Depth-Latency Tradeoffs in Multicast Tree Algorithms Michael T. Helmick and Fred S. Annexstein Department of Computer Science University of Cincinnati Cincinnati, OH 45221 USA
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Abstract
ogy information. Peer-to-peer overlay networks provide an attractive option because of the topology information that is either available at the overlay level or can be inferred by what is known, and several such models exist [3, 6, 11, 13, 14]. Multicast distribution routes are represented by a rooted, directed spanning tree. The problem of determining such a rooted tree which covers all subscribers is complicated by the need to balance network resources while optimizing the serving of the communication group. In order to enable real-time or near real-time delivery for bandwidth intensive multicast communication streams (such as a live video broadcast), the multicast distribution tree must have a bounded out-degree in order to allow for smooth playback/presentation of the material for subscribers. For example, a client with a peak upload bandwidth of 128Kbps participating in an overlay multicast requiring 48Kbps can only relay the communication to two other participants. This leads us to study the minimum average-latency degree-bounded directed spanning tree problem, a well known NP-hard problem [5]. Recent works propose approximate solutions to this problem [3, 13, 14]. Our work differs from previous work in that we focus on the additional constraint of minimizing the depth or hop count from the source in the message distribution. Hop count and time-to-live (TTL) bounds are important in the design of communication protocols, particularly at the application-level, in order to minimize congestion in unstructured networks (see e.g., [2]). Here we present a new heuristic algorithm, called DBSPT, that improves upon previous solutions, both in terms of empirical evaluation and theoretical worst-case approximation factors for degree and depth-bounded minimum average-latency spanning trees. We present an empirical analysis in which we compare our DBSPT algorithm against other solutions to the problem including the KLS algorithm [10], OMNI [3], and MDDBST [14]. Our algorithm consistently outperforms all previous algorithms when we consider the combined product measure of average-latency times the maximum depth.
The construction of multicast trees is complicated by the need to balance a number of important objectives, including: minimizing latencies, minimizing depth/hops, and bounding the degree. In this paper we study the problem of determining a degree-bounded directed spanning tree of minimum average-latency in a complete graph where the inter-node latencies determine a metric. In particular, we focus on measuring the effects on average latency when imposing depth constraints (i.e., bounds on hop count) on degree-bounded spanning trees. The general problem is a well known NP-hard problem, and several recent works have proposed approximate solutions which aim at minimizing either depth or latency. In this work, we present a new heuristic algorithm which improves upon previous solutions by considering both depth and latency and the tradeoffs between them. Our algorithms are shown to improve the theoretical worst-case approximation factors, and we show improvements under empirical evaluation. Our experiments examine and analyze several different topologies, including, low-dimensional random geometric networks, random transit-stub networks, and high-dimensional hypercube networks. We show how our solutions can be applied in the context of enabling multicasting support in locality aware peer-to-peer overlay networks.
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Introduction
The problem of multicasting, that is, disseminating the same information from a single source to many receivers, is a well studied problem at the network level and at the application level [4, 6, 8, 11, 13]. Although IP level multicast has not been widely adopted [9], gains can be made by deploying multicast enabled applications on the internet. Potential multicast applications include multimedia distribution and news/event notification. When constructing application-level multicast routes, we are able to leverage the availability of gathered topol-
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tion to the degree-bounded spanning tree with a O(logn) approximation for n-node networks. However, the tree only covers the MSNs in the network, and not all the nodes representing the end systems. Kanemann et al. [10] present an algorithm (hereafter referred to as KLS) for approximating the degree-bounded minimum diameter spanning tree problem. The mechanism used in the KLS algorithm works by a process of clustering and filtering, and for complete metric networks produces a spanning tree with an approximation ratio of √ O( logB n) for diameter, where B is the degree bound and n is the number of nodes. The MDDBST algorithm presented by Shi et al. [14] describes a heuristic algorithm for the degree-bounded minimum spanning tree problem. MDDBST places an emphasis on obeying the bandwidth constraints (degreebound) at each vertex while working to minimize the average latency in the network. Our experimental study of Section 6 shows results for our DBSPT algorithm and compares these against the algorithm discussed here, including, OMNI, KLS, MDDBST, and various hybrid algorithms.
