Oblivious Buy-at-Bulk in Planar GraphsThis work is supported by NSF grant CNS-084608.

Oblivious Buy-at-Bulk in Planar Graphs1

Abstract

In the oblivious buy-at-bulk network design problem in a graph, the task is to compute a fixed set of paths for every pair of source-destinations in the graph, such that any set of demands can be routed along these paths. The demands could be aggregated at intermediate edges where the fusion-cost is specified by a canonical (non-negative concave) function . We give a novel algorithm for planar graphs which is oblivious with respect to the demands, and is also oblivious with respect to the fusion function . The algorithm is deterministic and computes the fixed set of paths in polynomial time, and guarantees a approximation ratio for any set of demands and any canonical fusion function , where is the number of nodes. The algorithm is asymptotically optimal, since it is known that this problem cannot be approximated with better than ratio. To our knowledge, this is the first tight analysis for planar graphs, and improves the approximation ratio by a factor of with respect to previously known results.

1 Introduction

A typical client-server model has many clients and multiple servers where a subset of the client set wishes to route a certain amount of data to a subset of the servers at any given time. The set of clients and the servers are assumed to be geographically far apart. To enable communication among them, there needs to be a network of cables deployed. Moreover, the deployment of network cables has to be of minimum cost that also minimizes the communication cost among the various network components. This is what we roughly call as a typical network design problem. The same problem can be easily applied to many similar practical scenarios such as oil/gas pipelines and the Internet.

The “Buy-at-Bulk” network design considers the economies of scale into account. As observed in [5], in a telecommunication network, bandwidth on a link can be purchased in some discrete units with costs respectively. The economies of scale exhibits the property where the cost per bandwidth decreases as the number of units purchased increases: . This property is the reason why network capacity is bought/sold in “wholesale”, or why vendors provide “volume discount”.

The generalized form of the buy-at-bulk problem is where there are multiple demands from sources to destinations, and it is commonly referred as Multi-Sink Buy-at-Bulk (MSBB). Typically, the demand flows are in discrete units and are unsplittable (indivisible), i.e., the flow follows a single path from the demand node to its destination. These problems are often called “discrete cost network optimization” in operations research.

As mentioned in [6], if information flows from different sources over a link, then, the cost of total information that is transmitted over that link is proportional to , where . The function is called a canonical fusion function if it is concave, non-decreasing, and has the subadditive property , . Generally, MSBB problems use the subadditive property to ensure that the ‘size’ of the aggregated data is smaller than the sum of the sizes of individual data.

We study the oblivious buy-at-bulk network design problem (MSBB) with the following constraints: an unknown set of demands and an unknown concave fusion cost function . An abstraction of this problem can be found in many applications, one of which is data fusion in wireless sensor networks where data from sensors is aggregated over time in multiple sinks. Other application include Transportation & Logistics (railroad, water, oil, gas pipeline construction) etc. Many of these problems are formulated as networks on a plane that can be mapped to planar graphs.

1.1 Problem Statement

Assume that we are given a weighted graph , with edge weights . We denote to be the weight of edge . Let be a unit of demand that induces an unsplittable unit of flow from source node to destination node . Let be a set of demands that are routed through paths in . It is possible that some paths may overlap. The flow of these demands forms a set of paths .

There is an arbitrary canonical function at every edge where data aggregates. This is same for all the edges in . Let denote the set of paths that use an edge . Then, we define the cost of an edge to be . The total cost of the set of paths is defined to be . For this set , there is an optimal set of paths with respective cost . The approximation ratio for the paths is defined as . The MSBB optimization problem on input is to find a set of paths that minimizes the approximation ratio. We note that MSBB is NP-Hard as the Steiner tree problem is its special case (when and when there is only one destination node) [15].

An oblivious algorithm for the MSBB problem, computes a fixed set of paths, denoted for every pair of source destination nodes in . Given any set of demands , the path for each , is the fixed path in from to . This gives a set of paths to route the demands . We define the approximation ratio of , as:

We aim to find algorithms that minimizes the above approximation ratio for any canonical function which is unknown to the algorithm. The best known oblivious algorithm is by by Gupta et al. [9] and provides approximation ratio for general graphs. No better result is known for planar graphs. This problem is NP-hard, since MSBB is NP-hard.

1.2 Contribution

We provide an oblivious algorithm FindPaths for MSBB problems in planar graphs. Our algorithm is deterministic and computes in polynomial time a fixed set of paths that guarantees -approximation ratio for any canonical function (where is unknown to the algorithm). We also give a lower bound for the approximation ratio for to be of , where is the number of nodes in the graph. A lower bound of for planar graphs is provided in the context of the online Steiner tree problem by Imase and Waxman [10]. Thus, our bound is tight with respect to planar graphs. It is also a factor improvement over the best previously known result [9].

