A) Introduction to the Simple Genetic Algorithm. B) Applications I: Applying Genetic Algorithms to Tackle NP-hard Problems.

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1 Genetic Algorithms Sami Khuri Department of Computer Science San José State University One Washington Square San José, CA U.S.A. khuri

2 Outline Outline A) Introduction to the Simple Genetic Algorithm. B) Applications I: Applying Genetic Algorithms to Tackle NP-hard Problems. C) Walsh and Haar Functions. D) Applications II: Other Applications of Genetic Algorithms: Bioinformatics. Tracking a Criminal Suspect. Genetic Programming. Aalto University: Summer

3 Introduction to GA [Gol89] Introduction to Genetic Algorithms GA s were developed by John Holland. GA s are search procedures based on the mechanics of natural selection and natural genetics. GA s make few assumptions about the problem domain and can thus be applied to a broad range of problems. GA s are randomized, but not directionless, search procedures that maneuver through complex spaces looking for optimal solutions. Aalto University: Summer

4 Introduction to GA [Gol89] Comparison with other optimization procedures Genetic Algorithm Traditional Algorithm works with coding of parameter set works with parameters themselves searches from population searches from a single of points point uses payoff information: objective function uses auxiliary knowledge: derivatives, gradients, etc... uses probabilistic transition values is deterministic Aalto University: Summer

5 Simple Genetic Algorithm [KB90] 1) Randomly generate m strings of length n each, to form the initial population: Pop(1). Let i 1 2) Compute fitness and relative fitness of each string in Pop(i). i i+1 3) Form mating pool from Pop(i) by spinning the weighted roulette wheel m times. no done? yes Solution: String in Pop(i+1) with best fitness value 4) Form Pop(i+1) by: a) randomly pairing the strings in the mating pool b) choosing crossing sites at random for each pair c) exchanging rightmost substrings of each pair d) performing bit wise mutation with a very small probability Aalto University: Summer

6 SGA Operators Example: Strings Fitness Relative Fitness Roulette Wheel: 15% 25% 20% 40% Mating Pool: Aalto University: Summer

7 Crossover and Mutation Crossover Choose 2 strings from mating pool. Choose a X-site: Children: Mutation: Flip a randomly chosen bit: Aalto University: Summer

8 The 0/1 Knapsack Problem The 0/1 Knapsack Problem Objects: 1... i... n Weights: w 1... w i... w n Profits: p 1... p i... p n Knapsack capacity: M Maximize n i=1 x i p i subject to n i=1 x i w i M, where x i {0,1} for 1 i n. Each string of the population represents a possible solution. If the j th position of the string is 1 i.e. x j = 1, then the j th object is in the knapsack; otherwise it is not. A string might represent an infeasible solution: the total sum of the weights of the objects exceeds the capacity of the knapsack. Aalto University: Summer

9 The 0/1 Knapsack Problem Problem Instance: Objects: Weights: Profits: Knapsack capacity : 80 String is of weight 46 and has a profit of 96. Two alternatives for the fitness functions are: f( x) = x i p i, i=1,n where x = (x 1, x 2,..., x n ) is a feasible solution; or f( x) = i=1,n x i p i penalty for any x, where infeasible strings are penalized. Aalto University: Summer

10 Sample Runs Aalto University: Summer

11 Sample Runs Aalto University: Summer

12 Algorithm GA Algorithm GA is t := 0; initialize P(t); evaluate P(t); while not terminate P(t) do t := t + 1; P(t) := select P(t 1); recombine P(t); mutate P(t); evaluate P(t); od end GA. Aalto University: Summer

13 Rank-Ordered Genetic Algorithm Encode the problem instance Form Pop(1) by: randomly generating m strings of length n computing the fitness value of each string sorting the strings from highest to lowest fitness value Set i 1 i done? i+1 no Randomly pick a pair of strings among the top 15% of Pop(i) Form Pop(i+1) by: yes Solution: strings with maximum fitness in Pop(i+1) choosing a crossing site at random for the chosen pair performing crossover computing the fitness values of the new strings placing the new strings in the sorted population deleting 2 randomly picked strings among the bottom 15% performing mutation Aalto University: Summer

