12. Let be any path in a tree and let be any subtree of which intersects with the path at node then the relisum of the new subtree is
We consider increasing by adding a path to a subtree . The following equation holds for the increase of the reliability by addition of a path to a subtree.
Lemma 13. Let be a path from vertex to vertex in a tree and is any subtree of such that
Definition 14. In , given a path having an even (odd) length , the midpoint of is a vertex (edge) whose removal divides into two paths with length .
3. The Reliable Tree Core Problem
For any subtree of with exactly leaves and with diameter , let be the unique center of and suppose that is a vertex. Let be the height of the subtree in and orient the tree into , then is a subtree rooted at with height and having leaves.
Let be the midpoint of a path between any two vertices and of length . The center of unweighted tree occurs at the midpoint of the longest path in the tree, this result is presented by Handler . The center of a tree is either of one vertex or an edge and it is unique. The following two sets and define the centers of a subtrees with diameters
Therefore, there exists a point in or an edge in that is the center of a tree core of . All the subtrees of the tree with diameter will be classified by the midpoints of their diameters. Then, we search for a leaves subtree in the class of the subtrees of having a given as a midpoint. In other words, each or is the center of a set of subtrees of with diameter , and the reliable tree core is one of these subtrees having leaves and maximum reliability sum.
For each path with length less than or equal and midpoint , the tree is going to be oriented into a rooted tree at and denote it by . If which is an edge, then we contract it into a vertex and we continue to call the resulting vertex and the resulting rooted tree .
to be the candidate set of leaves for the subtree rooted at , where if is the center of an even length path and if is the center edge of an odd length path. Note that it is not necessary that the vertices of are leaves in , and it is the set of vertices at a distance or less from the root of the tree such that their children vertices are more than distance away from . For example, consider the rooted tree in Figure 3. If , we have . All leaves of the rooted reliable tree core of are vertices in Every vertex is called a candidate with respect to . The selected candidates from can not be contained in the same subtree of Otherwise, is not contained in the induced subtree by these vertices.
The stochastic optimization problem considered in this paper is the following: Given a tree network with unreliable edges and given two parameters and , the problem is to identify a subtree with exactly leaves, diameter and the reliability sum is maximized. If exceeds the diameter of , then the reliable tree core problem is just the tree core problem.
Problem 15. Given a rooted tree , find a reliable tree core rooted at of with maximum reliability sum.
Let be a subtree rooted at a vertex in the tree If is a vertex with , the candidate dominates if for all or and This means that the path has a maximum reliability saving among all paths from to vertices in . If the vertex dominates the vertex , then every vertex on the path is also dominated by In constructing the reliable subtree tree core we define to be the farthest ancestor vertex of that is dominated by it. Given , the reliability saving of paths in is defined bySorting the vertices in according to their reliability saving , The rank of a vertex is denoted by and it is the number of vertices such that either or and We denote the vertex with rank in by ,
The following Lemma states a formula for calculating the reliability sum from the vertices of to the subtree induced by the set of vertices .
Proof. Let By definition, dominates the root Thus, and is the path connecting the root with the vertex , For any differs from only in one path By Lemma 12, for Recursively, it is easy to note that .
If the nonincreasing set of vertices is contained in , then the subtree induced by will be a subtree of rooted at vertex and does not include the root . So, let us define
represents the maximum reliability saving of a path connecting the root to a vertex in which is denoted by . represents the second maximum reliability saving of a path connecting the root to a vertex in which is denoted by , where . If , then the induced subtree represents the required reliable tree core of If , then is included in one subtree , so is the subtree rooted at and the root is not included in the required subtree. To avoid this situation, in will be replaced by , where is the smallest integer such that
Note that maximizes over all subsets such that and the candidates in are not all contained in the same subtree of The following lemma gives the reliability sum of the optimal subtree among all induced subtrees of subsets
Proof. if , which means the subtree induced by the subset of vertices contains the first and the second reliability saving paths passing through the root , thus the lemma holds by using the previous lemma. Let , then differs from in only the path . Then, where , and the path has the highest reliability saving rank among all vertices that are not contained in , Hence,
Also, can be calculated by a different formula which is stated in the next lemma.
Lemma 18. The reliability sum of the subtree can be computed using the following formula:
The terms of candidates and values can be generalized to the case that both the root of the tree and the height of the subtrees to be considered are not fixed. The root of the tree can be any vertex in and the subtrees considered are those of height . For a given pair of and , we define
The value of the candidate is denoted by which is defined similarly as , the maximum reliability saving of a path from a vertex to a vertex such that
For each , we define the candidate set
The following lemma shows that the reliability saving value of any path can be computed recursively by bottom up procedure.
Lemma 19. Let be a vertex in and . We have (a) (b)
Proof. From (41), the candidate set of leaves consists of the set of vertices such that and for every child of as shown in Figure 2. Since the length of the edge is , a vertex and if and only if . This shows that consists of the union of over all Part follows immediately from definition (33) of and Lemma 9.
For any rooted subtree of , the set of paths with maximum reliability saving from vertices in to the vertices of can be constructed by using the definition of in part of the previous lemma.
The vertices in must be sorted in a descending order according to their reliability saving values and for any subset , denotes the rank of the vertex . A reliable tree core of a rooted tree must contain as a subtree a reliable tree core.
In this section, an algorithm for finding the location of a reliable tree core facility in a tree network with unreliable edges is presented. The algorithm first uses the dynamic programming technique to compute a table in bottom–up fashion. Then, it uses the table to compute for each efficiently. Finally, it finds a reliable tree core of by determining the induced subtree of the top ranked vertices of that maximizes . Given a tree network with unreliable edges, the proposed algorithm identifies a subtree which satisfies the following:(1)Orient the input tree into a rooted tree for every .(2) is the root of and has degree at least 2.(3) has exactly leaves.(4) has height at most (5)The reliability sum of the subtree is maximized.
Remark 20. Each reliable tree core must contain the two paths and with the maximum and the second maximum reliability saving such that .
The algorithm uses dynamic programming technique to compute a table entry which stores a set of up to candidates and their values .
For the subtree and , a recursive formula of is given as follows: