Chapter 3 Page 1 of 28
Chapter 3
Hamilton Circuits
Spring 2016 1/8/16
Learning Objectives: Students will determine the number of Hamilton circuits in a complete
graph and demonstrate the ability to use various methods to find
Hamilton circuits in graphs. Students will know when a Hamilton
circuit is appropriate to use in solving a real life problem. Students
will understand the notion of an “optimal” Hamilton circuit and
recognize the limitations of certain methods for finding optimal
Hamilton circuits.
Chapter 3 Page 2 of 28
Chapter 3 - Hamilton Circuits
Hamilton Circuit - a route which must start and end at same vertex, visit each vertex only once. Think of the
vertices as cities. (different than Euler circuit; Euler visited each edge only once.
Numbers are weights (cost, time, etc.).
A
B
126
116
88
107
113
There are many Hamilton circuits in this graph
96
C
D
A
B
126
ABCDA is one Hamilton circuit
It weighs 126+116+96+88 = 426
116
88
ADCBA (reverse) also weighs 426
96
C
A
D
B
ADBCA is another Hamilton circuit
It weighs 88+107+116+113 = 424
116
88
107
113
ACBDA is its reverse, also weighs 424
C
D
Hamilton Path - visit each vertex (city) only once, but don’t start and end at start vertex.
(different than Euler Path; Euler visited each edge (street) only once.)
A
B
126
116
88
107
113
A
96
C
There are many Hamilton paths in this graph
126
B
D
ABCD is one Hamilton path
116
C
D
96
Chapter 3 Page 3 of 28
Complete graphs - every vertex is adjacent to (touches) every other vertex.
A complete graph has (N-1)! Hamilton circuits.
N = number of vertices
! means factorial = N(N-1)(N-2)….etc.
If N = 5, N! = (5)(4)(3)(2)(1) = 120
(N-1)! = (4)(3)(2)(1) = 24
X
A
B
K
L
C
H
V
XX
C
D
M
complete graph with
3 vertices N = 3
(N-1)! = (3-1)! = 2
Hamilton Circuits
complete graph with
4 vertices N = 4
(4-1)! = (3)! = 6
Hamilton Circuits
S
complete graph with
6 vertices
6-1)! = (5)! = 5X4X3X2X1= 120
Hamilton Circuits
Find cheapest Hamilton Circuit?
4 Algorithms: 1.) Brute Force
2.) Nearest Neighbor
3.) Repetitive Nearest Neighbor
4.) Cheapest Link
(optimal, inefficient b/c it’s too long)
(approximate, efficient b/c it’s short)
(approximate, efficient b/c it’s short)
(approximate, efficient b/c it’s short)
Chapter 3 Page 4 of 28
BRUTE FORCE ALGORITHM
Only algorithm that always yields optimal answer, but is time
consuming.
Procedure:
1.) Pick any start vertex.
2.) List all (N-1)! Hamilton Circuits from start vertex. No duplicates allowed!
3.) Select the cheapest of all.
Simple example:
A
4
5
This graph has 6 Hamilton circuits
3
B
6
7
D
9
C
Pick any start vertex…I’ll pick A.
List all (N-1! ) Hamilton circuits that start (and end) at A. (N-1)! = (4-1)! = 3 X 2 X 1 = 6
List all 6 Hamilton circuits starting and ending at A.
Write all 6 that start at A no duplicates
A
A
A
A
A
A
D C B A = 5 + 9 + 7 + 4 = 25
B C D A = 4 + 7 + 9 + 5 = 25 (reverse)
B D C A = 4 + 3 + 9 + 6 = 22
C D B A = 6 + 9 + 3 + 4 = 22 (reverse)
D B C A = 5 + 3 + 7 + 6 = 21
C B D A = 6 + 7 + 3 + 5 = 21 (reverse)
cheapest Hamilton circuit
is ADBCA or ACBDA = 21
A
5
3
B
6
7
C
D
Chapter 3 Page 5 of 28
Rewriting a circuit - any Hamilton circuit can be rewritten without resolving again..
A
ADBCA or ACBDA or
BCADB or BDACB or
CADBC or CBDAC or
DBCAD or DACBD
5
3
B
6
7
D
C
these are
all the same Hamilton
circuit costing 21
Chapter 3 Page 6 of 28
Another example with 5 vertices; N=5
A
A
133
A
B
C
D
E
185
119
200
E
B
152
121
199
185
119
152
133
B
185
121
150
200
C
119
121
174
120
120
150
174
D
C
Pick any start vertex…I’ll pick A.
List all (N-1!) Hamilton circuits that start (and end) at A. (N-1)! = (5-1)! = 4(3)(2)(1) = 24
Write all 24 Hamilton circuits starting at A. No duplicates.
