This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
LECTURE 17-18. Course: “Design of Systems: Structural Approach”
Dept. “Communication Networks &Systems”, Faculty of Radioengineering & Cybernetics
Moscow Institute of Physics and Technology (University)
Spanning by two-connected graph:Revelation of two 3-node cliques (centers)
Spanning (illustration): 2-connected graph
Spanning by two-connected graph:Connection of each other node with the two centers
Traveling salesman problem
a0
a1
a2
a3
a4
a6
a5
a7
a8
a92
1
2
4
4
1
3 4
3
2
4 3
2
FORMULATION:Set of cities: A = { a1 , … , ai , … , an }Distance between cities i and j : ( ai , aj ) is set of permutations of elements of A,permutations* = < a(s*[1]), … ,a(s*[i]), … ,a(s*[n]) >
L = < a0,a1,a3,a5,a7,a9,a8,a4,a2,a6>2+1+3+4+2+2+3+4+4+4
Traveling salesman problem
ALGORITMS:1.Greedy algorithm 2.On the basis of minimal spanning tree3.Branch-And-BoundEtc.
VERSIONS (many):1.Cycle or None2.m-salesmen3.asymetric one (i.e., distances ( ai , aj ) and ( aj , ai ) are different ones )4.Various spaces (metrical space, etc.)5.Multicriteria problemsEtc.
Assignment problem
a3
a1
a2
an
b1
FORMULATION:Set of elements: A = { a1 , … , ai , … , an }Set of positions B = { b1 , … , bj , … . bm } (now let n = m)Effectiveness of pair i and j is: z ( ai , bj ) = {s} is set of permutations (assignment) of elements of A into position set B: s* = < (s*[1]), … ,(s*[i]), … ,(s*[n]) > , i.e., element ai into position s[i] in B The goal is:
max ni=1 z ( i, s[i])
b2
b3
bm
. . . . . .
Assignment problem
ALGORITMS:1.Polynomial algorithm ( O(n3) )
VERSIONS:1.Min max problem2.Multicriteria problemsEtc.
VERSIONS:1.Dynamical problem (multiple track assignment)2.Problem with errors 4.Problem with uncertainty (probabilistic estimates, fuzzy sets)Etc.
Multiple matching problem
Recent applied example: usage of assignment problem(s) to define velocity of particles
FRAME 1 FRAME 2 FRAME 3
VELOCITY SPACE
MODELS & ALGORITMS:1.Correlation functions (from radioengineering: signal processing)2.Assignment problem between two neighbor frames(algorithm schemes: genetic algorithms, other algorithms for assignment problems,hybrid schemes)3.Multistage assignment problem (e.g., examination of 3 frames, etc.)(algorithm schemes: genetic algorithms, other algorithms for assignment problems, hybrid schemes)