Our algorithm is unique in that it utilizes a heuristic that ensures each vertex in the multicast tree is connected through a vertex that is closer to the root (the source of the communication). This constraint is imposed and evaluated at each level, for each vertex. This heuristic differs from the OMNI solution, for example, which only applies this constraint during an initialization phase. The clustering method we use was inspired by the KLS algorithm of [10]; however, the KLS algorithm does not concern itself with minimizing the average-latency, but rather minimizing diameter. Out-degree and depth constraints significantly restrict the placement of individual nodes within the spanning tree and can significantly impact the performance as measured by average latency. The resulting stretch, what we call distortion, is the measure of the deviation from the shortest distance of the distance within the spanning tree. This distortion is particulary apparent in ”higher dimensional” networks; and we present in Section 4 lower bounds on the amount of distortion required by B-ary spanning trees on n-dimensional Hypercube-like networks. Finding an algorithm to match this bound is an open problem, to our knowledge. We provide experimental results showing that our DBSPT algorithm performs near to this bound. The paper is organized as follows. We provide an overview of the related work in Section 2. Section 3 formally describes the spanning tree problems we consider. The topology models used for the empirical study are described in Section 4. Our solution to the problem is fully detailed in Section 5, with experimental results shown in Section 6.
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Problem Formation
On the internet, two end systems are connected by multiple physical links spanned by middleboxes [1] such as routers, switches, firewalls, and network address translators. For purposes of computation, the path between two end systems is abstracted to a single virtual link and assigned a representative weight based on the latency (minimum round trip time) between the two hosts. The multicast tree will be calculated on a particular set of nodes where the latency can be measured between all nodes, yielding a complete graph. We study the problem using a graph model where the input is a complete graph, with the graph represented as G = (V, E). V is the set of all vertices (end systems in the network) and E is the set of weighted, undirected edges between all nodes. We consider only networks in which all nodes are subscribers of the multicast group or one in which non-subscribers can be ignored. The edges in the graph are undirected, indicating that there is the potential for information to flow in either direction. Edges are assigned weights corresponding to the latency between the nodes they connect. We are using the result of Kanemann et al. [10] as a base case for experimentation in relation to the degreebounded minimum diameter spanning tree problem. A solution to this problem will return a spanning tree T over the graph G with a minimum diameter and a maximum out-degree of B at any node. This is one of the criteria to be achieved. Our solution to this problem can be used with the network clustering and filtering of [10] or can be used
Related Work
Related work for this research falls into two categories, systems research on overlay network multicast solutions and research on the degree-bounded spanning tree problem. Scribe [6] is an existing overlay multicast network that relies on Pastry [12] to route messages in the multicast tree rather than calculating new routes to maximize multicast performance. Scribe over Pastry uses reversepath forwarding [6, 7] to distribute multicast content, a technique that does not necessarily observe degree constraints [16]. Borg is a hybrid multicast distribution scheme [16] that is also built upon Pastry and alleviates some of the impact when using reverse-path forwarding. OMNI [3] is a multicast overlay infrastructure which utilizes multicast service nodes (MSNs) in a hierarchical overlay network. Individual end systems subscribe to the multicast by subscribing through a MSN. The group of MSNs (a predetermined group) are formed into a distribution tree using a distributed multi-round algorithm that responds to client churn. OMNI forms a tree that is a solu-