1.)
3.)
5.)
7.)
9.)
11.)
13.)
15)
17.)
19.)
21.)
23.)
A
A
A
A
A
A
A
A
A
A
A
A
BCDEA and 2.) AEDCBA (reverse);
BCEDA “
4.) ADECBA
“
BDCEA “
6.) AECDBA
“
BDECA “
8.) ACEDBA
“
BECDA “ 10.) ADCEBA
“
BEDCA “ 12.) ACDEB
“
CBDEA “ 14.) AEDBCA
“
CBEDA “ 16.) ADEBCA
“
CDBEA “ 18.) AEBDCA
“
CEBDA “ 20.) ADBECA
“
DBCEA “ 22.) AECBDA
“
DCBEA “ 24.) AEBCDA
“
weight =
“
“
“
“
“
“
“
“
“
“
“
812 each
777 each
762 each
773 each
831 each
877 each
722 each cheapest
791 each
776 each
741 each
676 each
780 each
Cheapest Hamilton Circuits: ADBCEA and AECBDA cost $676 each.
Atlanta
AD
DB
BC
CE
EA
133
Erie
152
Buffalo
150
Denver
120
121
$152
$150
$121
ADBCEA
$120
$676
$133
$676
Note: ADBCEA ......Can re-write it as
DBCEAD or
BCEADB or
CEADBC or
EADBCE or others using different
start vertices...
each costs $676
Chicago
Video of Brute Force and Nearest Neighbor Algorithms
http://mediaplayer.pearsoncmg.com/assets/cV_e4_OrKFiYHQs1QelD1N_KI2SVybPW
D
152
150
174
199
E
133
200
120
199
Chapter 3 Page 7 of 28
Solutions
Practice problems page 98:
27.) Brute Force algorithm from A:
A
22
28
D
48
20
B
32
18
C
Brute Force Algorithm
Pick any start vertex…I’ll pick A.
List all (N-1! ) Hamilton circuits that start (and end) at A. (N-1)! = (4-1)! = 3 X 2 X 1 = 6
List all 6 Hamilton circuits starting and ending at A. No duplicates
A
A
A
A
A
A
D C B A = 22 + 18 + 32 + 48 = 120
B C D A = 48 + 32 + 18 + 22 = 120 (reverse)
B D C A = 48 + 20 + 18 + 28 = 124
C D B A = 28 + 18 + 20 + 48 = 124 (reverse)
D B C A = 22 + 20 + 32 + 28 = 102
C B D A = 28 + 32 + 20 + 22 = 102 (reverse)
cheapest Hamilton circuit
A
22
28
D
20
B
32
C
ACBDA or ADBCA = 102
Chapter 3 Page 8 of 28
29.) Brute Force Algorithm from A
A
35
18
40
18
E
B
21
Procedure: Pick a start vertex …I’ll pick A
List all (N-1)! from A = 24 circuits
No duplicates
Select cheapest
32
9
22
23
25
D
1.)
3.)
5.)
7.)
9.)
11.)
13.)
15.)
17.)
19.)
21.)
23.)
C
A
A
A
A
A
A
A
A
A
A
A
A
DBCEA
CEBDA
CDBEA
CBEDA
BCDEA
BCEDA
BDCEA
BDECA
BEDCA
BECDA
EBCDA
CBDEA
and
“
“
“
“
“
“
“
“
“
“
“
2.) AECBDA (reverse);
4.) ADBECA
“
6.) AEBDCA
“
8.) ADEBCA
“
10.) AEDCBA
“
12.) ADECBA
“
14.) AECDBA
“
16.) ACEDBA
“
18.) ACDEBA
“
20.) ADCEBA
“
22.) ADCBEA
“
24.) AEDBCA
“
weight = 93 each
“
120 each
“
122 each
“
121 each
“
119 each
“
121 each
“
122 each
“
151 each
“
150 each
“
123 each
“
92 each
“
120 each
Cheapest = AEBCDA or mirror image ADCBEA = 92 weight
A
18
18
E
B
….could be rewritten as BCDAEB = 92 = BEADCB
CDAEBC = 92 = CBEADC
DAEBCD = 92 = DCBEAD
EBCDAE = 92 = EADCBE
9
22
25
D
C
Supercomputer - The world's fastest supercomputer (Tianhe-2 in China) has achieved a performance
33,900 trillion floating point operations per second.
