Post on 24-Mar-2018
Advanced Algorithmics (4AP)Search
Jaak ViloJaak Vilo
2009 Spring
1MTAT.03.190 Text Algorithms Jaak Vilo
SearchSearch
• for what?– the solution
– the best possible (approximate?) solution
f h ?• from where? – search space (all valid solutions or paths)
• under which conditions?compute time space– compute time, space, …
– constraints, …
Objective functionObjective function
• An optimal solution– what is the measure that we optimise?p
• Any solution (satisfiability /SAT/ problem)– does the task have a solution?
– is there a solution with objective measure better than X?
• Minimal/maximal cost solution
• A winning move in a game
• A (feasible) solution with smallest nr of constraintviolations (e.g. course time scheduling)
Search spaceSearch space
• Linear (list, binary search, …)
• Trees, Graphs
• Real nr in [x y)Real nr in [x,y) – Integers
• A point in high‐dimensional space
ConstraintsConstraints
• Time, space…– if optimal cannot be found, approximatep , pp
All ki d f d h t i ti• All kinds of secondary characteristics
• Constraintssometimes finding even a point in the valid search– sometimes finding even a point in the valid searchspace is hard
OutlineOutline
B fi h• Best‐first search• Greedy best‐first search• A* search• Heuristics• Local search algorithms• Hill‐climbing searchc b g sea c• Simulated annealing search• Local beam searchLocal beam search• Genetic algorithms
Classes of Search TechniquesClasses of Search Techniques
Calculus based techniques Guided random search techniques Enumerative techniques
Search techniques
Direct methods Indirect methods
Calculus-based techniques
Evolutionary algorithms Simulated annealing
Guided random search techniques
Dynamic programming
Enumerative techniques
Finonacci Newton Evolutionary strategies
Parallel Sequential
Genetic algorithms
Centralized Distributed Steady-state Generational
11Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Local search algorithmsLocal search algorithms
I ti i ti bl th th t th l• In many optimization problems, the path to the goal is irrelevant; the goal state itself is the solution
• State space = set of "complete" configurations• Find configuration satisfying constraints e g n‐• Find configuration satisfying constraints, e.g., n‐queens
• In such cases, we can use local search algorithms• keep a single "current" state, try to improve itkeep a single current state, try to improve it
Example: n queensExample: n‐queens
• Put n queens on an n × n board with no two queens on the same row, column, or diagonalq g
• may get stuck…may get stuck…
Tree/Graph searchTree/Graph search
• order defined by picking a node for expansionorder defined by picking a node for expansion
• BFS, DFS
• Random, Best First, …– Best – an evaluation function
Id l i f i f( ) f h d• Idea: use an evaluation function f(n) for each node– estimate of "desirability"– Expand most desirable unexpanded nodeExpand most desirable unexpanded node
• Implementation:Order the nodes in fringe in decreasing order of desirabilityPriority queue
• Special cases:greedy best first search f(n) h(n) heuristic e g estimate to goal– greedy best‐first search f(n) = h(n) heuristic, e.g. estimate to goal
– A* search
A*A
• f(n) = g(n) + h(n)
– g(n) – path covered so far in graph
h( ) i d di f l– h(n) – estimated distance from n to goal
F
A C20
20 10
A CB
15
D25
Admissible heuristicsAdmissible heuristics
• A heuristic h(n) is admissible if for every node n,
h(n) ≤ h*(n), where h*(n) is the true cost to reach the goal state from n.
• An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic
• Example: hSLD(n) (never overestimates the actual road d ) ( h l d )distance) (SLD – shortest linear distance)
( ) *• Theorem: If h(n) is admissible, A* using TREE-SEARCH is optimal
Optimality of A* (proof)Optimality of A (proof)
b l l h b d d h f• Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
• f(G2) = g(G2) since h(G2) = 0
• g(G2) > g(G) since G2 is suboptimal
• f(G) = g(G) since h(G) = 0
• f(G ) > f(G) from above• f(G2) > f(G) from above
Optimality of A* (proof)Optimality of A (proof)
b l l h b d d h f• Suppose some suboptimal goal G2 has been generated and is in the fringe. Let n be an unexpanded node in the fringe such that n is on a shortest path to an optimal goal G.