14 Other Representations and Stochastic Operators Representations, Operators and Techniques: Integer, real and Gray code representations. Randomly generated and seeded initial populations. Inversion and mutation swap. 2-point, k-point, uniform, edge recombination, and partially mapped crossovers. Elitism. Aalto University: Summer

15 Partially Mapped Crossover Partially Mapped Crossover (PMX) PMX is used with permutations (eg. TSP). Parents: Choose 2 X-sites Children: Swap middle strings, then fill the rest Aalto University: Summer

16 The Schema Theorem Why does the GA work? H = is a schema, hyperplane partition, or similarity template describing { , ,...}. There are 3 l 1 schemata in the search space, where l is the length of the encoded string. Every string in the population belongs to 2 l 1 different schemata. Aalto University: Summer

17 The Schema Theorem The 3 and 4 dimensional Hypercubes [Whi94] Aalto University: Summer

18 The Schema Theorem Implicit Parallelism When sampling a population, we are sampling more than just the strings of the population. Many competing schemata are being sampled in an implicitly parallel fashion when a single string is being processed. The new distribution of points in each schema should change according to the average fitness of the strings in the population that are in the schema. Aalto University: Summer

19 The Schema Theorem Schema Properties Order: o(h) Number of fixed positions in the schema H. o( ) = 4. Defining length: δ(h) Distance between leftmost and rightmost fixed positions in H. δ( ) = 5. Aalto University: Summer

20 The Schema Theorem The Building Block Hypothesis When f(h) > f, the representation (instances) of H, in generation t + 1, is expected to increase, if δ(h) and o(h) are small. Aalto University: Summer

21 The Schema Theorem [Hol75] The Schema Theorem m(h, t + 1) m(h, t) f(h,t) f(t) [ 1 pc δ(h) l 1 p mo(h) ] where H : schema t : generation number f(h, t) : average fitness of H f(t) : fitness average of strings m(h, t) : expected number of instances of H δ(h) : defining length of H o(h) : order of H p c : crossover probability p m : mutation probability l : length of strings Aalto University: Summer

22 The Schema Theorem [Hol75] Limitations of the Schema Theorem Computing average fitnesses to evaluate f(h) at time t is often misleading in trying to predict f(h) after several generations. The Schema Theorem cannot predict how a particular schema behaves over time. Other models, of a more dynamic nature, have been proposed. Aalto University: Summer

23 Application Problems Applications I Maximum Cut Problem. Minimum Tardy Task Problem. Minimum Set Covering Problem. Terminal Assignment Problem. Edge Coloring Problem. Aalto University: Summer

24 Different Approaches: Applications: The Principles Knowledge-based restrictions on search space and stochastic operators. In the coming examples, only the fitness function is problem-dependent. Strategy for Fitness Functions: The fitness function uses a graded penalty term. The penalty is a function of the distance from feasibility. Infeasible strings are weaker than feasible strings. Note: The infeasible string s lifespan is quite short. Aalto University: Summer

25 Maximum Cut Problem [KBH94] Maximum Cut Problem Problem Instance: A weighted graph G = (V, E). V = {1,..., n} is the set of vertices and E the set of edges. w ij represents the weight of edge i, j. Feasible Solution: A set C of edges, the cut-set, containing all the edges that have one endpoint in V 0 and the other in V 1, where V 0 V 1 = V, and V 0 V 1 =. Objective Function: The cut-set weight W = i,j C w ij, which is the sum of the weights of the edges in C. Optimal Solution: A cut-set that gives the maximum cut-set weight. Aalto University: Summer

26 Maximum Cut Problem [KBH94] GA encoding for the Maximum Cut Problem: Encode the problem by using binary strings (x 1, x 2,..., x n ), where each digit corresponds to a vertex. The fitness function to be maximized is: f( x) = n 1 i=1 n j=i+1 w ij [x i (1 x j )+x j (1 x i )]. In absence of test problems of significantly large sizes use scalable test problems. Aalto University: Summer

27 Maximum Cut Problem [KBH94] Test Problems for the Maximum Cut Problem Consider randomly generated graphs of small sizes. The global optimum can be a priori computed by complete enumeration. Construct two graphs of n = 20 vertices each. Vary the probability of placing an edge between arbitrary pairs of vertices. Weights of edges are uniformly chosen in the range [0,1]. a) Sparse graph: cut Probability of an edge is 0.1 b) Dense graph: cut Probability of an edge is 0.9. Aalto University: Summer

28 Scalable Graph Aalto University: Summer

29 Maximum Cut Problem [KBH94] The Scalable Graph For n = 10, the string (or its complement) yields the optimum cut-set with value: f = 87. Graph can be scaled up, for any even n. The optimal partition is described by the (n 2)/4-fold repetition of the bit pattern 01, followed by a 0, then another copy of the (n 2)/4-fold repetition of the bit pattern 01, followed by a 0 if n = 4k + 2 and the n/2-fold repetition of 01 if n is a multiple of 4. The objective function value is: f = (n 4) for n 4. cut100 is of size n = 100. Aalto University: Summer

30 Maximum Cut Problem [KBH94] Runs for the Maximum Cut Problem: Recall: cut is a sparse graph, and cut is a dense graph. Perform a total of 100 runs with population size: 50. Use 200 generations for the small graphs, and 1000 generations for the large one. cut cut cut100 f( x) N f( x) N f( x) N Aalto University: Summer

31 Maximum Cut Problem [KBH94] Conclusions: Maximum Cut Problem 1. With the dense and the sparse graphs, the global optimum is found by more than two thirds of the runs. 2. For the dense and sparse graphs, the genetic algorithm performs 10 4 function evaluations, and the search space is of size Thus, the genetic algorithm searches less than one percent of the search space. 3. Similarly for cut100. The optimum is found in six of the runs. Average value from remaining runs: f = , which is about 5% from the global optimum. Only ( )/ % % of the search space is explored. Aalto University: Summer

32 The Minimum Tardy Task Problem Minimum Tardy Task Problem Problem instance: Tasks: n, i > 0 Lengths: l 1 l 2... l n, l i > 0 Deadlines: d 1 d 2... d n, d i > 0 Weights: w 1 w 2... w n, w i > 0 Feasible solution: A one-to-one scheduling function g defined on S T, g : S Z + {0} that satisfies the following conditions for all i, j S: 1. If g(i) < g(j) then g(i) + l i g(j). 2. g(i) + l i d i. Objective function: The tardy task weight W = i T S w i, which is the sum of the weights of unscheduled tasks. Optimal solution: The schedule S with the minimum tardy task weight W. Aalto University: Summer

33 The Minimum Tardy Task Problem Fact S is feasible if and only if the tasks in S can be scheduled in increasing order by deadline without violating any deadline. Example Tasks: Lengths: Deadlines: Weights: Aalto University: Summer

34 The Minimum Tardy Task Problem Example Tasks: Lengths: Deadlines: Weights: S = {1,3,5,6} S is feasible with W = 74 S = {2,3,4,6,8} g(4) + l 4 = and d 4 = 8 i.e. g(4) + l 4 > d 4 Therefore S is infeasible. Aalto University: Summer