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topologies share some common properties in that they are all weighted, undirected, complete graphs. The minimum one-way communication cost between any two end systems in the network is represented by the distance function d(x, y). Each of the graph models provides a consistent metric function which obeys the triangle inequality, such that d(a, b) ≤ d(a, c) + d(c, b) ∀ a, b, c ∈ V where a 6= b 6= c. The three topology models are; (1) 2dimensional random geometric, (2) transit-stub using the Georgia Tech Internetwork Topology Models (GT-ITM) generator [15], and (3) n-dimensional Euclidean hypercube topologies. 2-dim random geometric topologies are generated using graphs that are embedded in the plane at random locations, and where Euclidian distance is used as the metric function. This topology generator is also capable of modeling behavior of networks where not all hosts are publicly addressable. e.g., firewall and network address translation nodes. GT-ITM [15] is used to generate transit-stub networks which simulate sub-domains connected to each other through an internet backbone. We have generated topologies of 600 nodes. The output graphs from GT-ITM are not complete graphs, so the missing edges are virtualized by computing the all-pairs shortest path. As stated earlier, this models real internet communication, as a message from a to b is abstracted to a single link while the actual message will traverse several links. We extend the GT-ITM model by generating degree constraints for the nodes in generated graphs. The third model, n-dimensional hypercube topologies, allows us to examine solutions to the degree-bounded spanning tree problem in high-dimensional metric spaces. We define the hypercube metric Hnp as follows, each point of the 2n point metric is labeled with a unique n-bit binary label. The distance between two points of Hnp is comP 1 puted using the `p norm ||x − y||p = ( |xi − yi |p ) p . The `1 norm is the ordinary hamming distance which is calculated as the number of positions that the two pointlabels differ in. The `2 -norm can simply be calculated as the square root of the hamming distance. For purposes of notation we refer to the distance function for hypercube networks as dp (x, y). Of course, these norms obey the triangle inequality. Given a discrete metric space M , the T -distortion induced by a rooted spanning tree T is simply the ratio of the average-latency (equivalently, total distance) to the root in the tree T divided by the average shortest distance to the root given by the metric M . Define the TB 4 Topology Models distortion of a metric M as the minimum value of the T distortion over all B-ary spanning trees (and thus satisfyExperiments are conducted over input sets from three difing a degree bound of B + 1). For the hypercube metferent topology models, providing three different graph rics we have the following lower bound results on the TB models for representing the network. All of the generated distortion of Hnp .
to calculate trees without the clustering of nodes. Using clustering to partition the network into smaller subsets reduces computation time and makes distributed calculation of multicast trees a possibility. While this is a good general goal for group communication, there are additional criteria that are important for enabling (near) real-time delivery for internet multicast applications. In their research Banerjee et al. [3], defined this as a bicriteria problem, which can be stated as a minimization of two criteria; (1) the average-latency and (2) the maximum-latency. Here, we define a slightly different bicriteria problem with these constraints; (1) minimize the maximum depth in the tree and (2) minimize the average-latency in the class of out-degree B spanning trees. As a metric we will compute the product of average-latency and maximum depth and show results when constraining the maximum depth to the minimum possible depth (given the degree bound), and allowing the height of the tree to change based on the graph. By examining the normalized product of these two metrics we are able to compare the blended result for the all tested algorithms. Bounding the maximum depth of the tree, and therefore bounding the maximum hops, is a meaningful metric for networks in which time to live (TTL) is a parameter on messages. Reducing the number of hops between the root and leaves also lowers the number of failure points along any given root to leaf path. For a given spanning tree T , let dT (v) denote the length of the shortest path from the root to the v in T , and we identify this length as the latency required for a message to reach v. This tree distance dT (v) is simply the sum of the metric distances of each of the edges on the unique path to v. Hence, we refer to the average latency of T based on these P tree distances, and this average-latency is calculated as n1 v∈T dT (v). For the hop measurement of depthT (v), we use each node’s level in the tree, with the convention that the root has depth = 1. We will use these measures of average latency and maximum depth to evaluate our algorithms for degree-bounded spanning tree constructions. In particular, our goal is to approximate the minimum average latency in a multicast distribution tree using a compact tree. We will use the term compact B-ary tree to refer to a tree of minimum possible depth given the degree bound B. Such trees have the property that all leaves reside on the deepest 2 levels of the tree. Compact B-ary trees formed on n nodes will have maximum depth at most dlogB ne.