• f(G2) = g(G2) since h(G2) = 0
• g(G2) > g(G) since G2 is suboptimal
• f(G) = g(G) since h(G) = 0
• f(G ) > f(G) from above• f(G2) > f(G) from above
• Hence f(G2) > f(n), and A* will never select G2 for expansion
GraphGraph
• A Virtual graph/search space– valid states of Fifteen‐gameg
– Rubik’s cube
1 2 3 4
5 6 7 85 6 7 8
9 10 11 12
13 14 15
SolveSolve
• Which move takes us closer to the solution?
• Estimate the goodness of the stateEstimate the goodness of the state
34 1314
5 6 7 8
9 10 12
1 2
9 10
11
12
15
• How many are misplaced? (7) 34
5 6 7 8
1314
• How far have they beeni l d? S f h i l
1 2
9 10
11
12
15
misplaced? Sum of theoreticalshortest paths to the correct place Search
A* h t d fi l l
space
• A* search towards a final goal
The Traveling Salesperson Problem (TSP)(TSP)
i i i i• TSP – optimization variant:• For a given weighted graph G = (V,E,w), find a g g g p ( ) fHamiltonian cycle in G with minimal weight,
• i e find the shortest round‐trip visiting eachi.e., find the shortest round trip visiting each vertex exactly once.
• TSP decision variant:• TSP – decision variant:• For a given weighted graph G = (V,E,w), decide whether a Hamiltonian cycle with minimal weight ≤ b exists in G.
Search Methods
• Types of search methods:
• systematic←→ local searchsystematic ←→ local search
• deterministic ←→ stochastic
• sequential ←→ parallel
Local Search (LS) Algorithms
• search space S
(SAT: set of all complete truth assignments to propositional variables)
• solution set SS '(SAT: models of given formula)
• neighborhood relation SxSN (SAT: neighboring variable assignments differ in the
truth value of exactly one variable)
• evaluation function g : S → R+
(SAT: number of clauses unsatisfied under given
assignment)
Local Search:Local Search:
• start from initial position
• iteratively move from current position toiteratively move from current position to neighboring position
l i f i f id• use evaluation function for guidance
Two main classes:
• local search on partial solutions
• local search on complete solutions
Local search for partial solutionsLocal search for partial solutions
• Order the variables in some order.
• Span a tree such that at each level a givenSpan a tree such that at each level a given value is assigned a value.
P f d h fi h• Perform a depth‐first search.
• But, use heuristics to guide the search. Choose , gthe best child according to some heuristics. (DFS with node ordering)(DFS with node ordering)
Construction Heuristics for partial solutions
• search space: space of partial solutions
• search steps: extend partial solutions with assignment for the next elementass g e t o t e e t e e e t
• solution elements are often ranked according t d l ti f tito a greedy evaluation function
DFSDFS
• Once a solution has been found (with the first dive into the tree) we can continue search the )tree with DFS and backtracking.
• In fact this is what we did with DFBnB• In fact, this is what we did with DFBnB.
• DFBnB with node ordering.
Iterative Improvement p(Greedy Search):
• initialize search at some point of search space• initialize search at some point of search space
• in each step, move from the current search position to a neighboring position with better evaluation function valueevaluation function value
Iterative Improvement for SAT
i i i li i d l h l• initialization: randomly chosen, complete truth assignment
• neighborhood: variable assignments are neighbors iff they differ in truth value of one g yvariable
• neighborhood size: O(n) where n = number ofneighborhood size: O(n) where n = number of variables
l ti f ti b f l• evaluation function: number of clauses unsatisfied under given assignment
Hill climbingHill climbing
Choose the nieghbor with the largest
improvment as the next state
f lf-value
f-value = evaluation(state
statesstates
while f value(state) > f value(next best(state))while f-value(state) > f-value(next-best(state)) state := next-best(state)
Hill climbingHill climbing
function Hill-Climbing(problem) returns a solution stateg(p )
current Make-Node(Initial-State[problem])( S [p ])loop do
next a highest-valued successor of currentnext a highest valued successor of currentif Value[next] < Value[current] then return currentcurrent nextcurrent next
end
Problems with local searchProblems with local searchTypical problems with local search (with hill climbing in particular)
• getting stuck in local optima• getting stuck in local optima
• being misguided by evaluation/objective function
Stochastic Local SearchStochastic Local Search
• randomize initialization step
• randomize search steps such thatrandomize search steps such that suboptimal/worsening steps are allowed
i d f & b• improved performance & robustness
• typically, degree of randomization controlled yp y, gby noise parameter
Stochastic Local SearchStochastic Local Search
PPros:• for many combinatorial problems more efficient than systematic searchsystematic search
• easy to implement• easy to parallelize• easy to parallelizeCons:• often incomplete (no guarantees for finding existing• often incomplete (no guarantees for finding existing solutions)
• highly stochastic behaviorhighly stochastic behavior• often difficult to analyze theoretically/empirically
Simple SLS methods
• Random Search (Blind Guessing):
• In each step, randomly select one element ofIn each step, randomly select one element of the search space.
