Fan Yang 6.338 Final Project

Professor Edelman

Outline 2

I.  Introduction A.  The Traveling Salesman Problem B.  Simulated Annealing C.  Genetic Algorithm

II.  Methods III.  Results

A.  Running Time B.  Parallel Speedup C.  Optimization D.  Rate of Convergence

IV.  Conclusion

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Find the shortest route that goes through a set of nodes (cities).

Traveling Salesman Problem 4

Figure 1: An example of the traveling salesman problem.

Simulated Annealing

1.  Start with a state with some initial energy 2.  Reduce the energy at each iteration via state

transitions (neighbor functions & acceptance function)

3.  Terminate after some iteration count or low energy threshold achieved

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Serial Simulated Annealing Algorithm

  Initialization: Generate a random candidate route and calculate energy for the route.

  Repeat following steps:   Neighbor function: Generate a neighbor route by

exchanging a pair of cities.   Acceptance function: Evaluate the neighbor route for

acceptance - if accepted, replace current route with neighbor route

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Parallel Simulated Annealing Algorithm

1.  On each thread:   Initialization: Generate a random candidate route

and calculate energy for the route.   Repeat following steps:

if ITERATION_COUNT != CONVERGING_COUNT   Neighbor function: Generate a neighbor route by exchanging a pair of

cities.   Acceptance function: Evaluate the neighbor route for acceptance - if

accepted, replace current route with neighbor route Else

  MPI_BARRIER: share the best result among threads by sending all the results to root and have root broadcast the best result

2.  MPI_BARRIER: return the best result among all the threads

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Genetic Algorithm

1.  Start with a population of routes 2.  At each iteration, replace the worst routes with the

children of the best routes 3.  Terminate after some iteration count

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Serial Genetic Algorithm

  Initialization: Generate N random candidate routes and calculate fitness value for each route.

  Repeat following steps:   Selection: Select two best candidate routes.   Reproduction: Reproduce two routes from the best

routes.   Generate new population: Replace the two worst

routes with the new routes.

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Parallel Genetic Algorithm

1.  On each thread:   Initialization: Generate N random candidate routes and

calculate fitness value for each route.   Repeat following steps: 

if ITERATION_COUNT != CONVERGING_COUNT   Selection: Select two best candidate routes.   Reproduction: Reproduce two routes from the best routes.   Generate new population: Replace the two worst routes with the

new routes. Else

  MPI_BARRIER: share the best result among threads by sending all the results to root and have root broadcast the best result

2.  MPI_BARRIER – return the best result among all the thread

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Methods

  C with MPI on Beowulf cluster (using 1 to 8 nodes)

  Genetic Algorithm:   Iterations: 100,000

  population count: 1000

  Simulated Annealing   Iterations: 1,000,000

  Start temperature: 10.0

  Cooling constant: 0.9999

  For each condition, we conducted a total of 10 trials.

  Distances were computed as Euclidean distances.

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CITIES COORDINATES (X, Y)

a (110, 54)

b (236, 110)

c (153, 151)

d (227, 49)

e (307, 176)

f (220, 211)

g (341, 90)

h (149, 91)

i (335, 40)

j (371, 150)

k (218, 161)

l (334, 239)

m (148, 227)

n (49, 128)

o (183, 39)

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Running time results 14

Figure 2: Simulated annealing algorithm running time. The running time decreases as the number of processors increases.

Figure 3: Genetic algorithm running time. The running time decreases as the number of processors increases.

Running time results 15

Parallel speedup results 16

Figure 4: Simulated annealing algorithm parallel speedup. There is a near-linear parallel speedup. The curve approximates an ideal linear line at all points.

Parallel speedup results 17

Figure 5: Genetic algorithm parallel speedup. There is a near-linear parallel speedup. The scalability deteriorates slightly at 7-8 processors.

Optimization results 18

Figure 6: Simulated annealing algorithm distance results. The distance of the shortest route decreases as the number of processors increases. The anomaly at processor #5 is most likely due to an outlier in the raw data points, which is also indicated by the large standard deviation.

Optimization results 19

Figure 7: Genetic algorithm distance results. The distance of the shortest route decreases as the number of processors increases.

Figure 8: Simulated annealing algorithm convergence running time. There is a gradual increase in running time as the rate of convergence increases. The running time exceeds the serial implementation at a convergence rate of 35%.

Rate of convergence: running time results 20

Figure 9: Genetic algorithm convergence running time. There is a gradual increase in running time as the rate of convergence increases. The running time exceeds the serial implementation at a convergence rate of 15%.

Rate of convergence: running time results 21

Figure 10: Simulated annealing convergence distance results. The distance of the shortest route decreases as we increase the rate of convergence. The shortest distance plateaus beginning at a 10% rate of convergence.

Rate of convergence: optimization results 22

Figure 11: Genetic algorithm convergence distance results. The distance of the shortest route decreases as we increase the convergence rate. The shortest distance plateaus beginning at a 5% rate of convergence.

Rate of convergence: optimization results 23

  Good parallel speedup in both algorithms   Genetic algorithm achieves slightly better

results o  More mutations in each iteration is better

  Higher rate of convergence improves route optimization results but also increases the running time.

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