Travelling Salesman Explained

8/12/2019 Travelling Salesman Explained

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The Traveling Salesman Problem

Definition: A complete graph KNis a graph with Nvertices and an edge between every two vertices.

Definition: A weighted graph is a graph in which each

edge is assigned a weight(representing the time, distance, orcost of traversing that edge).

Definition: A Hamilton circuit is a circuit that uses everyvertex of a graph once.

Definition: The Traveling Salesman Problem (TSP) isthe problem of finding a minimum-weight Hamilton circuitin KN.


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Example: Willy decides to visit every Australian cityimportant enough to be listed onthis Wikipedia page.
http://en.wikipedia.org/wiki/States_and_territories_of_Australia#Distance_tablehttp://en.wikipedia.org/wiki/States_and_territories_of_Australia#Distance_table


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To avoid rental-car fees, he must finish the tour in the samecity he starts it in.


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What route minimizes the total distance he has to travel?


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What route minimizes the total distance he has to travel?

I.e., in this weighted K16,which Hamilton circuit has thesmallest total weight?


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The Brute-Force Algorithm

Willy could solve the problem by brute force:

Make a list of all possible Hamilton circuits.


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Calculate the weight of each Hamilton circuit by addingup the weights of its edges.


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Calculate the weight of each Hamilton circuit by adding

up the weights of its edges.

Pick the Hamilton circuit with the smallest total weight.

BIG PROBLEM:


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Calculate the weight of each Hamilton circuit by adding

up the weights of its edges.

Pick the Hamilton circuit with the smallest total weight.

BIG PROBLEM: There are 16 vertices, so there are

(16 1 )! = 1 5 ! = 1, 307, 674, 368, 000

Hamilton circuits that each need to be checked.


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The Nearest-Neighbor Algorithm

The result: The Nearest-Neighbor algorithm, using Sydney asthe reference vertex, yields the Hamilton circuit

SY CN ML HO AD AS UL BM KU

DA MI CS MK BR AL PE SY

whose total weight is 21,049 km.

A randomly chosen Hamilton circuit would have averaged40,680 km, so this is pretty good.

But can Willy do better?


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The Repetitive Nearest-Neighbor Algorithm

Observation: Willy can use any city as the reference vertex!

That is, Willy can execute the Nearest-Neighbor Algorithm

sixteen times, using each city once as the reference vertex.

Then, he can pick the Hamilton circuit with the lowest totalweight of these sixteen.

This is called the Repetitive Nearest-Neighbor Algorithm(RNNA).


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Ref. vertex Hamilton circuit Weight

AD AD,ML,HO,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,PE,AL,AD 18543AL AL,PE,BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,AL 19795AS AS,UL,BM,KU,DA,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,AS 18459BR BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,ML,HO,CN,SY,AL,PE,BR 22113BM BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,PE,AL,BM 19148

CS CS,MK,BR,SY,CN,ML,HO,AD,AS,UL,BM,KU,DA,MI,PE,AL,CS 22936CN CN,SY,ML,HO,AD,AS,UL,BM,KU,DA,MI,CS,MK,BR,AL,PE,CN 21149DA DA,KU,BM,UL,AS,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,DA 18543HO HO,ML,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,AL,PE,HO 20141KU KU,DA,AS,UL,BM,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,KU 18785MK MK,CS,MI,AS,UL,BM,KU,DA,AD,ML,HO,CN,SY,BR,AL,PE,MK 23255ML ML,HO,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,AL,PE,ML 20141MI MI,AS,UL,BM,KU,DA,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,MI 20877PE PE,AL,BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,PE 19148SY SY,CN,ML,HO,AD,AS,UL,BM,KU,DA,MI,CS,MK,BR,AL,PE,SY 21049 (NNA)UL UL,AS,MI,CS,MK,BR,SY,CN,ML,HO,AD,BM,KU,DA,PE,AL,UL 20763


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Ref. vertex Hamilton circuit Weight

AD AD,ML,HO,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,PE,AL,AD 18543AL AL,PE,BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,AL 19795AS AS,UL,BM,KU,DA,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,AS 18459 (best)BR BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,ML,HO,CN,SY,AL,PE,BR 22113BM BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,PE,AL,BM 19148

CS CS,MK,BR,SY,CN,ML,HO,AD,AS,UL,BM,KU,DA,MI,PE,AL,CS 22936CN CN,SY,ML,HO,AD,AS,UL,BM,KU,DA,MI,CS,MK,BR,AL,PE,CN 21149DA DA,KU,BM,UL,AS,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,DA 18543HO HO,ML,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,AL,PE,HO 20141KU KU,DA,AS,UL,BM,MI,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,KU 18785MK MK,CS,MI,AS,UL,BM,KU,DA,AD,ML,HO,CN,SY,BR,AL,PE,MK 23255ML ML,HO,CN,SY,BR,MK,CS,MI,AS,UL,BM,KU,DA,AD,AL,PE,ML 20141MI MI,AS,UL,BM,KU,DA,CS,MK,BR,SY,CN,ML,HO,AD,AL,PE,MI 20877PE PE,AL,BM,KU,DA,AS,UL,AD,ML,HO,CN,SY,BR,MK,CS,MI,PE 19148SY SY,CN,ML,HO,AD,AS,UL,BM,KU,DA,MI,CS,MK,BR,AL,PE,SY 21049 (NNA)UL UL,AS,MI,CS,MK,BR,SY,CN,ML,HO,AD,BM,KU,DA,PE,AL,UL 20763


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Apparently, using Alice Springs (AS) as the reference vertexyields the best Hamilton circuit so far, namely

AS UL BM KU DA MI CS MK BR SY CN ML HO AD AL PE AS


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Apparently, using Alice Springs (AS) as the reference vertexyields the best Hamilton circuit so far, namely

AS UL BM KU DA MI CS MK BR SY CN ML HO AD AL PE AS

Remember: Willy can still start anywhere he wants!For instance,

SY CN ML HO AD AL PE AS UL BM KU DA MI CS MK BR SY

represents the same Hamilton circuit.


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Randomly chosen Hamilton circuit: 40,680 kmHamilton circuit using NNA/Sydney: 21,049 kmHamilton circuit using RNNA: 18,459 km


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In general, theres no way of knowing in advance whichreference vertex will yield the best result.


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This algorithm is still efficient, but . . .


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This algorithm is still efficient, but . . .

Is it optimal? That is,Can Willy do even better?


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If we look at the map (warning: not quite to scale!) itbecomes clear that the RNNA has not produced an optimalHamilton circuit.


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Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


25/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


26/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


27/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


28/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


29/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


30/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


31/80

Oops.


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Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


33/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


34/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


35/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


36/80

Now the algorithm is stuck some very expensive edges arerequired to complete the Hamilton circuit.


37/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


38/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


39/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

The Repetitive Nearest Neighbor Algorithm


40/80


Starting from Uluru would have created a different problem.


41/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


42/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


43/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


44/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)



45/80


Starting from Alice Springs would have created a differentproblem (but a less harmful one).


46/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


47/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


48/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


49/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


50/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


51/80

It is easy for a human to look at this Hamilton circuit and finda way to improve it.


52/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


53/80

It is easy for a human to look at this Hamilton circuit and finda way to improve it.

But how do you make such improvements part of thealgorithm?


54/80

Does starting How about

The Cheapest-Link Algorithm


55/80

Idea: Start in the middle.

http://www.math.ku.edu/~math105-f11/Lectures/cheapest-link.xls


56/80


Find the single edge that would be cheapest to add.



57/80



Keep doing this until you have a Hamilton circuit.



58/80



Keep doing this until you have a Hamilton circuit. Make sure you add exactly two edges at each vertex.



59/80



Keep doing this until you have a Hamilton circuit. Make sure you add exactly two edges at each vertex.

This is called the Cheapest-Link Algorithm, or CLA.Here is an example.
http://www.math.ku.edu/~math105-f11/Lectures/cheapest-link.xlshttp://www.math.ku.edu/~math105-f11/Lectures/cheapest-link.xls


60/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


61/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


62/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


63/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


64/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

D D


65/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


66/80

Darwin(DA)


67/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


68/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


69/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Darwin

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(DA)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


70/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

( )

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


71/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

( )

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


72/80

Darwin(DA)


73/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


74/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)

Darwin(DA)


75/80

Hobart

Canberra

Sydney

Brisbane

Mackay

Cairns

Mount Isa

Alice Springs

Kununurra

Perth

Albany

(SY)(PE)

(MI)

(CN)

(HO)

(KU)

(MK)

(AL)

(AS)

(CS)

(BR)

(BM)Broome

Melbourne (ML)

Adelaide (AD)

Uluru(UL)


76/80

Comparing Algorithms


77/80

Randomly chosen Hamilton circuit: 40,680 kmHamilton circuit using NNA/Sydney: 21,049 kmHamilton circuit using RNNA: 18,459 kmHamilton circuit using CLA: 18,543 km

This didnt help in this case. But it might help in a differentexample (next time).


78/80

The Bad News


79/80

There is no known algorithm to solve theTSP that is bothoptimalandefficient.

The Bad News


80/80

There is no known algorithm to solve theTSP that is bothoptimalandefficient.

Brute-force is optimal but not efficient.

NNA, RNNA, and CLA are efficient but not optimal.

Travelling Salesman Explained

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