Investigating how Distributed Computation Pruning performs on the Generalized Linear Preference model

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DEGREE PROJECT, IN COMPUTER SCIENCE , FIRST LEVEL STOCKHOLM, SWEDEN 2015

Investigating how Distributed

Computation Pruning performs on the

Generalized Linear Preference model

PATRIK ACKLAND AND LASSE BERGLUND

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Investigating how Distributed Computation

Pruning performs on the Generalized Linear

Preference model

PATRIK ACKLAND LASSE BERGLUND

DEGREE PROJECT IN COMPUTER SCIENCE, FIRST CYCLE (DD143X) Bachelor’s Thesis at Computer Science and Communication

Supervisor: Danupon Nanongkai Examiner: Örjan Ekeberg

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Abstract

Finding all shortest paths in a distributed dynamic network has many practical applications, but it is perhaps most important in network routing. To this end a number of algorithms and tech-niques have been developed. This paper investigates one such technique called Distributed Computation Pruning (DCP), that utilizes a property found in many real-world networks, including the Internet, called Power Law degree distribution. DCP has pre-viously been shown to improve All Pairs Shortest Paths algorithms on networks generated by the Barabási-Albert model. In this pa-per we extend the expa-perimental evidence by evaluating the pa- perfor-mance of algorithms combined with DCP on networks generated by the Generalized Linear Preference model (GLP). This model has previously been shown to generate networks that are more representative of the Internet than those generated by previous models. We found that algorithms that are combined with DCP see a distinct decrease in the amount of messages sent across the network, suggesting that DCP is a viable technique for improving existing routing algorithms.

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Referat

Att hitta den kortaste vägen i ett distribuerat dynamiskt nätverk har många praktiska tillämpningsområden, men är kanske vikti-gast inom nätverksrouting. För detta ändamål har flera algorit-mer och tekniker utvecklats. Denna rapport undersöker en sådan teknik som heter Distributed Computation Pruning (DCP), som utnyttjar en egenskap som återfinns i många riktiga nätverk, in-klusive internet, som kallas Power Law gradfördelning. Tidigare studier har visat hur DCP förbättrat routing-algoritmer på nät-verk som genererats av Barabási-Albert-modellen. I den här rap-porten utökar vi de experimentella bevisen för DCP genom att utvärdera dess effektivitet på nätverk som genererats av Genera-lized Linear Preference-modellen (GLP). GLP har tidigare visat sig generera nätverk som är mer representativa för internet än ti-digare modeller. Vi fann att algoritmer som kombineras med DCP skickar betydligt färre meddelanden, vilket antyder att DCP är en värdefull teknik för att förbättra existerande routing-algoritmer.

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Acknowledgements

We would like to extend our appreciation to our supervisor Danupon Nanongkai, for his advice and feedback. To our opponents Hampus Fristedt and Joakim Hedlund for their comments. Finally, to Mattia D’Emidio for being incred-ibly helpful, patient, and for supplying the implementations of the routing algorithms tested in this report.

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Introduction

The shortest path is important and an easy concept to grasp for most people. Being able to choose the shortest path is a way to save time and money: the shortest path to work, the shortest route for a delivery or the shortest path for data in a computer network. Being able to compute the shortest path is of great interest to many companies and researchers for planning delivery routes or reduce the daily commute of people using the subway.

There has been considerable research done to find the shortest paths in a dynamic distributed network [1, 2, 3, 4, 5, 6]. Being able to efficiently compute the shortest paths has several practical applications for network routing proto-cols and route planning in road networks. When evaluating the performance of distributed shortest-paths algorithms compared to traditional shortest-paths algorithms one has to account for additional factors. In addition to compu-tation time, it is also important to consider memory usage for each node and the amount of messages sent between nodes. Recently there has been a trend focusing on utilizing the structure of the network to improve algorithms. One such improvement is Distributed Computation Pruning (DCP) [7] which is the focus of this report.

Previous experimental evaluation of DCP used the Barabási-Albert (BA) model [7]. However, the BA model has been shown to produce networks that fail to capture some characteristics of the Internet [8]. This report will investigate how DCP and its predecessor Distributed Leaves Pruning (DLP) perform on graphs generated by the Generalized Linear Preference (GLP) model. Both DLP and DCP were developed to reduce the number of messages sent by routing algorithms during the calculation of shortest paths. The result of our simulations are compared to the previous results.