We build the set of paths based on sparse covers (see [14] for an overview of sparse covers). A -cover consists of clusters where for each node there is some cluster that contains its -neighborhood. We construct levels of covers with exponentially increasing locality parameter . For every cluster we elect a leader. For any pair of nodes we identify an appropriate common lowest-level cluster that contains both and , and the cluster has a respective common leader . Then the path from to is formed by connecting successive path segments emanating from both and and using intermediate leaders of lower level clusters until the common leader is reached.

In the analysis, we introduce the notion of coloring sparse covers, where two clusters that are close receive different color. We show the existence of a sparse cover with constant coloring (based on the sparse covers in [4]). This enables us to obtain optimal approximation at every level. When we combine all the levels, we get an approximation.

1.3 Related Work

Oblivious Network Design

Below, we present the related work on oblivious network design and Table 1 summarizes some results and compares our work with their’s. What distinguishes our work with the others’ is the fact that we provide a set of paths for the MSBB problem while others provide an overlay tree for SSBB version.

Related Work Algorithm Type Graph Type Oblivious Function Oblivious Sources Approx Factor
Lujun Jia
et al. [12] Deterministic Random
Deployment ×
Lujun Jia Deterministic Arbitrary Metric
et al. [11] Deterministic Doubling Metric
Ashish Goel
et al. [6] Randomized General Graph -inequality
Ashish Goel
et al. [7] Probabilistic General Graph
Anupam Gupta Randomized General Graph
et al. [9] Randomized Low Doubling
This paper Deterministic Planar
Table 1: Our results and comparison with previous results for data-fusion schemes. is the total number of nodes in the topology, is the total number of source nodes.

Goel et al. in [6] build an overlay tree on a graph that satisfies triangle-inequality. Their technique is based on maximum matching algorithm that guarantees -approximation, where is the number of sources. Their solution is oblivious with respect to the fusion cost function . In a related paper [7], Goel et al. construct (in polynomial time) a set of overlay trees from a given general graph such that the expected cost of a tree for any is within an -factor of the optimum cost for that .

Jia et al. in [12] build a Group Independent Spanning Tree Algorithm (GIST) that constructs an overlay tree for randomly deployed nodes in an Euclidean 2 dimensional plane. The tree (that is oblivious to the number of data sources) simultaneously achieves -approximate fusion cost and -approximate delay. However, their solution assumes a constant fusion cost function. We summarize and compare the related work in Table 1.

Lujun Jia et al. [11] provide approximation algorithms for TSP, Steiner Tree and set cover problems. They present a polynomial-time -partition scheme for general metric spaces. An improved partition scheme for doubling metric spaces is also presented that incorporates constant dimensional Euclidean spaces and growth-restricted metric spaces. The authors present a polynomial-time algorithm for Universal Steiner Tree (UST) that achieves polylogarithmic stretch with an approximation guarantee of for arbitrary metrics and derive a logarithmic stretch, for any doubling, Euclidean, or growth-restricted metric space over vertices. They provide a lower bound of for UST that holds even when all the vertices are on a plane.

Gupta et al. [9] develop a framework to model oblivious network design problems (MSBB) and give algorithms with poly-logarithmic approximation ratio. They develop oblivious algorithms that approximately minimize the total cost of routing with the knowledge of aggregation function, the class of load on each edge and nothing else about the state of the network. Their results show that if the aggregation function is summation, their algorithm provides a approximation ratio and when the aggregation function is , the approximation ratio is . The authors claim to provide a deterministic solution by derandomizing their approach. But, the complexity of this derandomizing process is unclear.

Non-Oblivious Network Design

There has been a lot of research work in the area of approximation algorithms for network design. Since network design problems have several variants with several constraints, only a partial list has been mentioned in the following paragraphs.

The “single-sink buy-at-bulk” network design (SSBB) problem has a single “destination” node where all the demands from other nodes have to be routed to. Network design problems have been primarily considered in both Operations Research and Computer Science literatures in the context of flows with concave costs. The single-sink variant of the problem was first introduced by Salman et al. [15]. They presented an -approximation for SSBB in Euclidean graphs by applying the method of Mansour and Peleg [13]. Bartal’s tree embeddings [2] can be used to improve their ratio to . A -approximation was given by Awerbuch et al. [1] for graphs with general metric spaces. Bartal et al. [3] further improved this result to . Guha [8] provided the first constant-factor approximation to the problem, whose ratio was estimated to be around 9000 by Talwar [16].