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p THEOREM 4.1 The TB -distortion of the metric Hnp is O( logB n ∆) on the diameter of the produced degree1 bounded spanning tree [10]. The final approximation Ω(n1− p /lnB). tree is composed of clusters containing only short edges, PROOF A lower bound for the TB -distortion is obtained joined together by long edges. by calculating the ratio of the sum of minimum possiOur expansion of this solution differentiates itself by ble path lengths in any B-ary spanning tree divided by changing how the clusters of short edges are organized, the sum of the shortest distances to the root node in the providing a significant improvement in runtime result acmetric. In any B-ary spanning tree of the metric Hnp the cording to the criteria set forth in Section 3. In their proof, minimum possible average hop-distance to the root is at Kanemann et al. [10] state that the cluster local structure least logB 2n − 2. Also, the average distance in the met- must be a spanning tree of the nodes belonging to the clus1 ric is seen to be at least 12 (n/2) p . This follows since at ter of minimum height. This relies on the fact that any least half the nodes of the hypercube are (metric) distance edge used to join nodes within a local cluster will not be 1 (n/2) p from the root node. Thus the TB -distortion is at longer then 2r (where r is the radius of the cluster, 3α) 1 least cn1− p /lnB for some constant c > 0; and the theo- as long as the triangle inequality is in effect. Theorem 5.1 states the upper bound on distance from the root, for any rem follows. node in a local cluster. The result above does not provide a bound for the TB THEOREM 5.1 Given any complete metric M and a distortion for the usual Hamming metric . In fact, for B = root node r, let d(v) = dM (v, r) be the distance from 2 the well known doubly-rooted binary tree embedding in v to r in M . There is a compact B-ary spanning tree such Hn1 (see [? ]) shows that the T2 -distortion of Hn1 is Θ(1). that every node v has a tree distance dT (v) to the root that It is not difficult to show that this Θ(1) result extends for is bounded by 2 ∗ depthT (v) ∗ d(v) = O(logB n) ∗ d(v). all B that are powers of 2. We leave as an open problem whether or not this results extends for other values of B. PROOF It is always possible to construct a B-ary span-
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ning tree with the property that each node’s parent is no further from the root than the node itself. This can be done by simply sorting the nodes by their distance to the root and creating a compact B-ary tree T whose breadth first traversal respects this ordering. Let us consider the distance dT (v). We have that dT (v) ≤ dM (v, u) + dT (u) ≤ dM (v, u) + 2 depth (u) ∗ d(u) using the inductive hypothesis, where u is the parent of v. By the triangle inequality we have that dM (v, u) ≤ dM (v, r) + dM (r, u). A direct calculation shows that dT (v) ≤ 2 ∗ depthT (v) ∗ d(v) and the result follows by induction. Again, note that compact B-ary trees formed on n nodes will have depth at most dlogB ne.
Solution
Here we present a new solution to the degree-bounded minimum average-latency spanning tree problem. Our new algorithm, DBSPT-Circular-PO uses a heuristic to form degree-bounded spanning trees by ensuring that all hosts connect to the source by connecting through a host that is closer to the root. This helps to reduce latency introduced by deviations from the optimal path. We begin by examining the KLS algorithm [10], which uses node clustering to segment the network, internal cluster organization, and then meta cluster organization for a final result. We examine the effects of using the DBSPTCircular-PO as the internal cluster organization method for KLS, as well as comparing the algorithms individually, and against other algorithms in the area. The KLS algorithm for solving the degree-bounded minimum diameter spanning tree problem begins by forming clusters of nodes through filtering. The algorithm proceeds by performing a binary search to determine the best value for ∆, the estimated diameter of the degreebounded spanning tree. Cluster representatives are then chosen iteratively by selecting the vertex that covers the most uncovered vertices with a radius of 3α, where α ← p ∆ . The α threshold represents the differlogB n ence between long and short edges in the graph, this allows a limit to be imposed on the number of short edges and long edges in the final product, giving a bound of
This allows us to state that the total number of edges that contribute to the diameter of a cluster is bounded by O(logB n). We have studied various methods for organizing the clusters in the KLS algorithm and examine the cost of building a tree that minimizes average-latency versus building a tree that minimizes the maximum depth for leaves in the tree. We have measured the performance of our DBSPT-Circular algorithm and its compact variant (DBSPT-Circular-PO) in the context of KLS clusters and on the entire network, without clustering. While clustering and filtering provides for lower computational complexity and aids in the ability to produce a distributed algorithm, the resultant trees do not exhibit lower averagelatency when compared to the non-clustering version of DBSPT-Circular and DBSPT-Circular-PO. We have discovered that forcing the upper levels of the tree to be full, but not forcing a tree of overall min4