In chapter 2 we introduce the network model as well as the definition of scale-free networks and describe the two models in more detail. We also describe the two routing algorithms used and give an overview of DLP and DCP. Chapter 2 also introduces the results from the previously mentioned report which used the BA model. In chapter 3 the simulation environment,

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CHAPTER 1. INTRODUCTION

simulations and graph generators are explained. Finally, the results are pre-sented, discussed, and compared with the previous simulations done using BA. We end with conclusions and directions for further work.

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Chapter 2

Background

2.1 Routing algorithms

Algorithms that find and maintain shortest paths are typically categorised as either distance-vector or link-state algorithms. In a distance-vector algorithm each node knows the length of the shortest path to all other nodes in the net-work, however they do not have knowledge of all the nodes on that path, they only know which the next step is. The paths are generally computed using a variant of the Distributed Bellman-Ford (DBF) algorithm [3]. Distance-vector algorithms based on DBF have two major issues called count to infinity and looping. A loop is a path specified in the nodes’ routing tables at a partic-ular point in time, such that the path visits the same node more than once before reaching the intended destination. A node counts to infinity when it increments its distance to a destination until it reaches a predefined maximum distance value [9].

In link-state algorithms each node has to know about the entire network topology, and uses some variant of Djikstra’s algorithm to calculate the short-est path to a dshort-estination node. In this way link-state algorithms basically circumvent the distributed nature of the network by instead having a large amount of communication [3]. This also means that whenever a change oc-curs in the network it has to be broadcasted to all nodes [10]. Because of the large memory overhead, and high communication volume, link-state algo-rithms might be infeasible for large, dynamic networks [11]. For these reasons the rest of the paper will be limited to only distance-vector algorithms.

2.2 Network model

The network is modeled as a connected undirected graph; where each node is a processor. Local computation is considered in this model to be cheap enough to not impact the overall running time of the algorithm. Each node knows the following things; the identity of all other nodes in the network,

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CHAPTER 2. BACKGROUND

which nodes it is connected to, and the costs associated with those edges. Edges are communication channels, with some finite delay. Communication is asynchronous, and there is no shared memory between the processors. Because the network is dynamic, edge costs may change over time, either increasing or decreasing.

The following notation is used; let G = (V, E, w) be the graph representing the network.

• G is connected, it consists of a single component. • V is a finite set of n nodes.

• E is a finite set of m edges, where an edge is a pair (u, v), u, v ∈ V . • w is a function defined as {w : (u, v) → N}, (u, v) ∈ E.

• N(v) for v ∈ V , is the set of nodes connected to v, and |N(v)| is the degree of v, defined as deg(v).

• A path between nodes u and v, P = {u, ..., v} has a weight defined as the sum of the weights of the edges in the path.

• The shortest path is defined as the path with the minimum weight, this weight is noted as the distance d(u, v), and the nodes in that path is called V IA(u, v).

2.3 Scale-free networks

Scale-free networks are networks where some nodes in the graph have higher than average degree. More formally, a network is scale-free if the degree of a given node k can be described by a power law distribution P (k) ∼ k−y

[12]. Scale-free networks arise naturally in communication networks and social graphs. There are several practical examples of scale-free networks, most importantly the Internet has been shown to have scale-free properties [13]. There exists a number of different models for generating scale-free networks.

2.3.1 Barabási-Albert model

One of the earliest models for generating networks that capture the scale-free property is the Barabási-Abert model [12]. The model is based on two main ideas, growth and preferential attachement. Previous models worked by start-ing with a fixed number of nodes, and then changstart-ing the connections between them. Barabási-Albert instead continously adds more nodes, a phenomenon which is commonly observed in real world networks. Preferential attachement describes how these new nodes are connected to the existing network. Unlike previous models, the probability of a new node getting attached to a node v

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2.4. ALGORITHMS

depends on the degree of v in Barabási-Albert. This means that a new node is more likely to get connected to a node of high degree. These two ideas enable Barabási-Albert to generate scale-free networks. More formally, the probability of a new node connecting with node i can be defined as shown in equation (2.1)

Y

(i) = deg(i)/X

j

(deg(j)) (2.1)

(2.1) denotes the probability that a new node will be connected to node i.