Organization

In the next section, we present some definitions and notations used throughout the rest of the paper. Section 3 describes the concept of sparse covers and shortest path clustering. In addition, we derive a coloring for the cover. In section 4, we describe FixedPaths and FindPaths algorithms that build a set of shortest paths between all pairs of nodes in . Section 5 provides the analysis of the FindPaths algorithm as well as the main theorem of this paper. Finally, we discuss our contribution and future work in section 6.

2 Definitions

Consider a weighted graph , where . For any two nodes , their distance is the length of the shortest path that connects the two nodes in . We denote by the -neighborhood of which is the set of nodes distance at most from . For any set of nodes , we denote by the -neighborhood of which contains all nodes which are within distance from any node in .

A set of nodes is called a cluster if the induced subgraph is connected. Let be a set of clusters in . For every node , let denote the set of clusters that contain . The degree of in is defined as , which is the number of clusters that contain . The degree of is defined as , which is largest degree of any of its nodes. The radius of is defined as .

Consider a locality parameter . A set of clusters is said to -satisfy a node in , if there is a cluster , such that the -neighborhood of , , (nodes within distance from v) is included in , that is, . A set of clusters is said to be a -cover for , if every node of is -satisfied by in . The stretch of a -cover is the smallest number such that .

We define the following coloring problem in a set of clusters . We first define the notion of the distance between two clusters , . We say that , if there is a pair of nodes and such that is -satisfied in , is -satisfied in , and . A valid distance- coloring of with a palette of colors , is an assignment of an integer to every , such that there is no pair of clusters , , with which receive the same color. The objective is to find the smallest that permits a valid distance- coloring.

3 Sparse Cover

A -cover is sparse if it has small degree and stretch. In [4, Section 5] the authors present a polynomial time sparse cover construction algorithm Planar-Cover for any planar graph and locality parameter , which finds a -cover with constant degree, , and constant stretch, . Here, we show that this cover also admits a valid distance- coloring with a constant number of colors .

For any node , we denote by the shortest distance between and an external node (in the external face) of . We also define . The heart of sparse cover algorithm in [4, Section 5] concerns the case where which is handled in Algorithm Depth-Cover. The general case, , is handled by diving the graph into zones of depth , as we discuss later. So, assume for now that .

The Algorithm Depth-Cover, relies on forming clusters along shortest paths connecting external nodes (in the external face) of . For every shortest path , Algorithm Shortest-Path-Cluster in [4, Section 3] returns a set of clusters around the neighborhood of with radius at most and degree . Then, and all its -neighborhood is removed from producing a smaller subgraph (with possibly multiple connected components). The algorithm proceeds recursively on each connected component of by selecting an appropriate new shortest path between external nodes of . The algorithm terminates when all the nodes have been removed. The initial shortest path that starts the algorithm consists of a single external node in . The resulting -cover consists of the union of all the clusters from all the shortest paths. The shortest paths are chosen in such a way that a node participates in the clustering process of at most 2 paths, and this bounds the degree of the -cover to be at most , and stretch .

The analysis in [4, Section 5.1.1] of Algorithm Depth-Cover relies on representing the clustering process of as a tree as we outline here. Each tree node represents a pair where is a planar subgraph of that is to be clustered, and is a shortest path between two external nodes of . The root of the tree is , where is a trivial initial path with one external node . The children of a node are all the nodes , such that is a connected component that results from removing and its -neighborhood from .

Next, we extend [4, Lemma 3.1] to show that we can color the clusters obtained by a shortest path clustering using a constant number of colors. The proof of the following three lemmas is given in the appendix.

Lemma 1

For any graph , shortest path , the set of clusters returned by Algorithm Shortest-Path-Cluster() admits a valid distance- coloring with colors.

We can obtain a coloring of by coloring the respective levels of the tree . Assume that the root is at level .

Lemma 2

The union of clusters in any level of the tree , admits a valid distance- coloring with 3 colors.

Lemma 3

Algorithm Depth-Cover returns a set of clusters which admits a valid distance- coloring with 6 colors.

We are now ready to consider the case . Algorithm Planar-Cover decomposes the graph into a sequence of bands, such that each band has depth . The bands are organized into zones, such that zone consists of three consecutive bands . Thus, zone overlaps with bands , , and . The algorithm invokes Depth-Cover for each zone giving a -cover with degree and stretch .