35 The Minimum Tardy Task Problem Example Tasks: Lengths: Deadlines: Weights: Fitness Function String Term 1 Term 2 Term w 1 + w 5 + w 6 8 i=1 w i w 4 +w 7 + w w 1 + w 5 + w 7 8 i=1 w i w w 1 + w 2 + w w 6 + w 7 + w w 1 + w 5 + w 6 + w 7 8 i=1 w i w w 1 + w 5 + w 7 + w 8 8 i=1 w i w 4 Aalto University: Summer

36 GA Specifications The Minimum Tardy Task Problem S can be represented by a vector x = (x 1, x 2,..., x n ) where x i {0,1}. The presence of task i in S means that x i = 1. The fitness function to be minimized is the sum of three terms: f( x) = n i=1 w i (1 x i ) + (1 s) n i=1 w i + n i=1 w i x i 1 R + ( li + i 1 j=1 l jx j d i ). In the third term, x j is for schedulable jobs only. The whole term keeps checking the string to see if a task could have been scheduled. It makes use of the indicator function: { 1 if t A 1 A (t) = 0 otherwise. s = 1 when x is feasible, and s = 0 when x is infeasible. Aalto University: Summer

37 Scalable Problem Instance The Minimum Tardy Task Problem Construct a problem instance of size n = 5 that is scalable. Tasks: Lengths: Deadlines: Weights: To construct a MTTP instance of size n = 5t from above model: The first five tasks of the large problem are identical to the 5-model problem instance. The length l j, deadline d j, and weight w j of the j th task, for j = 1,2,..., n, is given by: l j = l i, d j = d i + 24 m and { wi if j 3 mod5 or j 4 mod 5 w j = (m + 1) w i otherwise, where j i mod 5 for i = 1,2,3,4,5 and m = j/5. The tardy task weight for the globally optimal solution of this problem is 2 n. Aalto University: Summer

38 The Minimum Tardy Task Problem Runs for the Minimum Tardy Task Problem mttp20 Population Size: 50 Number of Generations: 200 Crossover: Uniform Probability of Crossover: 0.6 Mutation: Swap Probability of Mutation: 0.05 A total of 100 runs is performed. f 1 ( x) d( x, opt) N Aalto University: Summer

39 mttp100 The Minimum Tardy Task Problem Scalable problem of size n = 100. Population Size: 200 Number of Generations: 100 Crossover: Uniform Probability of Crossover: 0.6 Mutation: Swap Probability of Mutation: 0.05 Note: About % of the search space is investigated. f 2 ( x) replaces infeasible strings by feasible ones. f 1 ( x) d( x, opt) N f 2 ( x) d( x, opt) N Aalto University: Summer

40 The Minimum Set-Covering Problem Minimum Set-Covering Problem Problem instance: A set of objects E = {e 1,..., e m } and a family of subsets of E, F = {S 1,..., S n }, such that n i=1 S i = E. Each S i costs c i where c i > 0 for i = 1,..., n. Or equivalently, a 0/1 m n matrix A = a a 1n a m1... a mn such that n j=1 a ij 1 for i = 1,2,..., m. Aalto University: Summer

41 The Minimum Set-Covering Problem Minimum Set-Covering Problem Feasible solution: A vector x = (x 1, x 2,..., x n ) where x i {0,1}, such that n j=1 for i = 1,2,..., m. a ij x j 1 Objective function: A cost function P( x) = n j=1 c j x j where x = (x 1, x 2,..., x n ) is a feasible solution. Optimal solution: A feasible vector that gives the minimum cost, i.e., a vector that minimizes the objective function. Aalto University: Summer

42 The Minimum Set-Covering Problem Minimum Set-Covering Problem Fitness Function B = n i=1 c i. U: the set of elements from E for which the subsets represented by x have failed to cover. The function considers the minimum cost required to transform the infeasible string into a feasible one: where and f( x) = s = n i=1 c i x i + s(b + m i=1 c i ) { 0, if x is feasible 1, if x is infeasible c i = { minei U{c j e i S j }, if e i U 0, if e i / U Aalto University: Summer