2.3.2 Generalised Linear Preference model

Another model for generating scale-free networks is the Generalized Linear Preference (GLP) model [8].

Y

(i) = (deg(i) − β)/X

j

(deg(j) − β) (2.2)

In (2.2) β ∈ (−∞, 1) indicates the preference for a new node connecting to more popular nodes. The GLP model starts with m0 vertices linked by

m0 −1 edges. The probability that vertex i increases its degree deg(i) is a

function of its degree. At each time-step one of the following operations will be performed:

• With probability p, n ≤ m0 edges are added. Vertex i is chosen for the

new edge with probability Q(i) as defined in (2.2). This leads to new

edges connecting popular vertices.

• With probability 1 − p a new vertex is added with m new edges. Each edge is connected to vertex i which is already in the graph with proba-bility Q(i) as defined in (2.2).

2.3.3 Comparison of GLP and BA

BA has been shown to produce a median power law exponent significantly less than measured from the Internet. In terms of power law exponents, character-istic path length, and clustering coefficient, GLP closer matches the observed structure of the Internet compared to BA [8].

2.4 Algorithms

This report will focus on the Diffuse Update Algorithm (DUAL) and the Loop Free Routing (LFR) algorithm. Both DUAL and LFR can be combined with two techniques known as Distributed Computation Pruning (DCP) as well as its predecessor Distributed Leaves Pruning (DLP) which decreases the number

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of messages sent by the original algorithms. The algorithms are described briefly. A more detailed explanation of each algorithm can be find in their respective sources which are referenced below.

2.4.1 Diffuse Update Algorithm

Diffuse Update ALgorithm (DUAL) is a distance-vector algorithm developed in 1993 by J. J. Garcia-Lunes-Aceves [3]. It is still used in CISCOs EIGRP protocol [14]. It avoids the looping problem mentioned above by using a technique called Diffusing Computations. However compared to some newer algorithms it uses more memory per node and may send more messages [7].

2.4.2 Loop Free Routing

Loop Free Routing (LFR) is a distance-vector algorithm introduced in 2012. LFR has been shown to use less memory compared to DUAL and also sent comparatively fewer messages between nodes when tested on real-world net-works. However, DUAL outperformed LFR in controlled scenarios [9].

2.4.3 Distributed Leaves Pruning

Distributed Leaves Pruning (DLP) was suggested by D’Angelo et al [15] as a way of reducing the number of messages sent during calculation of the all pairs shortest paths by a distance-vector algorithm. It is developed for scale-free networks and uses the fact that there is a low number of central nodes with high degree and that nodes on the edge of the network with low degree are not often used in the computation. In a normal routing algorithm every node performs the same operations. When combined with DLP, central and non-central nodes have different behaviour. Central nodes detect changes on all edges while non-central edges only detect changes on peripheral and cyclic edges.

DLP requires that each node in the graph stores additional information. Each node has to store and update information about non-central paths. It can be shown that the worst case overhead per node due to DLP is O(n). However, DLP can also reduce the memory usage per node when compared to the original algorithm.

Classification

In an undirected weighted graph G = (V, E, w), the core Gc of G is the

maxi-mal connected subgraph of G where all nodes have a degree higher than one. A node v is classified as follows:

• central if v ∈ Gc

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2.4. ALGORITHMS

Figure 2.1. DLP classification of a graph

• peripheral otherwise

An example of the classification is given in figure 2.1 where the grey nodes are central and the white nodes are peripheral.

2.4.4 Distributed Computation Pruning

Distributed Computation Pruning was introduced by D’Angelo et al [7] as an extension of DLP, it further divides the network into more fine-grained classes to take further advantage of the power law degree distribution in scale-free networks. These classifications are described below.

Classification

In an undirected weighted graph G = (V, E, w), a node v ∈ G is classified as follows:

• central if deg(v) ≥ 3

• semi-peripheral if deg(v) = 2 • peripheral if deg(v) = 1

A path P = {v0, v1, ..., vj} ∈ Gis classified as follows:

• central if for all vi ∈ P is a central node.

• semi-peripheral if v0 and vj are two distinct central nodes, and vi is

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Figure 2.2. DCP classification of a graph

• peripheral if v0 is central, vj is peripheral, and vi is semi-peripheral for

each 1 ≤ i ≤ j.