Algorithm 1 DBSPT-Circular( Vi , vi , B ): Organize the cluster Vi with cluster representative vi and maximum out-degree B by optimizing the shortest path tree and enforcing that each node has a parent closer to the root than itself. 1: shortestPathTree( Vi , vi ) 2: overloadedSet ← findOverloadedVertices( Vi ) 3: level ← 1 4: while overloadedSet 6= ∅ do 5: curLevelSet ← nodesInLevel( level ) 6: if overloadedSet ∩ curLevelSet 6= ∅ then 7: Set fixed, toMove ← new Set 8: for all currentVertex ∈ curLevelSet do 9: if currentVertex ∈ overloadedSet then 10: The closest B children of currentVertex are added to the fixed set, the rest are added to the toMove set. 11: else if currentVertex.outDegree < B then 12: fixed.add( currentVertex ) 13: end if 14: end for 15: for all moveVertex ∈ toMove do 16: Connect moveVertex to the closest vertex in the fixed set that is also closer to the root. 17: if moveVertex can not be moved then 18: Swap moveVertex with a sibling that is farther from the root than moveVertex. 19: end if 20: end for 21: else 22: level ← level + 1 23: end if 24: overloadedSet ← findOverloadedVertices( Vi ) 25: end while
imum height, we are able to achieve better experimental results in terms of average-latency measurements. Of the algorithms presented here, quick construction forces a tree of minimum height without regard to minimizing latency, DBSPT-Circular concentrates on producing a minimum average-latency without forcing a tree of minimum height, and DBSPT-Circular-PO derives a tree of low average-latency while guaranteeing a tree of minimum height (dlogB ne levels). We examined the effects of producing a compact tree over clusters, producing degreebounded spanning trees without regard to hop count, and enforcing a strict bound on the maximum hop count. Below, we describe two different implementations of local cluster organization; a naive algorithm for quick construction and our new heuristic algorithm referred to as DBSPT-Circular-PO, which stands for degree-bounded, shortest path tree, with circular organization, and post optimizations. The DBSPT-Circular-PO algorithm takes an additional parameter indicating if the tree should be compacted after construction or not. If the tree is not forced to be compact, the post optimization step is not run. Each of these algorithms is presented in the following sections.
5.1
Quick Construction
The quick construction algorithm uses the fact that no matter which edges are used in the construction of the local tree, the maximum distance from the root will still be O(rdlogB ne) since the tree is compact. In this case, the tree is built by adding nodes to the tree in the order of their distance from the cluster representative (breadthfirst search order). Consequently, this organization algorithm is runtime efficient, forming an out-degree B tree of minimum height with a runtime of O(n). However, the average-latency of trees formed through quick construction are not as efficient as they can be. The resultant trees Throughout the process, nodes that are over their dedo fall within the mathematical bounds set forth above and gree bound are relieved of this load by clearing the curin [10] for local clusters and feed into the larger problem rent search space (current level) and forcing the load one when the clusters are combined to form a final tree. level lower. While this guarantees that the current level contains no nodes above the maximum out-degree, nodes 5.2 DBSPT-Circular-PO in the previous level may now have this characterization The DBSPT-Circular algorithm for approximating and need to be cleared of extra load. Initially, all nodes in the shortest path tree will either degree-bounded spanning trees begins by calculating the shortest path tree from the root. The shortest path tree be connected directly to the root, or have a path to the root is then modified using the heuristic whereby we state that connects only through nodes closer to the root (if they that each node must be connected to a parent that is are in the same axis from the root in the `1 metric space). closer to the root than itself, that is if a’s parent is b, then For both the random distribution and hypercube models, dM (b, root) ≤ dM (a, root). Visually, this is represented the shortest path tree is a simple 2 level star topology inby concentric circles are drawn for each radius of a node stead of the calculated shortest path tree. Forcing the calfrom the root, and all nodes are connected to a parent in a culation of the star topology, rather than computing the circle of equal to, or small radius than the ring they are a shortest path tree provides a reduction in computational complexity. member of.