43 The Minimum Set-Covering Problem Minimum Set-Covering Problem Repair heuristic: Transform an infeasible string into a feasible one by changing x j = 0 to x j = 1 when c i = min e i U{c j e i S j } is calculated. The repaired string is always feasible; the fitness function in this case is simply f ( x) = n i=1 c i x i. Aalto University: Summer

44 The Minimum Set-Covering Problem GA Stochastic Operators and Test Problems The GA is used with the one-point, two-point, uniform, and fusion crossover operators. Fusion crossover: If a = (a 1, a 2,..., a n ) and b = (b1, b 2,..., b n ) are the parent strings with fitness f a and f b, then the offspring c = (c 1, c 2,..., c n ) is determined in the following fashion. for all i {1,..., n} if a i = b i then c i := a i ; else a c i := i with probability p = f b f a +f b b i with probability 1 p Aalto University: Summer

45 The Minimum Set-Covering Problem GA Stochastic Operators and Test Problems The crossover rate is 0.6 Mutation rate p m = 1/n. Population size λ = 100. Each experiment was run for 5,000 generations. Problem m n Density (%) a.1 a b.1 b c.1 c d.1 d e.1 e Aalto University: Summer

46 The Minimum Set-Covering Problem Experimental Results Prob 1-pt 2-pt unif fusion greedy bt kn a a a a a Aalto University: Summer

47 The Minimum Set-Covering Problem Experimental Results - Continued Prob 1-pt 2-pt unif fusion greedy bt kn b b b b b c c c c c d d d d d e e e e e Aalto University: Summer

48 Conclusions on the MSCP The Minimum Set-Covering Problem Repairing infeasible strings according to the minimum cost principle yields better results - the final values are close to the best known solutions. The repair procedure competes well with the greedy heuristic and yields better solutions in most cases. The choice of the crossover operator has no significant impact on the quality of the results. Our simple greedy heuristic always outperforms the genetic algorithm without repairing. (Table not shown here) Aalto University: Summer

49 Terminal Assignment Problem [KC97] Terminal Assignment Problem Problem Instance: Terminals: l 1, l 2,..., l T Weights: w 1, w 2,..., w T Concentrators: r 1, r 2,..., r C Capacities: p 1, p 2,..., p C Feasible Solution: Vector: x = x 1 x 2... x T x i = j: i th terminal assigned to j. Assign all terminals to concentrators without exceeding the concentrators capacities. Objective Function: A function Z( x) = T i=1 cost ij, where cost ij is the distance between terminal i and concentrator j. Optimal Solution: A feasible vector x that yields the smallest possible Z( x). Aalto University: Summer

50 Encoding for TA: Terminal Assignment Problem [KC97] Use non-binary strings: x 1 x 2... x T The value of x i is the concentrator to which the i th terminal is assigned. Example: l 8 r 3 l 9 l 2 l 3 l 7 l 1 l 6 l 5 l 10 r 1 l 4 r 2 Aalto University: Summer

51 Fitness Function: It is the sum of two terms: Terminal Assignment Problem [KC97] 1. the objective function which calculates the total cost of all connections. 2. a penalty function used to penalize infeasible strings. It is the sum of two terms: offset term: the product of the number of terminals and the maximum distance on the grid. graded term: the product of the sum of excessive load of concentrators and the number of concentrators that are in fact overloaded. Aalto University: Summer

52 Experimental Runs Experimental Runs Ten problem instances of 100 terminals each. Each concentrator has capacity between 15 and 25. The number of concentrators is between 27 and 33. Number of generations: 20,000 Population size: 500 Crossover rate: 0.6 Mutation rates: Between and 0.1 for LibGA and for GeneSyS. Aalto University: Summer

53 Experimental Runs The results for the Genetic Algorithm are the best obtained after 20,000 generations on each problem instance. Probably for 100 1, 100 2, and 100 3, the solutions obtained by LibGA (and Procedure Greedy) are the global optima. Problem Greedy Genetic Algorithms Instances Algorithm GENEsYs LibGA Aalto University: Summer

