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Lecture 2_Graph_2

# Lecture 2_Graph_2 - Graph_2 Minimum Spanning Tree Minimum...

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1 Graph_2: Minimum Spanning Tree & Minimum path finding Graph_2 Tommy W S Chow Jan 2011

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The Traveling Salesman Problem (TSP) - definition Given : A finite set of points(cities) V and the costs(distances) C ij between each pair of points i,j \in V A tour : A circuit which passes exactly once through each point in V TSP : Find a tour of minimum cost (distance) A B C F D E G H I 1 6 2 4 3 3 1 1 1 1 2 2 2 4 4 2 3 4 3 A B C F D E G H I 1 3 3 1 1 2 2 2 3
TSP- Industrial Logistic problem Minimum Spanning Tree & Traveling Salesman Problem (TSP) This is a very classical problem, But this has also become a popular industrial logistic problem, and many other IT, business problems Here we rely to use MST for finding an close optimal solution of the TSP Finding the best (optimal) solution requires complicated computational method, NP (Non polynomial) hard problem In the past year, some FYP used GA to solve it. Does GA gives the best possible and efficient solution?

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4 First: understanding the TSP problem Ways of solving it Straightforward: Brute force it: Work out all possible cases, find the shortest path. Computer now is so powerful , it can sort out for us, no problem, maybe give computer 1 hr, or 5 hours at most 1 day, okey! No big deal! No, Brute force won’t work!! Why?
5 Why brute force won’t work If there are n cities, it has to go to other (n-1) cities. Thus, there are (n-1)! Paths. Let n = 26, lot a huge number. 25! Is huge!! 25! Approximate = 1.55x10 paths It is approximated that a current PC can work out about 1x10 possible paths per second, There are 3.15x10 sec in 1 year, Thus, 1 year can work out 3.15x 10 paths, Thus, it takes (1.55x10 )/(3.15x10 )year to work out all possible paths That is about 5x10 year , There is still the sorting process. So Brute force won’t work for n =26, let alone n=100 25 7 13 6 25 13 11

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6 NP-completeness Do your best then.
7 Coping With NP-Hardness Brute-force algorithms. Develop clever enumeration strategies. Guaranteed to find optimal solution. No guarantees on running time. Heuristics. Develop intuitive algorithms. Guaranteed to run in polynomial time. No guarantees on quality of solution. Approximation algorithms. Guaranteed to run in polynomial time. Guaranteed to find "high quality" solution, say within 1% of optimum. Obstacle: need to prove a solution’s value is close to optimum, without even knowing what optimum value is!

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Another approach Bounce-and-bound You read it if you are interested FYI only 8
9 Bounce-and-bound (tree approach) The path is represented by the following tree Example: n=4, find the lowest cost of different paths Eliminate different branches (tree) according to the lowest cost, this will cut down the number of search hugely We illustrate this by the following example, and makes things more realistic, P1-P4 may not = P4 –P1, (single way) Start 1 2 3 4 use ∞ to represent we don’t need those 1 3 9 7 i.e., 1 – 2 – 3 – 4 – 1 = 3+6+6 +9= 24 2 3 6 5 we mininize the cost by – the smallest values 3 5 6 6 of each row, & col 4 9 7 4

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