We can obtain the following coloring result. Using Lemma 3, for every zone we can get a valid distance- coloring with a palette of colors. This implies that we can obtain a valid distance- coloring for the zone with at most colors. Zones and do not overlap and any two nodes satisfied in them (one from each zone) with respect to must be more than distance apart. Therefore, we can color all the zones with three different palettes each consisting of 6 colors, so that zone , uses the th palette. The coloring can be found in polynomial time. Therefore, we obtain:

Theorem 3.1

Algorithm Planar-Cover produces a set of clusters which has degree , stretch , and admits a valid distance- coloring with colors.

4 Algorithm

We describe how to find the shortest path between a given pair of nodes in a graph . To find such paths, we use Algorithm FindPaths (Algorithm 2) which relies on Algorithm AuxiliaryPaths (Algorithm 1).

Input: Graph
Output: Set of auxiliary paths for all nodes in
// is a -cover of , (, specified in Section 5)
// Assume each cluster has a designated leader node
;    // set of auxiliary paths for all nodes in
foreach  do
       ;    // auxiliary path for node from level to
       ;
       for  to do
             Let be a cluster that -satisfies ;
             leader of ;
             shortest path from to ;
             concatenate and ;
             ;
            
      ;
      
return ;
Algorithm 1 AuxiliaryPaths
Input: Graph .
Output: Set of paths between all pair of nodes in
// Let covers be as in Algorithm AuxiliaryPaths
// Let leader be as defined in Algorithm AuxiliaryPaths;
// Let auxiliary path be as computed in Algorithm AuxiliaryPaths
// Let be the auxiliary path segment from to ;
;    // set of paths for all pairs of nodes in
foreach pair  do
       Let be the smallest locality parameter such that ;
       Let be a cluster that -satisfies (and hence -satisfies );
       Let be the leader of (common leader of );
       concatenate shortest paths from to to ;
       concatenate , , and ;
       ;
      
return ;
Algorithm 2 FindPaths

Both algorithms use covers , where in every node in is a cluster, and is a -cover of , for , where the parameters and are defined in Section 5. We refer to the cover as the level cover of . We assume that each cluster in the covers has a designated leader node. There is a unique cluster, containing all nodes in , and leader node at level .

Algorithm AuxiliaryPaths computes a auxiliary path from every node to . The auxiliary paths are built in a bottom-up fashion. A auxiliary path from any node at level is built recursively. In the basis of the recursion, we identify a cluster , which -satisfies node . Let denote the leader . We now compute a shortest path, denoted , from to . This forms the first path segment of . Suppose we have computed up to level , . We now want to extend this path to the next higher level . To compute the path segment from level to level , we repeat the process of finding a cluster that -satisfies node . Let denote the leader . We compute the shortest path, denoted from to . We then append this new path segment to to form the current extended path . The path building process terminates when the last leader reaches level .

We are now ready to describe how Algorithm FindPath computes the shortest paths between all pair of nodes in . For pair of nodes , let be the distance between them. Let be the smallest locality parameter such that . Let be the cluster that satisfies , and let be the respective leader of . Note that by the way that we have chosen , cluster also -satisfies . Let denote the segment of the auxiliary path from to . We concatenate , with a shortest path from to , with a shortest path from to , and . This gives the path .

5 Analysis

Let be a planar graph with nodes. In this section we use the following parameters:

//highest cluster level in
//cover degree bound
//cover stretch bound
//locality parameter of level cover
//coloring of level

Consider levels of covers , where in each node in is a cluster, and each , , is a -cover of which is obtained from Theorem 3.1. Thus, each , , has degree at most , stretch at most , and can be given a valid distance- coloring with colors.

Let denote an arbitrary set of demands. For any demand let be the path given by Algorithm FindPaths. Suppose that the common leader of and is . The path consists of two path segments: the source path segment , from to , and the destination path segment from to . We denote by the subpath between level and level (we call this the level subpath).

Let denote the cost of optimal paths in . Let denote the cost of the paths given by our algorithm. We will bound the competitive ratio . For simplicity, in the approximation analysis, we consider only the cost of the source path segments . When we consider the destination segments the approximation ratio increases by a factor of 2.

The cost can be bounded as a summation of costs from the different levels as follows. For any edge let be the set of layer- subpaths that use edge . Denote by the cost on the edge incurred by the level- subpaths. Since is subadditive, we get . Let denote the cost incurred by the layer- subpaths. Since , we have that:

(1)

For any cluster let denote the set of demands with source in whose paths leave from the leader of toward the leader of a higher level cluster.

Lemma 4

For any , , , where .