54 The Edge Coloring Problem [KWS00] The Edge Coloring Problem Problem Instance A graph G = (V, E), V = {v 1,..., v n } is the set of vertices, E = {e 1,..., e m } is the set of edges. Feasible Solution Vector y = y 1 y 2... y m, y i = j means that the i th edge e i is given color j, 1 j m. The components of y are essentially the colors assigned to the m edges of G, where: Two adjacent edges have different colors. The colors are denoted by 1, 2,..., q, where 1 q m. Aalto University: Summer

55 The Edge Coloring Problem Objective function For a feasible vector y, let P( y) be the number of colors used in the coloring of the edges represented by y. More precisely, P( y) = max{y i y i is a component of y}. Optimal solution A feasible vector y that yields the smallest P( y). Aalto University: Summer

56 The Edge Coloring Problem Applications Practical applications include scheduling problems, partitioning problems, timetabling problems, and wavelength-routing in optical networks. NP-Hard Holyer proved that the edge coloring problem is NP-hard. Definition is maximum vertex degree of G. χ (G): the chromatic index of G, is the minimum number of colors required to color the edges of G. Vizing s Theorem If G is a nonbipartite, simple graph, then χ (G) = or χ (G) = + 1. Aalto University: Summer

57 Bipartite Graphs Edge-Coloring Bipartite Graphs Journal of Algorithms, February 2000 Article by A. Kapoor and R. Rizzi: O(mlog 2 + m log 2 m log 2 2 ) SIAM, Journal on Computation, 1982 Article by Hopcroft and Cole: O(mlog 2 + m log 2 m log 2 3 ) Aalto University: Summer

58 The Genetic Algorithm The Genetic Algorithm Encoding: Use non-binary strings of length m (number of edges): y 1 y 2... y m. The value of y j represents the color assigned to the j th edge. For example, if a graph has 12 edges then the following string is a possible coloring, where edge e 1 has color number 5, edge e 2 has color number 3, and so on Aalto University: Summer

59 The Genetic Algorithm The GA Operators generational genetic algorithms roulette wheel selection uniform crossover uniform mutation Each string in the population occupies a slot on the roulette wheel of size inversely proportional (since the edge coloring problem is a minimization problem) to the fitness value. Aalto University: Summer

60 The Genetic Algorithm Fitness Function The fitness function of y = y 1 y 2... y m is given by: f( y) = P( y) + s[( + 1) + T( y)] It is the sum of two terms: 1. the objective function P( y), which calculates the total number of colors used to color the edges of the graph. 2. a penalty term used to penalize infeasible strings. It is the sum of two parts: The first part is the maximum vertex degree to which one is added, i.e., + 1. It is an offset term. The second part of the penalty term, T( y), is the number of violated pairs. Aalto University: Summer

61 Grouping Genetic Algorithm [Fal93] Grouping Problems Many problems consist in partitioning a given set U of items into a collection of pairwise disjoint subsets U j under some constraint. In essence, we are asked to distribute the elements of U into small groups U j. Some distributions (i.e., groupings) are forbidden. Some examples: Bin packing problem, terminal assignment, graph coloring. The objective is to minimize a cost function over the set of all possible legal groupings. Aalto University: Summer

62 Grouping Genetic Algorithm The Grouping Genetic Algorithm Encoding To capture the grouping entity of the problem, the simple (standard) chromosome is augmented with a group part. For example, the chromosome of our hypothetical 12-edge graph: A B 7 C D 9 10 E The colors are labeled with letters, the edges with numbers. The first half is the object part (the entire chromosome of the previous section), the second half is the group part. Aalto University: Summer

63 Grouping Genetic Algorithm Fitness Function The fitness function of a string considers the object part of the string only and computes the fitness value as explained before. Aalto University: Summer