An example of the classification is given in figure 2.2 where the black nodes are central nodes, the grey nodes are semi-peripheral nodes and white nodes are peripheral nodes. Path {c1, c2, c3, c4} is a central path, {c5, sp1, sp2, sp3} is a semi-peripheral path and {c2, sp4, p1} is a peripheral path.

2.5 Our contributions

In this report we will extend the experimental evidence for the effectiveness of DCP by examining the performance of DUAL-DCP and LFR-DCP on net-works generated by the GLP model. We choose GLP because as mentioned above it generates networks that are more representative of the Internet than those generated by the BA model. We will compare the results by running DUAL-DLP and LFR-DLP as well as DUAL and LFR on the same networks. The results from running these six algorithms on dynamic networks will then be analyzed and discussed. The implementations are not ours and have been provided by the research team that developed them [7].

2.6 Previous results

The previous results where DUAL and LFR were combined with DLP and DCP are presented below. The results in more detail can be found in [7]. The simulations were run on a subgraph of the CAIDA Internet topology of

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2.6. PREVIOUS RESULTS

8000 nodes as well as a graph generated by the BA model with 8000 nodes. The number of changes in the network during the simulation were values in {5, 10, ..., 200}. The graphs are denoted G_t−n where t is the model or type of graph (IP or BA) and n is number of nodes. The presented results were from averages over five different experiments for each graph, resulting in a total of 200 runs.

The number of messages sent by DUAL-DCP compared to DUAL on

GBA−8000is between 22%–47% wheras the number of messages sent by

DUAL-DCP compared to DUAL-DLP is between 66%–96%. For LFR, the number of messages sent by DCP compared to LFR is within 26–34%. For LFR-DCP and LFR-DLP the number of messages are 65-85%

For GIP −8000 the number of messages sent by DUAL-DCP compared to

DUAL is 3%–16% and for DUAL-DCP compared to DUAL-DLP the number of messages are within 11%–40%. For LFR, the number of messages sent by LFR-DCP are within 10%–26% and by LFR-DLP they are within 21%–58%.

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Chapter 3

Method

3.1 Experiments

The purpose of these experiments are to provide further evidence that DCP is an improvement of earlier techniques. Earlier work tested DCP on the BA model [7]. This report continues their work by testing DCP on the GLP model. The implementations of DUAL, LFR, DLP, and DCP were provided by Mattia D’Emidio, a member of the research team that developed DLP and DCP. They also wrote the report on using DCP with the BA model.

3.1.1 Simulation environment

The simulations were run using the OMNet++ discrete event simulator tool [16] on Ubuntu 10.04 kernel 2.6.32.73 with 4 GB of memory. OMNet++ is widely used as a simulation tool for computer networks.

3.1.2 Graph generator

To generate the networks a generator called aSHIIP (Autonomous Supelec Hierarchical Interdomain Infering Program) was used [17]. aSHIIP enables a user to choose several parameter values when generating graphs. For GLP the parameters where chosen as follows:

• p = 0.55 • β = 0.75 • m0 = 20

• m = 1

These parameters have been shown to give the topologies that are most rep-resentative for the Internet. m and m0 used default values in aSHIIP because

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CHAPTER 3. METHOD 3.1.3 Network

Memory constraints limited us to running our experiment on a graph with 1200 nodes. However, the similiar study in [7] reported that results did not vary all that much for graph sizes in {1200, 5000, 8000}. The edge weights were randomly selected as non-negative integers between 0–1000000. The changes are applied to randomly selected edges, and a change is modeled as the weight being set to a new random number between 0–1000000, thereby leading to random increases and decreases in edge weights. Each change is applied after a random number of discrete steps between 500000–1000000, this is to avoid simultaneous updates to one edge that might otherwise break the simulation environment. We ran our simulations with the 1200 node graph with k = {10, 20, ..., 100} changes. This results in a total of 60 simulated runs.

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Chapter 4

Results

Figure 4.1. Results of DUAL simulations

In figure 4.1 the result of running the three different DUAL algorithms are shown. The number of messages sent by DUAL-DLP and DUAL-DCP as a percentage of the number of messages of DUAL were observed to lie within 3%–37% for DLP and within 3%–16% for DCP. When comparing DUAL-DLP to DUAL-DCP, DUAL-DCP used between 11%–94% messages of DUAL-DLP with an average of 45%.