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X
Y
X Z
Y
Z
Figure 1: Example correction, when a node’s B closest children are not the 3 closest nodes to the root (with a maximum out-degree of 3). The relationship of each node always connecting to the root through nodes closer to the root is kept in effect by correcting the tree one level at a time. For example, starting at the root (level 1), the closest B children at each vertex are kept as children, while the remaining children (up to n − B − 1) are made children of the closed B children. The process is then repeated until all nodes in the tree have ≤ B children at all levels. There are scenarios that arise where the B closest children to a vertex are not the B children that are also closest to the root as demonstrated in Figure 1. This scenario does arise in some graphs due to the relaxed rules of the heuristic. In this example, node Y would normally be the third child of node X, and node Z would be made a grandchild of X. However, this would violate the organizational rule stating that each vertex be connected to a vertex that is closer to the root than itself. This is solved by swapping the tree position of Y and Z. Node Z is made a child of X, and Y is made a grandchild X and a child of Z. The entire DBSPT-Circular algorithm is presented as Algorithm 1. The DBSPT-Circular-PO algorithm is an extension to the DBSPT-Circular algorithm in which the tree is forced to be of minimum possible height, dlogB ne. The PO in this algorithm name stands for post optimization. After the DBSPT-Circular algorithm has completed, the tree is compacted by examining the tree in breadth-first search order, starting with the root. Any vertex encountered that has an out-degree < B is brought up to an out-degree of B by promoting a sub-tree. If the node being examined is at level i, sub-trees rooted in levels ≥ i + 2 are considered potential children. The promotion which causes the least increase to average-latency is the transfer enacted in
the current tree. This procedure guarantees a final tree of minimum height, at the expense of increased averagelatency.
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Empirical Study
We have conducted an empirical study aimed at measuring the performance of these solutions, compared against existing algorithms for networks of different sizes, degree bounds, and topology models. We have constructed a simulator known as GraphSim, integrated with the tunable topology generator (Section 4) that allows for automated, repetitive execution and measurement of graph algorithms on many different random graphs in a short period of time. Two graph implementations are provided, one fully object-oriented where vertices and edges are all instantiated as objects in memory, and another where graph information is synthesized as needed (for modeling large hypercube networks). For experimentation, we use a version of GraphSim capable of running on a high performance computing cluster. Experiments have been conducted for completely random networks of sizes ranging from 100 nodes through 1600 nodes and with a degree bound varying from 3 through 5. This allows us to benchmark the performance of our approximation algorithms under varying conditions. The same algorithms have been tested using 600 node internetwork topologies generated by GT-ITM and imposing out-degree constraints of 3, 4, and 5. Hypercube networks have been examined for up to 2048 nodes (11 bits used to represent each vertex), using the same degree-bounds (3,4,5), using both the `1 and `2 metrics.
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Table 1: Latency, compared against depth. Random Network: N = 600, B = 3. Product of average-latency and maximum depth is normalized. average-latency Max Depth Normalized Algorithm Compact Mean StdDev Mean StdDev avg. latency ∗ max depth DBSPT-Circular-PO Yes 1590.33 91.51 7.00 0.00 0.271 DBSPT-Circular No 1329.08 42.33 8.80 0.40 0.285 KLS w/ DBSPT-Circular-PO Yes 1631.38 92.42 10.20 1.08 0.405 MDDBST No 1342.65 66.22 13.80 0.87 0.451 OMNI Yes 2051.23 558.79 10.60 1.36 0.529 KLS Yes 2977.76 430.81 9.10 0.83 0.659 Table 2: Latency, compared against depth. Transit-Stub Network: N = 600, B = 3. Average-latency Max Depth Normalized Algorithm Compact Mean StdDev Mean StdDev avg. latency ∗ max depth DBSPT-Circular-PO Yes 280.16 35.59 6.00 0.00 0.259 DBSPT-Circular No 210.14 31.25 7.67 0.75 0.248 KLS w/ DBSPT-Circular-PO Yes 210.14 31.25 7.67 0.75 0.248 MDDBST No 222.37 46.16 10.67 1.37 0.365 OMNI Yes 257.62 28.09 7.33 0.47 0.291 KLS Yes 608.99 76.74 7.50 0.76 0.703
