reducing the time of heuristic algorithms for the symmetric tsp

8/8/2019 Reducing the time of heuristic algorithms for the Symmetric TSP

http://slidepdf.com/reader/full/reducing-the-time-of-heuristic-algorithms-for-the-symmetric-tsp 1/79

Available at http://goo.gl/rjSw

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http://goo.gl/rjSw



Available at http://goo.gl/rjSw

http://goo.gl/rjSw

http://goo.gl/rjSw



!N

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2 N



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G r e e

d y

A l g o r i t h m

1naive-greedy-nn (start , n, C )

be T [1 . . n ] a new array

visited = ∅; curr = start ; cost = 0for i = 1 to n − 1T [i] = curr ; visited = visited ∪ {curr }min = ∞ ; next = nil

forj = 1

tonif j ∈visited and C (curr , j ) < min

min = C (curr , j )next = j

cost = cost + min

curr = next

T [n ] = curr

cost = cost + C (curr , start )return (T, cost )



80000 * 80000 * 8 = 51200000000 Bytes51200000000 / 1024 / 1024 / 1024 ≈ 47.6837158 GB



Average of 5 executions, user time + system time



* Along with operations that make good use of the structure



vm1084d2103

d15112

d18512

pla33810

pla85900

0,01 0,1 1 10 100

0,17

0,07

0,04

0,04

0,01

0,01

56,04

8,58

2,94

2,15

0,04

0,02

Excution time (seconds)

1

127

254

380

vm1084 pla85900

329,64

122,57

73,5053,75

4,002,00

Improvement (times)



0%

25%

50%

75%

100%

d15112 d18512 pla33810 pla85900

12%11%11%9%

22%20%18%15%

15%20%25%37%

17%16%16%14%

34%32%30%25%

Activity/Time in the improved Greedy



0

2,5

5,0

7,5

10,0

pla85900 rl2-1116700 rl2-2147500 rl2-3178300 rl2-4209100

y = 0,0966x - 0,1196

R ² = 0,9988

Time for running Greedy NN

Time (s) Tendency



0

2,5

5,0

7,5

10,0

pla85900 rl2-1116700 rl2-2147500 rl2-3178300 rl2-4209100

y = 0,0408x 1,4135

R ² = 0,9941

Time for structure initialization

Time (s) Tendency



0

2,5

5,0

7,5

10,0

pla85900 rl2-1116700 rl2-2147500 rl2-3178300 rl2-4209100

Initialization and Greedy tour

Init time (s) Greedy time (s)



Example construction

algorithmbuild (P,l ,u,d )node = kdnode-new ()if u − l = = 0

node . val = P [l]else

m = ( l + u)/ 2if (d mod 2) = = 0

sort (P,l,u, X )

node . cut = P [m]. x else

sort (P,l,u, Y )node . cut = P [m]. y

node . left = build (P,l ,m,d + 1)

node . right =build

(P, m + 1 ,u ,d + 1)return node

Maximum amount of pointsin a bucket; 1 here

Beware of the cost



Construction example

XY

350 5 10 15 20 25 30

35

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5

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25

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5,5

25,10

20,15

10,20

30,25

10,30

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)



XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25) >Sort by x

<(5;5), (25;10), (20, 15), (10;20), (30;25), (10;30) >

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30

Sorted by x




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25) >

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30

Sorted by x




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25)>Sorted by y

350 5 10 15 20 25 30

35

0

5

10

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20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25)>Sorted by x

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25)>

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (25;10), (20;15), (30;25) >Sorted by y

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25) >Sorted by x

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

<(5;5), (10;20), (10;30), (20;15), (25;10), (30;25) >

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30




XY

tree = BUILD(<(5;5), (25;10), (20;15), (10;20), (30;25), (10;30)>, 1, 6, 0)

350 5 10 15 20 25 30

35

0

5

10

15

20

25

30

5,5

25,10

20,15

10,20

30,25

10,30 10

20

X

Y

5X

5;

5

10;

20

10;

30

15

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20;

15

25;

10

30;

25

C u t d i m e n s i o n

Another representation


rnn (root , d )



Example algoritm

for Top-Down NNsearch

kdtree-nn (root , nntarget )target = nntarget dist = ∞

nn = NIL

rnn (root , 0)return nn

( , )if root . val = NIL

if root . val = = target return

thisdist =

euc-2d

(root . val , target

)if thisdist < dist dist = thisdist nn = root . val

else

cutval = root . cut if (d mod 2) = = 0

thisval = target . x else thisval = target . y if thisval ≤ cutval

rnn (root . left , d + 1)if (thisval + dist ) > cutval

rnn (root . right , d + 1)else

rnn (root . right , d + 1)if (thisval − dist ) < cutval

rnn (root . left , d + 1)



Example NN search

10

20

X

Y

5X

5;

5

10;

20

10;

30

15

20

20;

15

25;

10

30;

25

Find the closest point to 10;30



10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10 <= 10

10;

30

Example NN search




10 <= 1030 > 20

10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10;

30

Example NN search




10 <= 1030 > 20

return30 - ∞ < 20

10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10;

30

Example NN search




10 <= 1030 > 20

return30 - ∞ < 20

10 > 5 dist = 10nn = 10;20

10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

Euclidean distance

dist = 10nn = 10;20

10;

30

Example NN search




10 <= 1030 > 20

return30 - ∞ < 20

10 > 5dist = 10nn = 10;2010 - 10 < 5

não altera dist

10

20

X

Y

5X

5;

5

10;

20

10;

30

15

20

20;

15

25;

10

30;

25

Euclidean distancedist unaffected

Example NN search




10 <= 1030 > 20

return30 - ∞ < 20

10 > 5dist = 10nn = 10;2010 - 10 < 5

dist unaffected10 + 10 > 10

10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10;

30

Example NN search




10 <= 1030 > 20

return30 - ∞ < 20

10 > 5dist = 10nn = 10;2010 - 10 < 5

dist unaffected10 + 10 > 10

30 > 15não altera dist30 - 10 ≮ 15

10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10;

30

Euclidean distance

dist unaffected30 - 10 ≮ 15

Example NN search




10

20

X

Y

5X

5;

5

10;

20

15

20

20;

15

25;

10

30;

25

10;

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Example NN search




* Implementation found in Robert Sedgewick’s book

for |sub-array| ≤ 16

BNDS_LEVEL: 3 XXX não uso no greedy



G r e e

d y

A l g o r i t h m

2 kdtree-greedy-nn (tree , start , n, C )be T [1 . . n ] a new arrayT [1] = start curr = start

cost = 0for i = 2 to n

kdtree-delete ( tree , curr )next = kdtree-nearest ( tree , curr , C )T [i] = next

cost = cost + C (curr, next )curr

=next

cost = cost + C (curr , start )kdtree-undelete-all (tree , n )return (T, cost )



Naïve Improved* Diff ~

vm1084 290.806 286.437 1,52%

d2103 86.504 86.765 -0,29%

d15112 1.960.503 1.921.015 2,05%

d18512 799.220 779.783 2,49%

pla33810 77.332.499 81.131.055 -4,68%

pla85900 163.516.994 174.486.522 -6,29%

* Best result found starting from every point

3 304

806

0

0

0

0

6523

761

10818

11394

38007

(Naïve / Improved - 1) * 100



Achieved Optimal Prox ~

vm1084 286.437 239.297 19,70%

d2103 86.504 80.450 7,52%

d15112 1.921.015 1.573.084 22,11%

d18512 779.783 645.238 20,85%

pla33810 77.332.499 66.048.945 17,08%

pla85900 163.516.994 142.382.641 14,84%



1 5 4 3 2 6 7 8 9a b c d

1 2 3 4 5 6 7 8 9a c b d

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56

7

8

9

2-exchange

naive-2opt (T,n, cost , C )



2 - O

p t A l g o r i t h m

( , , , )repeat

gain = 0for i = 1 to n

for j = i + 2 to nif (( j + 1) mod n ) = = i

continuea = T [i]; b = T [(i + 1) mod n ]

c = T [ j ]; d = T [( j + 1) mod n ]new -gain = C (a, c ) + C (b, d )−

(C (a, b ) + C (c, d ))if new -gain < gain

gain = new -gain best = [b, c]

if gain < 0cost = cost + gain invert (T, best [1], best [2])

until gain = =

0



dist(a,b)



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9a

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c

d

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2 -

O p t



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1 2 6 7 8 9 5 4 3

1 2 9 6 7 8 5 4 3a d



2-Opt 2. 1/4-Opt Prox.