Proof

Let to be the set of clusters at level which receive color . Consider a cluster . Consider a demand . Since the common leader of and is at a level or higher. From the algorithm, . Consider the subpaths from of length up to . In the best case, these subpaths from may be combined to produce a path with smallest possible total cost . Any two nodes and , where and , have , since each node is -satisfied in its respective cluster and and receive the same color in the distance- coloring of . Therefore, the subpaths of lengths up to from the demands and cannot combine. Consequently, where . Let . We have that . Since . We obtain , as needed.

We also get the following trivial lower bound for the special case where , which follows directly from the observation that each demand needs to form a path with length at least 1.

Lemma 5

For any , , .

We obtain the following upper bound.

Lemma 6

For any , , where .

Proof

For any cluster , we can partition the demands , where , , according to leader at level that they use, so that all demands in use the same leader in , and and use a different leader of . Next, we provide a bound on .

Consider any two demands and . Let be the common leader at level . Since and are -satisfied by the cluster of , they are both members of that cluster. Therefore, , and . Thus, . Suppose that demand chooses leader at level with respective cluster . Since is at least -satisfied in , is a member of . Since any node is a member of at most clusters at level , it has to be that the number of different level leaders at level that the demands in select is bounded by . Consequently, .

Since is subadditive and for any demand , . Therefore, . Which gives: , as needed.

Lemma 7

For any , .

Proof

From Lemma 6, for any , . From Lemma 4, for any , . Note that . Therefore, . For , we use the lower bound of Lemma 5, and we obtain .

We now give the central result of the analysis:

Theorem 5.1

The oblivious approximation ratio of the algorithm is .

Proof

Since the demand set is arbitrary, from Lemma 7 and Equation 1 we obtain oblivious approximation ratio bounded by . When we take into account the source path segments together with the destination path segments, the approximation ratio bound increases by a factor of 2, and it becomes . Since, , , , are constants and , we obtain approximation ratio .

6 Conclusions

We provide a set of paths for the multi-sink buy-at-bulk network design problem in planar graphs. Contrary to many related work where the source-destination pairs were already given, or when the source-set was given, we assumed the obliviousness of the set of source-destination pairs. Moreover, we considered an unknown fusion cost function at every edge of the graph. We presented nontrivial upper and lower bounds for the cost of the set of paths. We have demonstrated that a simple, deterministic, polynomial-time algorithm based on sparse covers and shortest-path clustering can provide a set of paths between all pairs of nodes in that can accommodate any set of demands. We have shown that this algorithm guarantees -approximation. As part of our future work, we are looking into obtaining efficient solutions to other related network topologies, such as minor-free graphs.

Appendix: Proofs

Proof of Lemma 1 The algorithm divides the path into consecutive disjoint subpaths each of length (except for the last subpath which may have shorter length). The algorithm builds a cluster around each subpath which consists of the -neighborhood of . We can show that . Suppose otherwise. Then, there are nodes , , which are -satisfied in their respective clusters and . Thus, . Then, there is a path of length at most that connects the two paths and which is formed through . However, this is impossible since the paths are at distance at least . Therefore, we can use a palette of at most colors to color the clusters, so that each cluster receives color .

Proof of Lemma 2 Consider a level of . From Lemma 1, the clusters produced in any node of level from path can be colored with colors. Consider now two nodes and in level of . Let be a cluster from . Any node which is -satisfied (with respect to ) in , cannot have in its -neighborhood any node in , since and are disjoint. Therefore, any two nodes which are -satisfied in the respective clusters of and have to be at distance more than from each other in . This implies that we can use same palette of 3 colors for each node in the same level of the tree.

Proof of Lemma 3 From Lemma 2, the clusters of each level of the tree can be colored with 3 colors. From the proof in [4, Lemma 5.4], any node is clustered in at most consecutive levels of , and does not appear in any subsequent level. Any node which is -satisfied in a cluster of level cannot be with distance of or less from , since doesn’t appear in the level subgraph of . Therefore, any node which is -satisfied in a cluster of level must be at distance more than than any node which is -satisfied in a cluster of level . Therefore the clusters formed at level are at distance at least from clusters formed at level . Consequently, we can use color palette for odd levels and color palette for even levels, using in total colors.

Footnotes

  1. thanks: This work is supported by NSF grant CNS-084608.
  2. email: {ssrini1, busch, iyengar}@csc.lsu.edu
  3. email: {ssrini1, busch, iyengar}@csc.lsu.edu
  4. email: {ssrini1, busch, iyengar}@csc.lsu.edu

References

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