(U i f d) R d W lk• (Uninformed) RandomWalk:
• In each step, randomly select one of the p, y fneighbouring positions of the search space and move thereand move there.
Randomized Iterative Improvement:
i i i li h i f h h• initialize search at some point of search space search steps:• with probability p, move from current search position to a
randomly selected neighboring positionrandomly selected neighboring position• otherwise, move from current search position to neighboring
position with better evaluation function value.• Has many variations of how to choose the randomly neighbor,
and how many of theml k d ( k• Example: Take 100 steps in one direction (Army mistake
correction) – to escape from local optima.
General iterative AlgorithmsGeneral iterative Algorithms
eneral and “eas ” to implement– general and “easy” to implement
– approximation algorithms
–must be told when to stop
hill li bi–hill‐climbing
– convergenceg
spie98‐48
General iterative searchGeneral iterative search
• Algorithm• Algorithm– Initialize parameters and data structures
– construct initial solution(s)
– Repeatp• Repeat
– Generate new solution(s)( )
– Select solution(s)
• Until time to adapt parameters
• Update parameters
– Until time to stop
spie98‐49
p
• End
Iterative searchIterative search
• Most popular algorithms of this class– Genetic Algorithmsg
• Probabilistic algorithm inspired by evolutionary mechanisms
– Simulated Annealing• Probabilistic algorithm inspired by the annealing ofProbabilistic algorithm inspired by the annealing of metals
– Tabu SearchTabu Search• Meta‐heuristic which is a generalization of local search
spie98‐50
Hill climbing searchHill‐climbing search
• Problem: depending on initial state, can get stuck in local maxima
•
Simulated annealing
Combinatorial search technique inspired by the physical process of annealingphysical process of annealing [Kirkpatrick et al. 1983, Cerny 1985]
O tliOutline Select a neighbor at random.
If better than current state go there.
Otherwise, go there with some probability.
Probability goes down with time (similar to
temperature cooling)temperature cooling)
Simulated AnnealingKirkpatrick, S., C. D. Gelatt Jr., M. P. Vecchi, "Optimization by Simulated
Annealing",Science, 220, 4598, 671‐680, 1983.
g
Pseudo codePseudo code
f ti Si l t d A li ( bl h d l ) tfunction Simulated-Annealing(problem, schedule) returns solution state
current Make Node(Initial State[problem])current Make-Node(Initial-State[problem])for t 1 to infinity
T h d l [ ] // T d dT schedule[t] // T goes downwards.if T = 0 then return currentnext Random-Successor(current) E f-Value[next] - f-Value[current]if E > 0 then current nextelse current next with probability eE/T
end
Example application to the TSP [Johnson & McGeoch 1997]McGeoch 1997]
b li i l ibaseline implementation:• start with random initial solution• use 2‐exchange neighborhood• simple annealing schedule; relatively poor performanceimprovements:p o e e s• look‐up table for acceptance probabilities• neighborhood pruningneighborhood pruning• low‐temperature starts
Summary‐Simulated Annealing
Simulated Annealing . . .