64 Grouping Genetic Algorithm Crossover Operator Two parents are randomly chosen from the mating pool and two crossing sites are chosen for each parent: A B 7 C D 9 10 E A B 8 11 C 2 9 D E 4 5 Inject the entire group part between the two crossing sites of the second parent into the first parent at the beginning of its crossing site: A B 8 11 C 2 9 D B 7 C D 9 10 E Aalto University: Summer

65 Grouping Genetic Algorithm Dealing with Redundancy Resolve redundancy by eliminating groups B, C, and D from their old membership (i.e., from the first parent string): A B 8 11 C 2 9 D E Remove duplicate edges (8 and 11) from their old membership (from the first parent string): A 5 B 8 11 C 2 9 D E Use the first fit heuristic to reassign lost edges (6 and 7): assign the edge to the first group that is able to service it, or to a randomly selected group if none is available. Reversing the roles of the two parents allows the construction of a second offspring. Aalto University: Summer

66 Grouping Genetic Algorithms The Mutation Operator Probabilistically remove some group from the chromosome and reassign its edges. Each edge is assigned to the first group that is able to take it. After crossover and mutation are performed on the group part, the object part is rearranged accordingly to reflect the changes that the group part has undergone. Aalto University: Summer

67 Experimental Runs Benchmarks Twenty eight simple graphs with the number of edges between 20 and 2596: Five book graphs from the Stanford GraphBase. Two miles graphs from the Stanford GraphBase (denoted by mls). Nine queen graphs from the Stanford GraphBase (q). Four problem instances based on the Mycielski transformation from the DIMACS collection (mcl). Seven complete graphs (not shown in tables). Fifteen multigraphs generated by repeatedly assigning edges to randomly chosen pairs of vertices until each vertex has a fixed degree. We denote the multigraphs we generated by rexmg10 x, rexmg20 x, and rexmg30 x, where 10, 20 and 30 are the number of vertices and x is the degree and has the values of 10, 20, 30, 40 and 50. Aalto University: Summer

68 Test results for the simple graphs Test results for the GraphBase and DIMACS problem instances Problem Instances Greedy Algorithm LibGA GGA Name Vertices Edges Density Best Freq. Mean Best Freq. Mean anna david homer huck jean gs mls mls q q q q q q q q q mcl mcl mcl mcl Aalto University: Summer

69 Test results for the multigraphs Test results for the multigraph problem instances Problem Instances Greedy Algorithm GGA Name Vertices Edges Density Best Freq. Mean Best Freq. Mean rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg rexmg Aalto University: Summer

70 Walsh and Haar Functions Walsh and Haar Functions Walsh Functions are used: as basis for the expansion of fitness functions to compute the average fitness values of schemata to design deceptive functions for the genetic algorithm. Haar Functions. Comparison between Walsh and Haar Functions. Fast Transforms. Aalto University: Summer

71 Walsh Functions Walsh Transforms: Orthogonal, rectangular functions. Practical as bases for fitness functions of genetic algorithms. Work with ±1 rather than 0 and 1. Associate with x = x l x l 1... x 2 x 1 an auxiliary string y = y l y l 1... y 2 y 1 y = aux(x l )aux(x l 1 )... aux(x 2 )aux(x 1 ) where { 1 if xi = 1 y i = aux(x i ) = 1 if x i = 0 The Walsh functions over auxiliary string variables form a set of 2 l monomials defined for y = aux(x): Ψ j (y) = Π l i=1 yj i i where j = j l j l 1... j 1 and j = Σ l i=1 j i2 i 1. Aalto University: Summer

