Tarry vs Awerbuchs

Shawn Biesan

Background

• Tarry’s Transversal Algorithm– Initiator forwards token to one of neighbors, each neighbor

forwards token to all other nodes and when done returns token– Complexity: 2 * [number of edges]– Constructs spanning tree

• Awerbuchs– Node notifies neighbors that it is visited by sending <vis> so

tokens are never sent over frond edges– Time complexity: 4 * [number of nodes] – 2– Constructs spanning tree

• Time Complexity - Number of causally related messages

Experiment

• 2 experiments comparing time complexities– Varying number of processes

• 5-50, varied by 5 node increments

– Varying density of partially connected graph• Probability that there is an edge between two nodes

varies from 30%-100%(fully connected) by increments of 10%, denoted as p

• Graph must be connected, if it isn’t the graph is regenerated until the created graph is connected

• Number of nodes is fixed at 10

– Each data point is the result of averaging 5 trials

Expected Results

• Tarrys time complexity will be better for sparse graphs– Its time complexity depends on number of

edges(2E) whereas Awebuchs depends on the number of nodes (4N -2)

• As p increases Awerbuchs will improve until it has a better time complexity than Tarrys– Greatest difference between them for fully

connected graphs

Result

5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

Varying Number of Nodes

vary node Tarry timevary node Awe time

Number of Nodes

Time Complexity

Result

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

Varying Edge Density

vary prob Tarry timevary p Awe time

Probability of an edge

Time Complexity

Interpretation

• Awerbuchs is indeed better than Tarrys as nodes scale in a completely connected graph– The number of edges grows much faster than

nodes ( #edges= (N(N-1))/2 )• Tarrys algorithm is indeed better for sparse

graphs up until about p=0.4– Expected value of number edges when p=0.4 is

18 so it matches up with theoretical

Code

• Most difficult/interesting part of the code was related to how the simulation engine was made– Each new algorithm must implement a specific

interface in order to be used– Made adding new algorithms less painful

Conclusions And Future Work

• In general Tarry’s algorithm should be used for sparse graphs with small number of nodes, otherwise use Awerbuchs– Easier to implement– Sends less messages – No overhead of <VIS> and

<ack> messages• Future Work– Explore different topologies– Explore Message Complexity deeper

• Questions?• Thank you