• is historically importantis historically important
• is easy to implement
• has interesting theoretical properties (convergence), but these are of very limited ( g ), ypractical relevance
hi d f ft t th t• achieves good performance often at the cost of substantial run‐times
Examples for combinatorial problems:
• finding shortest/cheapest round trips (TSP)
• finding models of propositional formulaefinding models of propositional formulae (SAT)
l i h d li i bli• planning, scheduling, time‐tabling
• resource allocation
• protein structure prediction
• genome sequence assembly
Tabu SearchTabu Search
C bi i l h h i hi h h il li• Combinatorial search technique which heavily relies on the use of an explicit memory of the search process [Glover 1989 1990] to guide search processprocess [Glover 1989, 1990] to guide search process
• memory typically contains only specific attributes of previously seen solutionspreviously seen solutions
• simple tabu search strategies exploit only short term memorymemory
• more complex tabu search strategies exploit long term memoryterm memory
Tabu search – exploiting short term memorymemory
i h t t b t i hb i l ti• in each step, move to best neighboring solution although it may be worse than current one
• to avoid cycles tabu search tries to avoid revisiting• to avoid cycles, tabu search tries to avoid revisiting previously seen solutions by basing the memory on attributes of recently seen solutions
• tabu list stores attributes of the tlmost recently visited
• solutions; parameter tl is called tabu list length or tabu tenurel ti hi h t i t b tt ib t• solutions which contain tabu attributes are
forbidden
Example: Tabu Search for SAT / MAX‐SATSAT
i hb h d i hi h diff i• Neighborhood: assignments which differ in exactly one variable instantiation
• Tabu attributes: variables• Tabu criterion: flipping a variable is forbiddenTabu criterion: flipping a variable is forbidden for a given number of iterations
• Aspiration criterion: if flipping a tabu variable• Aspiration criterion: if flipping a tabu variable leads to a better solution, the variable’s tabu t t i iddstatus is overridden
[Hansen & Jaumard 1990; Selman & Kautz 1994]
• Bart Selman, Cornellwww‐verimag.imag.fr/~maler/TCC/selman‐tcc.pptg g f / / / pp
Ideas from physics statistics combinatoricsIdeas from physics, statistics, combinatorics, algorithmics …
Fundamental challenge: Combinatorial Search Spacesg p
• Significant progress in the last decade.
• How much?
• For propositional reasoning:• ‐‐We went from 100 variables, 200 clauses (early 90’s)• to 1,000,000 vars. and 5,000,000 constraints in• 10 years. Search space: from 10^30 to 10^300,000.y p ,
• ‐‐ Applications: Hardware and Software Verification,• Test pattern generation Planning Protocol DesignTest pattern generation, Planning, Protocol Design, • Routers, Timetabling, E‐Commerce (combinatorial• auctions), etc.•••
• How can deal with such large combinatorial spaces and• still do a decent job?
• I’ll discuss recent formal insights into• combinatorial search spaces and their• practical implications that makes searching• such ultra‐large spaces possible.
• Brings together ideas from physics of disordered systems • (spin glasses), combinatorics of random structures, and• algorithms• algorithms.
• But first, what is BIG?
What is BIG?
Consider a real-world Boolean Satisfiability (SAT) problem
I.e., ((not x_1) or x_7)((not x_1) or x_6)
etc.
x_1, x_2, x_3, etc. our Boolean variables(set to True or False)
Set x_1 to False ??
10 pages later:p g
I.e., (x_177 or x_169 or x_161 or x_153 …x_33 or x_25 or x_17 or x_9 or x_1 or (not x_185))
…
_ _ _ _ _ ( _ ))
clauses / constraints are getting more interesting…
Note x_1 …
Finally 15 000 pages later:Finally, 15,000 pages later:
C bi t i l h f t th i t HOW?Combinatorial search space of truth assignments: HOW?
Current SAT solvers solve this instance in approx. 1 minute!
Progress SAT SolversProgress SAT Solvers
Instance Posit' 94
2670 136 40 66
Grasp' 96
1 2
Sato' 98
0 95
Chaff' 01
0 02ssa2670-136 40,66s
bf1355-638 1805,21s
1,2s
0,11s
0,95s
0,04s
0,02s
0,01s
pret150_25 >3000s
d b i 100 3000
0,21s
11 85
0,09s
0 08
0,01s
0 01dubois100 >3000s
aim200-2_0-no-1 >3000s
11,85s
0,01s
0,08s
0s
0,01s
0s
2dlx_..._bug005 >3000s
6288 3000
>3000s
3000
>3000s
3000
2,9s
300073
c6288 >3000s >3000s >3000s >3000sSource: Marques Silva 2002
• From academically interesting to practicallyFrom academically interesting to practically relevant.
