A New Evolutionary Approach for the Optimal Communication Spanning Tree Problem

1
A New Evolutionary Approach for the Optimal
CommunicationSpanning Tree Problem

• Sang-Moon Soak
• Speaker ???????????????

2
Outline

• OCST problem
• Evolutionary Algorithm
• 4 kinds of encodings for the Spanning tree
• Proposed encodings evolutionary algorithm
  design
• Experiment conclusion

3
Outline

• OCST problem
• Evolutionary Algorithm
• 4 kinds of encodings for the Spanning tree
• Proposed encodings evolutionary algorithm
  design
• Experiment conclusion

Speaker ???
4
OCST Problem

• Goal a spanning tree with minimum communication
  cost
• dT(i, j) distance between i and j on T
• R(i, j) communication requirements
• associated with each pair of nodes
• e.g. of telephone calls between two cities

5
Example
Example tree
6
How to find it?

• NP-hard problem!
• Search problem
• Search for a better solution

7
Evolutionary Algorithm

• One kind of local search
• Genetic algorithm
• the most popular type of EA
• Uses some mechanisms inspired by biological
  evolution
• Reproduction, mutation, recombination, natural
  selection, survival of the fittest
• Add some random mechanism

8
Evolutionary Algorithm

• Population Start with k randomly generated
  individuals (i.e. states)
• Individual each is represented as a string over
  a finite alphabet
• Fitness function evaluation of the goodness of
  a given state.
• A successor is generated by combining two parents
  from the current population.
• Produce the next generation of states by
  selection, crossover, and mutation

Chapter 4 Local Search, Jane Hsu, Artificial
Intelligent, 2006
9
Evolutionary Algorithm
next generation
fitness function
24
2 4 7 4 8 5 5 2
3 2 7 5 2 4 1 1
3 2 7 4 8 1 5 2
20
3 2 7 5 2 4 1 1
2 4 7 4 8 5 5 2
2 4 7 5 2 4 1 1
16
2 4 4 1 5 1 2 4
3 2 7 5 2 4 1 1
3 2 2 5 2 1 2 4
11
3 2 5 4 3 2 1 3
2 4 4 1 5 1 2 4
2 4 4 1 5 4 1 7
Initial population
selection
cross-over
mutation
Chapter 4 Local Search, Jane Hsu, Artificial
Intelligent, 2006
10
Evolutionary Algorithm

• Initialize the population
• Evaluate initial population
• Repeat
• Perform competitive selection
• Apply genetic operators to generate new solutions
• Evaluate solutions in the population
• Until some convergence criteria is satisfied

11
OCST Use EA

• Encode spanning tree to a string
• Generate valid initial population
• Design selection method
• Design crossover and mutation operator
• To generate new spanning tree
• Evaluate the cost

3 1 5 2 5 1 4
12
OCST Use EA

• Encode spanning tree to a string
• Generate valid initial population
• Design selection method
• Design crossover and mutation operator
• To generate new spanning tree
• Evaluate the cost

13
OCST Use EA

• Encode spanning tree to a string
• Generate valid initial population
• Design selection method
• Design crossover and mutation operator
• To generate new spanning tree
• Evaluate the cost

14
OCST Use EA

• Encode spanning tree to a string
• Generate valid initial population
• Design selection method
• Design crossover and mutation operator
• To generate new spanning tree
• Evaluate the cost

15
OCST Use EA

• Encode spanning tree to a string
• Generate valid initial population
• Design selection method
• Design crossover and mutation operator
• To generate new spanning tree
• Evaluate the cost

Cost 93
Cost 75
Cost 86
better solutions!!
16
Outline

• OCST problem
• Evolutionary Algorithm
• 4 kinds of encodings for the Spanning tree
• Proposed encodings evolutionary algorithm
  design
• Experiment conclusion

Speaker ???
17
Encodings for the Spanning Tree

• Prüfer
• The Prüfer Encoding
• LNB
• The Link and Node Biased Encoding
• NetKey
• The Network Random Key Encoding
• ES
• The Edge Set Encoding

18
Prüfer

• Vertex set is 1, 2 ,3, 4, 5, 6, 7, 8
• Encode in Prüfer sequence
• 6 3 2 5 4 1 (a line)
• 5 6 3 2 1 8
• 1 1 1 3 3 3 (two-star)
• 1 1 1 1 1 1 (star)

19
Prüfer Decoding

• P (3, 3, 4, 5, 4, 6)
• V 1, 2, 3, 4, 5, 6, 7, 8

20
LNB The Link and Node Biased Encoding

• Encode in the bias value
• Decode by calculating the modified cost function
• Cmax is the maximum link cost
• b(i, j) is the link bias associated with the
  edge from i to j
• bi is the node bias associated with node i

21
LNB Decoding

• Use Prims algorithm with the modified cost
  matrix C

22
NetKey The Network Random Key Encoding

• Encode in key sequence
• Construct the permutation
• 10 ? 8 ? 6 ? 9 ? 2 ? 7 ? 1 ? 5 ? 4 ? 3

23
NetKey Decoding

• Use Kruskals algorithm
• 10 ? 8 ? 6 ? 9 ? 2 ? 7 ? 1 ? 5 ? 4 ? 3

24
ES The Edge Set Encoding

• Degree constrained minimum spanning tree
• ES with heuristic (ESWH)
• ES without heuristic (ESWOH)
• Encode in edge set
• (a, c), (c, d), (b, d), (d, e)

25
Evolutionary Algorithm Design Crossover and
Mutation

• Prüfer
• Mutation Swap
• LNB
• Crossover One-Point
• Mutation Random Perturbation
• NetKey
• Crossover Uniform
• Mutation Swap
• ES
• Crossover KruskalRST
• Mutation Edge Insertion

26
Outline

• OCST problem
• Evolutionary Algorithm
• 4 kinds of encodings for the Spanning tree
• Proposed encodings evolutionary algorithm
  design
• Experiment conclusion

Speaker ???
27
A New Encoding for the OCST

• Each spanning tree is encoded as a string of
  length 2(N-1)
• Each elements is in the range 1, N
• Each adjacency pair in the string represents an
  edge in G
• Defines the possible edge set on the spanning
  tree

28
How to decode the string?

• Cycle-Free Tree Construction Routine
• Add edges that do not induce a cycle
• Cycle-Breaking Tree Construction Routine
• Add edges one by one, on cycle remove the edge
  with the largest cost.