T (s) Exchgs. Cost T (s) Exchgs. Cost ~*

vm1084 0,00 153 260.184 0,00 190 254.355 6,29%

d2103 0,00 147 82.751 0,00 174 82.402 2,42%

d15112 0,07 3.564 1.686.957 0,07 4.228 1.666.102 5,91%

d18512 0,08 4.023 692.396 0,09 4.949 680.798 5,51%

pla33810 0,17 6.002 72.031.744 0,24 7.830 70.744.397 7,11%pla85900 0,64 16.845 153.558.764 0,70 20.844 150.777.638 5,90%

* Considering best obtained



0

38

75

113

150

pla85900 rl2-515400 rl2-944900

y = 1,3369x 2 - 7,1294x + 10,326R ² = 0,9936

Approximated 2-Opt time

2-Opt (s) Tendency

0

75000

150000

225000

300000

pla85900 rl2-515400 rl2-944900

y = 20546x + 252,05R ² = 0,999

Exchanges in the approximated 2-Opt

2-Opt swaps Tendency

0E+00

1,3E-04

2,5E-04

3,8E-04

5,0E-04

pla85900 rl2-515400 rl2-944900

y = 4,722E-6x 2 - 1,905E-5x + 7,019E-5R ² = 0,9957

“2-exchange” time

Time/Exchange (s) Tendency



0

38

75

113

150

pla85900 rl2-515400 rl2-944900

y = 1,7278x 2 - 9,8544x + 14,61R ² = 0,9848

Approximated 2.1/4-Opt time

2.1/4-Opt (s) Tendency

0

75000

150000

225000

300000

pla85900 rl2-515400 rl2-944900

y = 25368x - 2354,1R ² = 0,9976

Exchanges in the approximated 2.1/4-Opt

2.1/4-Opt swaps Tendency

0E+00

1,3E-04

2,5E-04

3,8E-04

5,0E-04

pla85900 rl2-515400 rl2-944900

y = 4,631E-6x 2 - 1,768E-5x + 6,255E-5R ² = 0,9852

“2.1/4-exchange” time

Time/Exchange (s) Tendency



0

38

75

113

150

pla85900 rl2-515400 rl2-944900

Approximated local search time

2-Opt (s) 2.1/4-Opt (s)

0

75000

150000

225000

300000

pla85900 rl2-515400 rl2-944900

Exchanges in the local search

2-Opt trocas 2.1/4-Opt trocas

0E+00

1,3E-04

2,5E-04

3,8E-04

5,0E-04

pla85900 rl2-515400 rl2-944900

Time/Exchange

2-exchange (s) 2.1/4-exchange (s)



kdtree-grasp (n )be B [1 . . n ] a new arraybest

-cost

=∞

while stopping conditions not metGT = kdtree-semigreedy-tour (random (n ))T, cost = kdtree-2opt (GT )if cost < best -cost

best -cost = cost

B = T return (B, best -cost )



“ KNN”



KG+2. 1/4 GRASP* Prox.T (s) Cost T (s) Execs. Cost ~

vm1084 0,00 254.355 >> 16.207 249.164 4,12%

d2103 0,00 82.402 >> 7.260 87.860 =

d15112 0,07 1.666.102 >> 6.451 1.668.978 =

d18512 0,09 680.798 >> 10.014 685.330 =

pla33810 0,24 70.744.397 >> 11.903 71.776.398 =pla85900 0,70 150.777.638 >>> 7.445 152.935.500 =

* Results found by using the rand48 family of functions; seed fixed at 0

Worse than Greedy-NN



* Considering all the choices implemented in this K-d treeand in this GRASP



Instance Cost Distance

a280 2.599 0,77%

berlin52 7.542 0,00%

ch130 6.117 0,11%

d15112 1.666.102 5,91%

d18512 680.798 5,51%

d2103 82.402 2,42%

dantzig42 699 0,00%

eil101 629 0,00%

eil51 426 0,00%

fri26 937 0,00%

GRASP

KG+2. 1/4

Instance Cost Distance

kroA100 21.282 0,00%

kroC100 20.749 0,00%

pla33810 70.774.397 7,11%

pla85900 150.777.638 5,90%

rat575 7.035 3,86%

rat99 1.211 0,00%

st70 675 0,00%



K-d tree

reducing the time of heuristic algorithms for the symmetric tsp

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