• We now have regular SAT solver competitionscompetitions.
• Germany ’89, Dimacs ’93, China ’96, SAT‐02, SAT‐03, SAT‐04, SAT05.
• E.g. at SAT‐2004 (Vancouver, May 04):• ‐‐‐ 35+ solvers submitted
500 i d t i l b h k• ‐‐‐ 500+ industrial benchmarks• ‐‐‐ 50,000+ instances available on the WWW.
•
Real‐World ReasoningTackling inherent computational complexity
DARPA ResearchProgram
Tackling inherent computational complexity
1M5M10301,020
mpl
exity
Multi-AgentSystems
0.5M 1M10150,500
Cas
e co Hardware/Software
Verification
50K1015,050W
orst
Chess
200K600K
Military Logistics
10K 50K
50K 200K
103010Deep space mission control
Chess
No. of atomson earth 1047
Hi h P f R iTechnology Targets
100200
50K
1030 Car repair diagnosis
mission controlSeconds until heat death of sun
Protein foldingcalculation
• High-Performance Reasoning• Temporal/ uncertainty reasoning• Strategic reasoning/Multi-player
75
Variablescalculation (petaflop-year) 100 10K 20K 100K 1M
Rules (Constraints)Example domains cast in propositional reasoning system (variables, rules).
A Journey from Random to Structured InstancesInstances
•• I ‐‐‐ Random InstancesI Random Instances• ‐‐‐ phase transitions and algorithms• ‐‐‐ from physics to computer science
• II ‐‐‐ Capturing Problem Structure• ‐‐‐ problem mixtures (tractable / intractable)
b kd i bl d h il• ‐‐‐ backdoor variables, restarts, and heavy tails
• III ‐‐‐ Beyond Satisfactiony• ‐‐‐ sampling, counting, and probabilities• ‐‐‐ quantification
Genetic Algorithms:Genetic Algorithms:ggA TutorialA Tutorial
“Genetic Algorithms are good at taking large,good at taking large,
potentially huge search spaces and navigating
them looking for optimalthem, looking for optimal combinations of things, solutions you might not
th i fi d iotherwise find in a lifetime.”
S l t M- Salvatore ManganoComputer Design, May 1995
77Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
The Genetic AlgorithmThe Genetic Algorithm Directed search algorithms based on
the mechanics of biological evolutiong Developed by John Holland, University
of Michigan (1970’s)of Michigan (1970 s) To understand the adaptive processes of
natural systems To design artificial systems software that
retains the robustness of natural systems
78Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
The Genetic Algorithm (cont )The Genetic Algorithm (cont.) Provide efficient, effective techniques
for optimization and machine learning p gapplicationsWidely used today in business Widely-used today in business, scientific and engineering circles
79Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Components of a GAComponents of a GAA problem to solve, and ... Encoding technique (gene chromosome) Encoding technique (gene, chromosome)
Initialization procedure (creation)
Evaluation function (environment)
Selection of parents (reproduction) Selection of parents (reproduction)
Genetic operators (mutation, recombination)
Parameter settings (practice and art)
80Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Simple Genetic AlgorithmSimple Genetic Algorithm{{
initialize population;evaluate population;evaluate population;while TerminationCriteriaNotSatisfied{{
select parents for reproduction;perform recombination and mutation;evaluate population;p p
}}
81Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
The GA Cycle of ReproductionThe GA Cycle of Reproduction
reproduction modificationchildren
parentsmodifiedchildren
population evaluation
d l t devaluated children
deleted members
discard
82Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Genetic algorithmsGenetic algorithms
• How to generate the next generation.
• 1) Selection: we select a number of states1) Selection: we select a number of states from the current generation. (we can use the fitness function in any reasonable way)fitness function in any reasonable way)
• 2) crossover : select 2 states and reproduce a child.
• 3) mutation: change some of the genues• 3) mutation: change some of the genues.