29
Cycle-Breaking Tree Construction Routine

• E (N1, N2), (N2, N3), , (N2n-2, N2n-1)
• U 1,2,,n, T
• For i 1..2n-2
• Remove (Ni, Ni1) from E
• If Ni and Ni1 not in U, put (Ni, Ni1) into T
  and remove Ni, Ni1 from U
• If Ni is in U but Ni1 is not, put (Ni, Ni1) in
  T and remove Ni1 from U
• If both Ni and Ni1 are not in U and (Ni, Ni1)
  is in T, do nothing
• If both Ni and Ni1 are not in U and (Ni, Ni1)
  is not in T, add (Ni, Ni1) into T and find out
  the largest edge in the cycle to remove

30
A New Encoding for OCST

• (1, 5, 2, 1, 4, 3, 2, 5)

5
1
5
1
4
1
5
4
1
5
2
7
2
7
8
8
3
2
3
2
9
9
31
Theorem 1

• Any string of 2(N-1) node identifiers, which
  contains each node at least once, encodes a valid
  spanning tree when using CB-TCR
• Each node appears in the string and will be added
  into T
• Once a node is added, it is not removed
• As cycle induced, some edge is removed from T to
  ensure cycle-free property

32
Theorem 2

• Given any spanning tree on N nodes, there is a
  string of at most 2(N-1) node identifiers which,
  when using CB-TCR, encodes precisely that tree
• Use induction construct tree of n nodes from
  tree of n-1 nodes by adding an edge with leaf
  node on the tree
• When k 2, the tree could be encoded in a string
  of length 2

33
Theorem 2 (cond.)

• Assume the when k n, each n-node spanning tree
  could be generated by a string of length at most
  2(n-1) without any edge removal in the
  tree-construction process
• At k n1, choose a leaf node y and its
  adjacency node x. If (a, x) is some edge in the
  representation of remaining n nodes, replace (a,
  x) by (a, x, y, x) to form a new string
• The new string has length at most
• 2(n-1) 2 2n

34
Evolutionary Algorithm Design

• How to do reproduction and crossover in the new
  coding scheme?
• Reproduction Real World Tournament Selection
  (RWTS)
• Crossover Adjacent Node Crossover (ANX)

35
Real World Tournament Selection

• Score each string and give higher score ones more
  probability to mate
• Pair each 2 strings in the last generation and
  choose the floor(m/2) higher score ones to be the
  level 1 winner
• Pair level i winner and choose the higher score
  ones to generate level i1 winner

36
Real World Tournament Selection (cond.)

• Algorithm RWTS
• i
• for i 1, 2,
• Generate level i winner and put it into I
• if level i winner 1 then break
• If I lt m, copy level i winner multiple times to
  fill up the slots.

37
Adjacent Node Crossover

• Construct the offspring in string format
  element-wise one by one
• Starting from some node c, check parent A1 and A2
  for nodes adjacent to c, choose one from them to
  get the next node
• Define Aij be the set of nodes adjacent to j of
  tree Ai

38
Adjacent Node Crossover

• Build adjacent map A1t and A2t
• Select starting point c randomly, O1 c
• For i 2 ..N
• S lt- A1c n A2c, V lt- A1c ? A2c
• If S is not empty then choose
• p argmin_j in S d(c, j)
• remove p from A1and2c
• remove c from A1and2p

39
Adjacent Node Crossover

• else if V is not empty then choose
• p argmin_j in V d(c, j)
• remove p from A1or2c
• remove c from A1or2p
• else
• random choose a valid p
• Oi lt- p, c lt- p

40
An example
41
An example
42
Some notes

• Is the generated tree is always a valid spanning
  tree?
• Not necessary!
• Some node may not be traversed at all!
• Solve it by post-processing to repair this

43
Outline

• OCST problem
• Evolutionary Algorithm
• 4 kinds of encodings for the Spanning tree
• Proposed encodings evolutionary algorithm
  design
• Experiment conclusion

Speaker ???
44
Experiment

• Each algorithm repeat 20 times
• Use published benchmark instances
• Symbol define
• Gap
• Gen mean of generations to achieve best sol
• Time cost time of achieving best sol
• Opt of trials that find best sol

45
Locality and Diversity

• Locality
• Phenotype distance
• Mutation innovation (MI)
• Compare with mutant and its parent
• Diversity
• The relation between phenotype and fitness
  distance
• Fitness distance

46
Experimental results
47
Comparison of Locality

1. Redundancy
2. Heuristic bias of encodings

High value represent low degree of locality
48
Diversity of CB-TCR
Solutions with small fitness values can easily
find the optimal solution
49
ESWH
ESWOH
LNB
NetKey
P24
B35U
50
Conclusion

• The optimal solution of the OCST problem is
  biased towards the MST
• Propose a new encoding CB-TCR, and use EA with
  RWTS strategy in addressing the OCST problem
• CB-TCR has better performance
• Relative high locality
• Preserve the population diversity for a long
  generation