A degree-bounded spanning tree has been produced for The resultant trees from simulation experiments are each of these graphs using 6 different algorithms or com- measured to obtain data on various aspects, most imporbinations of algorithms (OMNI is not examined for hyper- tantly the average-latency and maximum depth (hop count cube networks). The algorithms considered are: + 1). Our best algorithm will be the one which provides the lowest product of average-latency and maximum 1. KLS [10] with Quick Construction (baseline). depth for a variety of networks. Table 1 shows the results for 10 simulations of random 2. KLS with DBSPT-Circular-PO. networks of 600 nodes and a degree bound of 3. The best 3. DBSPT-Circular without post optimizations. algorithms for average-latency are MDDBST [14] and the DBSPT-Circular algorithm presented in this paper. Ex4. DBSPT-Circular-PO (tree of minimum height). perimentally, DBSPT-Circular shows an improvement in average-latency between 1.0% and 8.2% for the networks 5. MDDBST [14]. tested (this includes all sizes and degree bounds tested). 6. OMNI [3]. DBSPT-Circular-PO performs the best in terms of the bicriteria problem (as described in Section 3) for the product Our implementation of OMNI is a centralized simuof average-latency and maximum depth. lation of the distributed algorithm described by Banerjee When the hop count is considered: KLS [10] (with et al. [3]. All 5 types of local transformations and the Quick Construction) and OMNI provide good results, probabilistic transformations are implemented. Our exwith DBSPT-Circular-PO providing the best result since periments utilize a Prand value of 0.1 and a simulated the algorithm always returns a tree of minimum height. annealing temperature parameter of 10, having chosen DBSPT-Circular is not by definition compact, but perboth of these values based on the results presented in [3]. forms well experimentally for both random and internetOMNI is not directly comparable because, by definition, work based topologies since the tree is formed from the MSNs are assigned a client load that may be above the root down by pushing nodes down the tree. While this degree-bound. The authors of OMNI do not specify if leaves the possibility for under-loaded nodes in the tree, it or how client load is scheduled, so we assume that they is unlikely to occur in a uniform random distribution. If are all serviced directly by their MSN. This, however, the tree is forced to be compact, and of minimum height, leaves MSNs above the degree-bound in some cases since the average-latency level is not maintained. With the degree-bound is enforced for MSN to MSN connections post optimize step enabled, DBSPT-Circular-PO calcuand does not seem to be enforced for MSN to client connections. 7
Table 3: Latency, compared against depth. Hypercube network: N = 2048 (11 bit representation), B = 3, for both the `1 -norm and `2 -norm metrics. Average-latency stretch is the distortion in average-latency when degree constraints are imposed over the average-latency in the shortest path tree. Algorithm Metric Average-latency Max Depth Normalized Product Distortion DBSPT-Circular-PO `1 -norm 7.53 8 0.287 1.370 DBSPT-Circular `1 -norm 7.29 10 0.347 1.325 KLS w/ DBSPT-Circular-PO `1 -norm 7.61 10 0.362 1.385 MDDBST `1 -norm 7.21 12 0.412 1.311 KLS `1 -norm 17.52 11 0.917 3.186 DBSPT-Circular-PO `2 -norm 6.37 8 0.298 3.319 DBSPT-Circular `2 -norm 6.33 8 0.300 3.298 KLS w/ DBSPT-Circular-PO `2 -norm 6.45 8 0.304 3.363 MDDBST `2 -norm 7.02 9 0.372 3.657 KLS `2 -norm 15.43 11 1.000 8.040 Table 4: Known theoretical bounds for the algorithms studied. For latency bounds, r is half the maximum distance between two vertices. For the KLS algorithm, ∆ is the estimated diameter of the spanning tree. Results for OMNI are presented in [3], and results for KLS are presented in [10]. Algorithm Depth (Hops + 1) Maximum Latency Average Latency DBSPT-Circular-PO O(logB n) O(logB n r) O(logB n r) DBSPT-Circular – O(logB n r) O(logB n r) KLS w/ DBSPT-Circular-PO O(logB n) – – MDDBST – – – OMNI O(log2 n) O(log n r) – 2 p KLS O(logB n) O(∆ logB n) – lates trees of the minimum height (dlogB ne) for all networks tested. For experiments using the transit-stub network topologies generated by GT-ITM, we focus on 600 node networks, with a degree bound of 3. For this set of experiments, we select a set of 128 nodes to serve as MSNs for execution of the OMNI algorithm. After OMNI is completed, we organize the same 128 nodes using the same algorithms as described before (requiring the tree to have at least dlog3 128e → 6 levels). This experimental configuration provides a clearer picture of the capabilities of each algorithm since client nodes are not considered after OMNI is finished. Organizing just the 128 nodes feeds into the use of a hierarchical distribution network as described for OMNI [3]. Hierarchical organization is desirable because smaller sub-graphs can be formed into distribution trees in less time and allows for distributed computation. Table 2 shows the results for 6 simulation runs on different 600 node transit-stub networks. We have demonstrated that our centralized DBSPT-Circular algorithm outperforms, in terms of average latency, KLS [10] when using quick construction, MDDBST [14], and OMNI [3]. Our results also indicate that there is a tradeoff between