Summary: Genetic AlgorithmsSummary: Genetic Algorithms
i l i hGenetic Algorithms• use populations, which leads to increased p psearch space exploration
• allow for a large number of differentallow for a large number of different implementation choices
• typically reach best performance when using• typically reach best performance when using operators that are based on problem h t i ticharacteristics
• achieve good performance on a wide range of problems
Classes of Search TechniquesClasses of Search Techniques
Calculus based techniques Guided random search techniques Enumerative techniques
Search techniques
Direct methods Indirect methods
Calculus-based techniques
Evolutionary algorithms Simulated annealing
Guided random search techniques
Dynamic programming
Enumerative techniques
Finonacci Newton Evolutionary strategies
Parallel Sequential
Genetic algorithms
Centralized Distributed Steady-state Generational
87Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
The GA Cycle of ReproductionThe GA Cycle of Reproduction
reproduction modificationchildren
parentsmodifiedchildren
population evaluation
d l t devaluated children
deleted members
discard
88Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
PopulationPopulationl tipopulation
Chromosomes could be: Bit strings (0101 ... 1100) Real numbers (43.2 -33.1 ... 0.0 89.2) Permutations of element (E11 E3 E7 ... E1 E15)( ) Lists of rules (R1 R2 R3 ... R22 R23) Program elements (genetic programming)g (g p g g) ... any data structure ...
89Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
ReproductionReproductionhild
reproductionchildren
population
parents
population
P t l t d t d ithParents are selected at random with selection chances biased in relation to chromosome evaluations.
90Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Chromosome ModificationChromosome Modification
modificationchildren
modified children
Modifications are stochastically triggered Operator types are:
Mutation Mutation Crossover (recombination)
91Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Mutation: Local ModificationMutation: Local ModificationBefore: (1 0 1 1 0 1 1 0)Before: (1 0 1 1 0 1 1 0)
After: (0 1 1 0 0 1 1 0)
Before: (1 38 -69 4 326 44 0 1)Before: (1.38 69.4 326.44 0.1)
After: (1.38 -67.5 326.44 0.1)
Causes movement in the search space(local or global)
Restores lost information to the population
92Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Crossover: RecombinationCrossover: Recombination*
P1 (0 1 1 0 1 0 0 0) (0 1 0 0 1 0 0 0) C1
P2 (1 1 0 1 1 0 1 0) (1 1 1 1 1 0 1 0) C2( ) ( )
Crossover is a critical feature of geneticalgorithms:
It greatly accelerates search early in g y yevolution of a population
It leads to effective combination of It leads to effective combination of schemata (subsolutions on different chromosomes)
93Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
chromosomes)
EvaluationEvaluation
evaluatedmodifiedchildren
evaluationchildren
Th l t d d h d The evaluator decodes a chromosome and assigns it a fitness measure
The evaluator is the only link between a classical GA and the problem it is solving
94Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
DeletionDeletionl ipopulation
discarded members
discard
discarded members
Generational GA: Generational GA:entire populations replaced with each iterationiteration
Steady-state GA:a few members replaced each generation
95Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
a few members replaced each generation
An Abstract ExampleAn Abstract Example
Distribution of Individuals in Generation 0f
Distribution of Individuals in Generation N
96Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
A Simple ExampleA Simple Example
“The Gene is by far the most sophisticated program around.”
- Bill Gates, Business Week, June 27, 1994
97Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
A Simple ExampleA Simple Example
The Traveling Salesman Problem:The Traveling Salesman Problem:
Find a tour of a given set of cities so thatFind a tour of a given set of cities so that each city is visited only once the total distance traveled is minimized
98Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
RepresentationRepresentationRepresentation is an ordered list of citynumbers known as an order-based GAnumbers known as an order based GA.
1) London 3) Dunedin 5) Beijing 7) Tokyo) ) ) j g ) y2) Venice 4) Singapore 6) Phoenix 8) Victoria
CityList1 (3 5 7 2 1 6 4 8)CityList2 (2 5 7 6 8 1 3 4)
99Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
CrossoverCrossoverCrossover combines inversion andrecombination:
* *Parent1 (3 5 7 2 1 6 4 8)Parent1 (3 5 7 2 1 6 4 8)Parent2 (2 5 7 6 8 1 3 4)
Child (5 8 7 2 1 6 3 4)
This operator is called the Order1 crossover.
100Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
MutationMutationMutation involves reordering of the list:
* *B f (5 8 7 2 1 6 3 4)Before: (5 8 7 2 1 6 3 4)
After: (5 8 6 2 1 7 3 4)
101Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
TSP Example: 30 CitiesTSP Example: 30 Cities
120
80
100
40
60y
0
20
00 10 20 30 40 50 60 70 80 90 100
x
102Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution (Distance = 941)Solution i (Distance = 941)TSP30 (Performance = 941)
100
120
TSP30 (Performance = 941)
80
100
40
60y
0
20
0 10 20 30 40 50 60 70 80 90 100x
103Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution (Distance = 800)Solution j(Distance = 800)4462 TSP30(Performance = 800)6269677864 100
120
TSP30 (Performance = 800)
6462544250
80
100
5040403821
40
60y
2135676060 0
20
40425099
0 10 20 30 40 50 60 70 80 90 100x
104Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution (Distance = 652)Solution k(Distance = 652)TSP30 (Performance = 652)
100
120
TSP30 (Performance = 652)
80
100
40
60y
0
20
00 10 20 30 40 50 60 70 80 90 100
x
105Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Best Solution (Distance = 420)Best Solution (Distance = 420)4238 TSP30Solution (Performance = 420)3835262135 100
120
TSP30 Solution (Performance = 420)
35327
3846
80
100
4644586069
40
60y
6976787169 0
20
67628494
0 10 20 30 40 50 60 70 80 90 100x
106Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Overview of PerformanceOverview of PerformanceT S P 30 - O verview o f P erfo rm an ce
1600
1800
T S P 30 - O verview o f P erfo rm an ce
1000
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nce
400
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0
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31B e st
W tG en erat io n s (1000)
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A ve ra g e
107Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Considering the GA TechnologyConsidering the GA Technology“Almost eight years ago ... lmost eight yea s ago ...people at Microsoft wrote
a program [that] uses some genetic things for
finding short code sequences Windows 2 0sequences. Windows 2.0
and 3.2, NT, and almost all Microsoft applications f ppproducts have shipped
with pieces of code created by that system.”
- Nathan Myhrvold, Microsoft Advanced Technology Group, Wired, September 1995
108Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Technology Group, Wired, September 1995
Issues for GA PractitionersIssues for GA Practitioners Choosing basic implementation issues:
representation population size, mutation rate, ... selection, deletion policies crossover, mutation operators
Termination Criteria Performance, scalability
Solution is only as good as the evaluation Solution is only as good as the evaluation function (often hardest part)
109Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Benefits of Genetic AlgorithmsBenefits of Genetic Algorithms Concept is easy to understand Modular separate from application Modular, separate from application Supports multi-objective optimization Good for “noisy” environments Always an answer; answer gets better Always an answer; answer gets better
with time Inherently parallel; easily distributed
110Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
B fit f G ti Al ith ( t )Benefits of Genetic Algorithms (cont.)
Many ways to speed up and improve a GA-based application as knowledge pp gabout problem domain is gainedEasy to exploit previous or alternate Easy to exploit previous or alternate solutions
Flexible building blocks for hybrid applicationsapplications
Substantial history and range of use
111Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
When to Use a GAWhen to Use a GA Alternate solutions are too slow or overly
complicated Need an exploratory tool to examine new
approaches Problem is similar to one that has already been
successfully solved by using a GA Want to hybridize with an existing solution Benefits of the GA technology meet key problem
requirements
112Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Some GA Application TypesSome GA Application TypesDomain Application TypesControl gas pipeline, pole balancing, missile evasion, pursuit
Design semiconductor layout, aircraft design, keyboardDesign semiconductor layout, aircraft design, keyboardconfiguration, communication networks
Scheduling manufacturing, facility scheduling, resource allocation
Robotics trajectory planning
Machine Learning designing neural networks, improving classificationalgorithms classifier systemsalgorithms, classifier systems
Signal Processing filter design
Game Playing poker, checkers, prisoner’s dilemmaGame Playing p , , p
CombinatorialOptimization
set covering, travelling salesman, routing, bin packing,graph colouring and partitioning
113Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
p
ConclusionsConclusions
Question: ‘If GAs are so smart, why ain’t they rich?’
Answer: ‘Genetic algorithms are rich - rich in application across a large and growing number of disciplines.’- David E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning
114Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial