Adaptive pyramid model for the Traveling Salesman Problem

Transcription

1 Adaptive pyramid model for the Traveling Salesman Problem Zygmunt Pizlo, Emil Stefanov & John Saalweachter Purdue University Yll Haxhimusa & Walter G. Kropatsch Vienna University of Technology Acknowledgment: Support: Zheng Li AFOSR

2 Find the shortest tour of N cities. Traveling Salesman Problem:

3 Traveling Salesman Problem: TSP is a difficult optimization problem.

4 Experiment 5 subjects Problem size: 6, 10, 20, random problems per size Problems were shown on a computer screen

5 6 city BSL, ZP optimal OSK, ZL 1.1%

6 10 city optimal BSL 0.7% OSK 2.8% ZL 1% ZP 2.2%

7 20 city optimal BSL 2.7% 10.7% OSK 12.9% ZL ZP 0.4%

8 optimal 4.8% 50 city OSK BSL 10.6% 7% ZL ZP

9

10

11

12

13 Model Multiresolution pyramid representation Top-down process of tour approximations

14 1D Pyramid architecture The number of nodes on layer i+1 is b times smaller than that on layer i. Receptive field on layer i+1 is b times larger than that on layer i. What is local close to the top, is global close to the bottom.

15 2D Pyramid Representation

16 Model Multiresolution pyramid representation Top-down process of tour approximations Pyramid with the fovea and with eye movements

17 Neuroanatomy of the visual system (Hubel & Wiesel, 1974) At each retinal location, there is a family of receptive fields with different sizes and resolutions. The size of the smallest field is a function of eccentricity.

18 Pyramid with Fovea Resolution of the finest representation decreases with the distance from the fixation point this corresponds to the non-uniform density of the receptors on the retina. Prevents from handling too much information at a time.

19 Model Multiresolution pyramid representation Top-down process of tour approximations Pyramid with the fovea and with eye movements Local search by means of cheapest insertion

20 Cheapest Insertion

21 Cheapest Insertion

22 Cheapest Insertion

23 Cheapest Insertion

24 Model Multiresolution pyramid representation Top-down process of tour approximations Pyramid with the fovea and with eye movements Local search by means of cheapest insertion Adaptive receptive fields

25 Blurring with Gaussian Filter

26

27 Min-Max Method for Determining Cluster Boundaries

28 Bisection Pyramid Top Layer (8)

29 Layer 7

30 Layer 6

31 Layer 5

32 Layer 4

33 Layer 3

34 Layer 2

35 Layer 1

36 Testing the Pyramid Model The model was run on the same problems that were used with the subjects The size k of the neighborhood for cheapest insertion was a free parameter Computational complexity of the model: between O(N) and O(N 2 ). Demo

37 Local Search in Cheapest Insertion 6 Ammount of Search (K) BSL OSK YH ZL ZP Problem Size

38

39

40 Large Problems

41 Large Problems

42 Large Problems

43 ZP solving large problems

44 Minimum Spanning Tree vs. TSP MST TSP

45 Psychophysics: MST vs. TSP

46 Psychophysics: MST vs. TSP

47 Psychophysics: MST vs. TSP

48 What is MST actually good for? Clustering? What type of clustering?

49 MST as line detector Perfect circle Less-than perfect circle

50 MST for a realistic example

51 TSP solutions Optimal Line Pyramid

52 Summary Computational complexity of the mental mechanisms is very low but TSP tours found by the subjects are close to optimal. Coarse-to-fine sequence of approximations produced by a pyramid algorithm provides a plausible model of the mental mechanisms involved in solving TSP. The TSP model simulates attention (visual acuity), as well as eye movements this minimizes the use of memory without slowing down the solution process. Simulated receptive fields are adaptive. The line detection mechanism is likely to be based on MST.

53 Next Step Test the model using TSP with obstacles.

54 Euclidean TSP with Obstacles (NE-TSP)

55 Maze Like Obstacles Visual spatial relations in the problem representation (proximities, directions) have to be modified by bottomup verification of availability of moves.

56 Metric Always Exists, but May be Difficult to Reconstruct