30 city sales territory = 280 million years using Brute Force Algorithm
Chapter 3 Page 9 of 28
Nearest Neighbor Algorithm - 2nd method (approximate)
- not as much work as Brute Force
- but, usually won’t give optimal answer
A
133
185
E
120
A
B
200
152
A
B
C
D
E
119
150
199
121
185
119
152
133
B
185
121
150
200
C
119
121
D
152
150
174
174
120
E
133
200
120
199
199
174
D
C
Procedure
1.) Pick a start vertex
2.) Travel on cheapest edge from start vertex to any other vertex. If there is a tie, pick one
at random
3.) Travel on cheapest edge from current vertex to another vertex except one you’ve visited
4.) same
5.) same
…….continue procedure until you return to start vertex
Step1.) Pick a start vertex…I’ll pick A. Draw a 5 vertex graph
A
A
133
185
200
E
120
152
B
E
B
119
150
199
121
D
D
174
C
C
Chapter 3 Page 10 of 28
Step 2.) - examine all edges from A going to any other vertex,
pick the cheapest = AC 119
A
A
133
E
185
E
152
119
150
199
D
119
B
200
120
B
D
C
1st edge chosen
121
C
174
Step 3.) - examine all edges from C going to any other vertex except back to A,
pick the cheapest = CE 120
A
133
A
185
E
B
200
120
152
119
B
120
150
199
119
E
121
D
C
2nd edge chosen
174
D
C
Step 4.) - examine all edges from E going to any other vertex except back to A, C
pick the cheapest = ED 199
A
A
133
185
E
B
200
120
152
119
150
199
199
B
120
121
D
C
3rd edge chosen
174
D
119
E
C
Chapter 3 Page 11 of 28
Step 5.) - examine all edges from D going to any other vertex except back to A, C, E
pick the cheapest = DB 150 there was no other choice
A
A
133
185
E
B
200
152
120
E
B
119
150
199
121
D
C
4th edge chosen
174
D
C
Step 6.) – There is no choice except to return to A from B to complete the Hamilton
Circuit = BA 185
A
133
A
185
200
E
120
152
B
E
B
119
150
199
121
D
C
finished circuit
174
D
C
Hamilton Circuit is ACEDBA or reverse ABDECA
Hamilton Circuit = ACEDBA = 119 + 120 + 199 + 150 + 185 = 773
AC
CE ED DB BA = ACEDBA
or you can rewrite it: BACEDB or CEDBAC or DBACED or EDBACE
or other ways following the sequence of edges you used… all weigh 773
Brute Force Algorithm on this graph yielded best answer = 676 weight
Improvement:
Percentage relative error = Approximate answer - Optimal answer = 773-676 = .14 = 14% improvement
Optimal answer
676
Video of Brute Force and Nearest Neighbor Algorithms
http://mediaplayer.pearsoncmg.com/assets/cV_e4_OrKFiYHQs1QelD1N_KI2SVybPW
Chapter 3 Page 12 of 28
RNN Algorithm - 3rd method (approximate)
- more work than NN, but not as much as Brute Force
- but, won’t always give optimal answer
Use RNN to find cheapest Hamilton Circuit from A:
A
133
185
E
B
200
120
152
119
150
199
121
A
A
B
C
D
E
185
119
152
133
B
185
121
150
200
C
119
121
D
152
150
174
174
120
E
133
200
120
199
199
174
D
Procedure:
1.) Perform NN algo from A.
2.) Then NN from B
3.) Then NN from C
4.) Then NN from D
5.) Finally NN from E
6.) Select the cheapest
C
NN algo
5 times
A
NN from A
1st: AC 119
2nd: CE 120
E
3rd: ED 199
4th: DB 150
5th: BA 185
D
773 ACEDBA
A
B
C
NN from B
1st: BC 121
2nd: CA 119 E
3rd: AE 133
4th: ED 199
5th: DB 150
D
722 BCAEDB
B
C
A
NN from C
1st: CA 119
E
2nd: AE 133
3rd: ED 199
4th: DB 150
5th: BC 121
D
722 CAEDBC
B
C
NN from D
1st: DB 150
2nd: BC 121
E
rd
3 : CA 119
4th: AE 133
5th: ED 199
722
D
DBCAED
A
B
C
Chapter 3 Page 13 of 28
NN from E
1st: EC 120
2nd: CA 119
E
rd
3 : AD 152
4th: DB 150
5th: BE 200
741 ECADBE
A
B
D
C
Now pick cheapest Hamilton Circuit:
(3) tied with weight of 722
BCAEDB
CAEDBC
all are the same
DBCAED
$722 circuit.....
different start
vertices
Re-write the circuit using A to start: ADBCEA or AECBDA (reverse)
video:
http://mediaplayer.pearsoncmg.com/assets/BySt7j_wWrnrmlYx2ac0LQoXJQdn0FgT
Chapter 3 Page 14 of 28
Practice problems
39a.) Nearest neighbor Algorithm from A
A
18
35
E
B
18
23
22
40
21
32
9
25
D
C
Procedure
Step1.) Pick cheapest edge from A =
Step2.) Pick cheapest edge from E but not back to A =
Step3.) Pick cheapest edge from B but not back to E or A =
Step4.) Pick cheapest edge from C but not back to B,E, or A =
Step5.) Pick cheapest edge from D no choice back to A =
AE = 18
EB = 18
BC = 9
CD = 25
DA = 22
92
A
E
B
D
C
ADCBEA = 92
ADCBEA can be re-written as DCBEAD or
CBEADC or
BEADCB or
EADCBE or
5 reverses
Chapter 3 Page 15 of 28
39b.) Nearest neighbor Algorithm from B but rewrite it from A
A
18
35
E
B
18
23
22
40
21
32
9
25
D
C
Procedure
Step1.) Pick cheapest edge from B =
BC = 9
Step2.) Pick cheapest edge from C but not back to B =
CE = 23
Step3.) Pick cheapest edge from E but not back to C or B = EA = 18
Step4.) Pick cheapest edge from A but not back to E,C, or B = AD = 22
Step5.) Pick cheapest edge from D no choice back to B =
DB = 21
93
A
E
B
D
C
BCEADB = 93
BCEADB re-written
as ADBCEA = 93
Chapter 3 Page 16 of 28
43.) RNN Algorithm
A
18
35
18
E
23
B
22
40
21
32
9
25
D
Procedure:
1.) Perform NN algo from A.
2.) Then NN from B
3.) Then NN from C
4.) Then NN from D
5.) Finally NN from E
6.) Select the cheapest
C
NN algo
5 times
NN algo from A = ADCBEA = 92 (problem 39a)
A
E
B
D
NN algo from B = BCEADB = 93 (problem 39b)
C
A
E
B
D
C
Chapter 3 Page 17 of 28
NN algo from C
A
18
35
E
B
18
22
23
40
21
32
9
25
D
C
A
CB = 9
BE = 18
EA = 18
AD = 22
DC = 25
92
E
B
D
CBEADC = 92
C
NN algo from D
A
18
35
18
E
23
B
22
40
21
32
9
25
D
C
A
DB = 21
BE = 18
EA = 18
AC = 40
CD = 25
122
E
B
D
C
DBEACD = 122
Chapter 3 Page 18 of 28
NN algo from E
A
18
35
E
B
18
22
23
40
21
32
9
25
D
C
EB = 18 or EA = 18? Which? Pick one at random…I’ll pick EB
A
EB = 18
BC = 9
CD = 25
DA = 22
AE = 18
92
E
B
D
EBCDAE = 92
C
Calculated 5 Hamilton Circuits, pick cheapest: CBEADC = 92
ADCBEA = 92
EBCDAE = 92
A
E
B
D
C
all 3 same Ham circuit
Chapter 3 Page 19 of 28
Cheapest Link Algorithm - 4th Method (approximate)
- more work than NN, but not as much as Brute Force
- but, won’t always give optimal answer
Procedure: 1.) Even if problem wants a Hamilton Circuit starting from a particular vertex,
don’t pick a start vertex...instead select cheapest edge anywhere
2.) Select next cheapest anywhere
3.) Select next cheapest edge anywhere, but no 3 edges from same vertex
and don’t complete the circuit until all vertices have been contacted
4.) Repeat step 3 until you have a Hamilton circuit
5.) Add the weights
6.) Write the Hamilton Circuit from the desired start vertex
Use Cheapest Link algorithm to find cheapest Hamilton Circuit from
A
133
185
E
B
200
120
D
152
119
150
199
121
A
A
B
C
D
E
185
119
152
133
B
185
121
150
200
174
D
C
Draw a 5 vertex graph:
A
E
B
D
when done, rewrite circuit from D
C
C
119
121
174
120
D
152
150
174
199
E
133
200
120
199
Chapter 3 Page 20 of 28
Don’t pick a start vertex…
Pick 1st cheapest edge anywhere = AC = 119 If there was a tie, 2 with same weight,
pick either one)
A
E
B
D
when done, rewrite circuit from D
C
Pick 2nd cheapest edge anywhere = EC = 120
A
E
B
D
C
when done, rewrite circuit from D
Pick 3rd cheapest edge anywhere = BC = 121 no good…3 edges from C
A
E
B
no good (3 edge rule)
D
when done, rewrite circuit from D
C
Chapter 3 Page 21 of 28
Pick 3rd cheapest edge anywhere again = AE = 133 no good…created circuit
A
no good (created circuit)
E
B
D
C
when done, rewrite circuit from D
Pick 3rd cheapest edge anywhere again = DB = 150
A
E
B
D
C
when done, rewrite circuit from D
Pick 4th cheapest edge anywhere = DA = 152
A
E
B
D
when done, rewrite circuit from D
C
Chapter 3 Page 22 of 28
Pick 5th cheapest edge anywhere = DC = 174 no good…created circuit and 3 edges from D and C
A
E
B
D
no good
C
when done, rewrite circuit from D
Pick 5th cheapest edge anywhere = AB = 185 no good…created circuit and 3 edges from A
A
no good
E
B
D
C
when done, rewrite circuit from D
Pick 5th cheapest edge anywhere = DE = 199 no good…created circuit and 3 edges from D
A
E
B
no good
D
when done, rewrite circuit from D
C
Chapter 3 Page 23 of 28
Pick 5th cheapest edge anywhere = BE = 200
A
E
B
D
C
when done, rewrite circuit from D
Write Hamilton circuit from D: DBECAD or reverse. It weighs:
AC = 119
EC = 120
DB = 150
DA = 152
BE = 200
741
Video for Cheapest Link Algorithm
http://mediaplayer.pearsoncmg.com/assets/cthirMi_nYN4OUkoI9qbrEyqM0hZd1Fp
Do practice problems page 101: 49,56a
This is the
order edges
were chosen
Chapter 3 Page 24 of 28
Practice problem page 101:
49.) Use Cheapest Link Algorithm to find circuit starting at A
A
when done, write circuit from A
18
35
18
E
B
22
23
40
21
32
9
25
D
C
Don’t pick a start vertex
Select cheapest edge anywhere. CB = 9
A
when done, write circuit from A
E
B
9
D
Select
C
next cheapest edge anywhere. If tie, pick one at random. EA = 18
A
when done, write circuit from A
18
E
B
9
D
C
Chapter 3 Page 25 of 28
Select
next cheapest edge anywhere. EB = 18
A
when done, write circuit from A
18
18
E
B
9
D
Select
C
next cheapest edge anywhere. BD = 21 NO, 3 edge rule
A
when done, write circuit from A
18
B
18
E
21
D
Select
9
C
next cheapest edge anywhere. AD = 22 OK
A
when done, write circuit from A
18
B
18
E
22
D
9
C
Chapter 3 Page 26 of 28
Select
next cheapest edge anywhere. EC = 23 No, 3 edge rule and circuit rule
A
when done, write circuit from A
18
B
18
E
9
23
22
D
Select
C
next cheapest edge anywhere. CD = 25
A
when done, write circuit from A
18
B
18
E
22
9
D
25
C
1st edge BC = 9
2nd edge EA = 18
3rd edge BE = 18
4th edge AD = 22
5th edge CD = 25
92
Write the circuit starting at A
ADCBEA or AEBCDA
Chapter 3 Page 27 of 28
56a.) Cheapest Link Algorithm start at A
A
A
B
C
D
E
A
B
C
D
E
1.2
0.7
1.0
1.3
B C
1.2 0.7
0.9
0.9
0.8 1.2
1.1 0.8
D
1.0
0.8
1.2
A
B
D
C
E
1.3
1.1
0.8
0.9
0.9
E
Ignore upper right
or
lower left (it’s duplicate data)
1.2
0.7 0.9
1.0 0.8 1.2
1.3 1.1 0.8 0.9
Don’t pick a start vertex
Select cheapest edge anywhere. AC = 0.7
Draw 5 vertices (any shape is OK). Label them ABCDE in any order.
B
E
D
0.7
C
A
Select next cheapest edge anywhere. If tie, pick one at random
DB = 0.8
B
0.8
E
D
0.7
C
A
Chapter 3 Page 28 of 28
Select next cheapest edge anywhere. EC = 0.8
B
0.8
E
D
0.8
0.7
C
A
Select next cheapest edge anywhere. If tie, pick one at random
ED = 0.9
B
0.8
0.9
E
D
0.8
0.7
C
A
Select next cheapest edge anywhere. If tie, pick one at random
CB = 0.9 No, 3 edge rule and circuit
DA = 1.0 No, 3 edge rule and circuit
EB = 1.1 No, 3 edge rule and circuit
DC = 1.2 No, 3 edge rule and circuit
AB = 1.2 OK
B
0.8
0.9
E
D
1.2
0.8
0.7
C
A
Write circuit from A: ABDECA = 1.2 + 0.8 + 0.9 + 0.8 + 0.7 = 4.4
or
reverse