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CHAPTER 4. RESULTS

Figure 4.2. Results of LFR simulations

In figure 4.2 the result of running the three different LFR algorithms are shown. For LFR, the message percentages observed were within 13%–53% for LFR-DLP and within 5%–29% for LFR-DCP. A similar comparison as was done with DUAL between LFR-DLP and LFR-DCP showed that LFR-DCP sent between 21%–93% of the messages that LFR-DLP sent, with an average of 57%.

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Chapter 5

Discussion

The results were in line with what we expected. As observed with the BA model, combining the algorithms with DCP is an improvement over the stand-alone algorithm as well as the algorithm combined with DLP.

For some changes, such as k = 30 there is a large increase in the number of messages for DUAL and LFR, whilst DLP and DCP do not see this increase. This can probably be attributed to the classification in DLP and even further classification in DCP. If the changes happen in non-central nodes, fewer mes-sages are sent. Since DUAL and LFR do not have these classifications, more messages are sent.

The smaller improvement gained when combining DCP with LFR can probably be attributed to the fact that LFR has been shown to send fewer messages than DUAL [9] so with fewer messages sent overall, there are fewer chances for DCP to eliminate redundant messages.

Table 5.1. Comparison of BA, GLP, and CAIDA for DUAL

DUAL-DCP DUAL-DCP / DUAL-DLP GLP 3%–16% 11%–94%

BA 22%–47% 66%–96% CAIDA 3%–16% 11%–40%

In table 5.1 the results of running DUAL-DCP each simulation is compared to running DUAL on the same graph. The results are compared to the earlier results using the BA model and on the CAIDA topology. The table shows that the results of running the algorithms on a GLP network are similar to those from the CAIDA simulations. This is expected since the GLP model generates networks that are more representative of the Internet topology. This is important as the algorithms in question are routing algorithms and are used on the Internet.

One point where our results diverged from the previously obtained results was in the relative improvement when going from DLP to DCP. As shown

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CHAPTER 5. DISCUSSION

in table 5.1 the improvement varied between 11%–94% for GLP, compared to 11%–40% for CAIDA. This means that DCP at times sent 94% of the messages of DLP which is a higher number than observed for the CAIDA simulations which was 40%. Presumably the improvement is large when the changes affect semi-peripheral paths, and small when there are no semi-peripheral paths involved.

Table 5.2. Comparison of BA, GLP, and CAIDA for LFR

LFR-DCP LFR-DCP / LFR-DLP GLP 5%–29% 21-%–93%

BA 26%–34% 65%–85% CAIDA 10%–26% 21%–58%

For LFR, the results, presented in table 5.2 are similar where the algo-rithms show a greater reduction of messages on GLP and CAIDA networks. Just as in the case of DUAL, the improvement of DCP when compared to DLP is very varied for GLP. The improvement for GLP is also slightly closer to CAIDA than BA.

We observed quite a bit of variance in the results; there seems to be certain changes that creates the need for a very large number of messages and takes a long time to execute. The effects of these outliers could possibly have been reduced by running a larger number of experiments and taking the average of the results, as in [7]. However it would be interesting to know what caused this behaviour, especially since they seem to disappear when the algorithms are combined with DLP or DCP.

Time limitations forced us to abort certain simulations after several hours. What we did instead was to replace the file that specified the edge updates with a new file on which the simulations were much faster. This means that the results presented above are from changes that took a resonable amount of time to run. Since we do not know what causes the longer run time for certain changes we cannot say what data has been excluded from the report. The longer run times were mostly when running the DUAL or LFR algorithms. Our primary theory is that something in the edge updates caused an error in the simulation environment. This should not invalidate the simulations that are shown in this report and were able to terminate in a reasonable amount of time.

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Chapter 6

Conclusions

When combined with DUAL and LFR, DCP is an improvement over both DLP and the stand-alone algorithms on graphs generated by the GLP model. Because GLP is a more accurate representation of the Internet compared to BA this result contributes to the empirical evidence for the effectiveness of DCP on real-world networks.

Some questions that might merit further investigation: Whether a tech-nique like DCP could be used to improve other distributed graph algorithms, to solve problems beyond All Pairs Shortest Paths, such as Minumum Cut. If the benefits of DCP scale to truly huge networks. It might also be meaning-ful to examine if classification techniques like DCP could be utilized in other networks, ones that do not exhibit a Power Law degree distribution.

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