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Traveling Salesman Problem
The travelling salesman problem (TSP) asks the following question: "Given a list of cities
and the distances between each pair of cities, what is the shortest possible route that
visits each city and returns to the origin city?" It is an NP-hard problem in combinatorial
optimization, important in operations research and theoretical computer science.
Approaches taken to solve the TSP:
• Heuristic approaches;
• Memetic algorithms;
• Ant colony optimizations;
• Simulated annealing;
• Genetic algorithms;
• Neural networks;
• And various other methods for more specific variations
of the TSP.
4
Elastic Neural Network
Ridge (L2) and Lasso (L1) regression are some of the simple techniques to reduce
model complexity and prevent over-fitting which may result from simple linear
regression.
Elastic net regularization method includes both Lasso (L1) and Ridge (L2)
regularization methods. Lasso Regression (Least Absolute Shrinkage and Selection
Operator) adds “absolute value of magnitude” of coefficient as penalty term to
the loss function. Ridge regression adds “squared magnitude” of coefficient as
penalty term to the loss function.
https://en.wikipedia.org/wiki/Regularization_(mathematics)
ENN for TSP
TSP with elastic net:
• An array of cities (y);
• An array of network points (x);
• Network points are located in form of a circle;
• The elastic network evolves iteratively;
• Two forces act on each point of the network at every evolution step:
1. The first force attracts points to cities - stretches out the network;
2. The second force pulls the closest points to each other - tightens the network.
ENN for TSP
TSP with elastic net:
• An array of cities (y);
• An array of network points (x);
• Network points are located in form of a circle;
• The elastic network evolves iteratively;
• Two forces act on each point of the network at every evolution step:
1. The first force attracts points to cities - stretches out the network;
2. The second force pulls the closest points to each other - tightens the network.
ya − Point(xa, ya)
xi − Cit y(xi, yi)
- coordinates of the net point
- coordinates of the city
K - coefficient determining the temperature of the system (decreases as the network evolves)
Δya - change point position in 1 step
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Recommended settings
α = 1. β = 1. initialK = 0.1 Kupdate period = 25 iterations
Knew
= max( 0.01, 0.99 *K );
Field size : X : [0,1]; Y : [0,1] - City coordinates are normalised
Points circle radius : 0.1
NofPoints = NofCities * numPointFactor; numPoimtFactor = 2.5
Stop conditions:
• By worst distance City<->closestPoint: 0.005 - 0.009;
• By maximal number of iterations: 10K - 50K
9
Scalar solution with graphical interface
IDE for your choice
Qt Creator is
recommended but
not required
You can use your existing code
code if you passed PRG-PR 2018
10
Qt Creator
Free download for open source users: https://www.qt.io/download
Wikipedia page: https://de.wikipedia.org/wiki/Qt_(Bibliothek)
Manual (German): https://de.wikibooks.org/wiki/Qt_für_C%2B%2B-Anfänger
11
Data for tests
http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsp/
Lists of cities with optimal solutions for some of them:
City list Optimal way
12
Implementation
1. With graphical interface
2. Console application for calculation speed tests
TSP EN -> implementation as a separate file / header is recommended
g++ consoleENTSP.cpp -O3 -fno-tree-vectorize -fopenmp -o run.out
Compilation flags are needed for console application
13
SIMDization
Optional:
• with given headers;
• with Vc;
• with basic SSE(or AVX) intrinsics;
• with your own headers.
Investigations:
• good/bad vectorizable code;
• data structures: SoA, AoSoA;
• the impact of the number of cities on the quality of vectorization
• …
14
Parallelization
Parallelization of the code with OpenMP
Investigations:
• effect of parallelization of different parts of the code;
• combination of SIMD and OpenMP;
• the impact of the number of cities on the quality of parallelization;
• speed up with different numbers of threads/cores used (if possible);
• …

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