Traveling Salesman

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    Traveling Salesman Problem

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    TSP

    The goal is, to find the most economical way for a

    select number of cities with the following

    restrictions:

    - Must visit each city once and only once

    - Must return to the original starting point

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    BASICS

    Complete Graph vertices joined by a single edge

    Weighted Graph edges carry a value

    Hamiltonian Circuit - connects all points on a graph, passes

    through each point only once, returns to origin

    Hamiltonian Path - A route not returning to the beginning

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    Finding an Approximate Solution

    Cheapest Link Algorithm

    Edge with smallest weight is drawn first

    Edge with second smallest weight is drawn in

    Continue unless it closes a smaller circuit or three

    edges come out of one vertex

    Finished once a complete Hamilton Circuit is drawn

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    Cheapest Link Algorithm

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    Cheapest Link Algorithm

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    Cheapest Link Algorithm

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    Cheapest Link Algorithm

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    Cheapest Link Algorithm

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    Finding an Approximate Solution

    Nearest Neighbor Algorithm

    Start at any given vertex

    Travel to edge that yields smallest weight and has not

    been traveled through yet

    Continue until we have a complete Hamilton circuit

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    Nearest Neighbor Algorithm

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    Nearest Neighbor Algorithm

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    Nearest Neighbor Algorithm

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    Nearest Neighbor Algorithm

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    Nearest Neighbor Algorithm

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    Comparing Approximating Methods

    Cheapest Link Nearest Neighbor

    Total weight for

    Cheapest Link 31

    Nearest Neighbor 33

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    17

    A 42-City Problem (The Nearest Neighbour Method)

    (Starting at City 1)

    1

    21

    20

    19

    29

    7

    30

    28

    3137

    3236

    11

    9

    34

    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    1523

    22

    213

    16

    3

    17 4

    18

    42

    40

    41

    5

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    18

    The Nearest Neighbour Method (Starting at City 1)

    1

    21

    20

    19

    29

    7

    30

    28

    3137

    3236

    11

    9

    34

    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    1523

    22

    213

    16

    3

    17 4

    18

    42

    40

    41

    5

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    19

    The Nearest Neighbour Method (Starting at City 1)

    Length 1498

    1

    21

    20

    19

    29

    7

    30

    28

    3137

    3236

    11

    9

    34

    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    1523

    22

    213

    16

    3

    17 4

    18

    42

    40

    41

    5

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    20

    29

    Remove Crossovers

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    21

    20

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    7

    30

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    31 37

    32 36

    11

    9

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    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    15 23

    22

    213

    16

    317 4

    18

    42

    40

    41

    5

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    21

    Remove Crossovers

    1

    21

    20

    19

    29

    7

    30

    28

    31 37

    32 36

    11

    9

    34

    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    15 23

    22

    213

    16

    317 4

    18

    42

    40

    41

    5

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    22

    Remove Crossovers Length 1453

    1

    21

    20

    19

    29

    7

    30

    28

    31 37

    32 36

    11

    9

    34

    10

    39

    38

    35

    12

    33

    8

    2726

    624

    25

    14

    15 23

    22

    213

    16

    317 4

    18

    42

    40

    41

    5

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    Applications of the TSP

    Computer Wiring - connecting together computer

    components using minimum

    wire length

    Archaeological Seriation - ordering sites in time

    Genome Sequencing - arranging DNA fragments in

    sequence

    Planning, logistics, and the manufacture of microchips

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    THANK YOU