minimizing the average-latency/maximum-latency versus minimizing the maximum depth in the tree. In order to gain a better understanding of the relationship between minimizing average-latency and minimizing maximum depth, we turn to experiments on hypercube graphs. Using well defined metrics, we are able to measure the optimal average-latency (shortest paths) for all nodes in the network. For hypercube experiments, we start by calculating the optimal path for each individual vertex. This is simply the shortest path tree calculated from the root, but gives us a good benchmark for comparing the performance of the algorithms because this is not a random graph. The degree-bounded spanning tree algorithms discussed thus far (except for OMNI) are tested on these hypercube configurations containing 2k nodes for k ∈ {3, 4, 5, 6, 7, 8, 9, 10, 11} using degree bounds of 3, 4, 5 and 6, executing the algorithms once for each metric function, d(x, y)1 and d(x, y)2 . Table 3 summarizes the results for a hypercube network of size 2k and degree bound 3, a representative case of the overall result set. DBSPT-Circular, with and without the post optimization step, is the best performing algorithm in terms of the average-latency, minimum depth bicriteria problem. For the particular networks shown here (11 bits for each vertex) using the `2 metric, the minimum 8
√ stretch for a B-ary spanning tree is 11/ ln 3 → 3.019. Experimentally, the DBSPT-Circular algorithm delivers an average-latency stretch of 3.298, or 9.2% over the minimum required stretch. These experiments show that there is very little distortion required for obtaining a degreebounded spanning tree of minimum average-latency in high dimensional spaces with a complete, uniform metric. Table 4 shows the currently known theoretical bounds for the algorithms studied Bounds are not known in all categories for all algorithms. The DBSPT-Circular algorithm in this paper has a latency bound of O(logB n r), where r is the radius of the graph, or half the distance of the longest edge in the complete graph. Forcing a compact tree with DBSPT-Circular-PO also gives a bound on the maximum depth (hop count + 1). This bound is O(logB n) and is equal to the minimum depth required to form a tree of out-degree B over n nodes, dlogB ne.
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able). Research is also planned on an incentive-based system to manage the churn of new subscribers and existing nodes unsubscribing from a multicast group, thus changing the structure of the degree-bounded spanning tree. This incentive system will function in a distributed fashion and will seek to keep the network operational while providing a path for a node to possibly receive better performance at the cost of willingness to share and repeated demonstration of such.
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Acknowledgment
This research is supported in part by NSF grant CNS0521189.
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Conclusions
This work provides an empirical study of existing and new algorithms for the degree-bounded minimum spanning tree problem with a focus on improving the result for average-latency within the constraint of tree compactness. We have presented a new algorithm for approximation of degree-bounded minimum spanning trees with and without hop (tree depth) constraints. Of the three algorithms tested, DBSPT-Circular-PO delivers the best result for the criteria of average-latency and maximum hop count while forming compact B-ary trees on different network topologies. While the results for KLS with DBSPT-Circular-PO do not consistently provide the best results, we consider it valuable because of the potential for a distributed implementation in the context of an overlay network. We have chosen to use the clustering as it lends itself to a potentially decentralized solution for use in a peer-to-peer multicast overlay network. These organization algorithms are suitable for implementation in a locality aware peerto-peer network for organization of these multicast routes. We also note the merit of the OMNI [3] and look towards a hierarchical solution for application-level and peer-topeer multicasting. Our future research plans involve integrating the algorithms discussed here into the framework of a hierarchical, locality aware peer-to-peer overlay network. A study is underway to measure the performance of these same algorithms within this framework with a comparison study of existing application level multicast systems. Modeling is underway to examine the best possible way to handle end systems connecting to the internet through network address translation devices (and thus not publicly address-
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