72 Walsh Functions: Basis Example In Ψ 3 (y) where l = 3, j = 3, and j 3 = 0, j 2 = 1, j 1 = 1. Ψ 3 (y) = y 0 3 y1 2 y1 1 = y 1y 2. If x = 101 then: Ψ 3 (aux(101)) = Ψ 3 (aux(1)aux(0)aux(1)) = Ψ 3 ( 11 1) = ( 1)(1) = 1 {Ψ j (x) : j = 0,1,2,...,2 l 1} forms a basis for the fitness functions defined on [0,2 l ). Consequently, any fitness function defined on binary strings of length l can be represented by a linear combination of dicrete Walsh functions. Aalto University: Summer

73 Walsh Functions: Basis Example continued f(x) = Σ 2l 1 j=0 w jψ j (x) w j : Walsh coefficients, Ψ j (x): Walsh monomials. The Walsh coefficients are given by: w j = 1 2 lσ2l 1 x=0 f(x)ψ j(x). Aalto University: Summer

74 Walsh functions on [0,8) Walsh functions on [0,8) x Ψ 0 (x) Ψ 1 (x) Ψ 2 (x) Ψ 3 (x) Ψ 4 (x) Ψ 5 (x) Ψ 6 (x) Ψ 7 (x) ind j j 3 j 2 j mono y3 0y0 2 y0 1 y0 3 y0 2 y1 1 y0 3 y1 2 y0 1 y0 3 y1 2 y1 1 y1 3 y0 2 y0 1 y1 3 y0 2 y1 1 y1 3 y1 2 y0 1 y1 3 y1 2 y1 1 y j 3 3 yj 2 2 yj y 1 y 2 y 1 y 2 y 3 y 1 y 3 y 2 y 3 y 1 y 2 y 3 Aalto University: Summer

75 Lowest-order Hadamard matrix: [ ] 1 1 H 2 =. 1 1 Recursive relation: H N = H N2 H 2 Hadamard Matrices : Kronecker product (also known as direct product). H N = H N 2 H N2 H N2 H N2 H 8 = Aalto University: Summer

76 Schema Average Fitness Average fitness value of a schema can be expressed as a partial signed summation over Walsh coefficients. A schema h (hyperplane): h l, h l 1,..., h 2, h 1 where h i {0,1, }. f(h) = 1 h Σ x hf(x). f(h) = 1 h Σ x hσ 2l 1 j=0 w jψ j (x) = 1 h Σ2l 1 j=0 w jσ x h Ψ j (x), The lower the order of the schema, the fewer the number of terms in the summation. Aalto University: Summer

77 Schema Average Fitness Schema Fitness Averages w 1 0 w 0 + w 1 1 w 0 w 4 01 w 0 w 1 + w 2 w 3 11 w 0 w 2 w 4 + w w 0 w 1 + w 2 w 3 + w 4 w 5 + w 6 w 7 Fitness averages as partial Walsh sums of some schemata. Aalto University: Summer

78 Easy and Hard Problems for the GA Goldberg s minimal deceptive functions using Walsh coefficients Two Types of Minimal Deceptive Functions. Fully 3 bit Deceptive Function: If 111 is the optimal point, then all order-one schemata lead away from it. In other words: f( 1) < f( 0), f( 1 ) < f( 0 ), f(1 ) < f(0 ). Similarly, for order-two schemata: f( 00) > f( 01), f( 00) > f( 10), f( 00) > f( 11). Aalto University: Summer

79 Goldberg s ANODE Analysis of Deception Goldberg used neighborhood arguments in: ANODE: analysis of deception: an algorithm for determining whether and to what degree a coding-function combination is deceptive. The diagnostic is problem-instance dependent. Aalto University: Summer

80 Haar Functions Haar Functions Functions were proposed by the Hungarian mathematician Haar in Haar got his PhD from Germany. Thesis title: Zur Theorie der Orthogonalen Funktionensysteme. Haar functions take values of 1, 0 and 1, multiplied by powers of 2. Aalto University: Summer

81 Haar Functions Haar Functions H 0 (x) = H 1 (x) = H 2 (x) = H 3 (x) =. H 2q +m(x) =. H 2l 1(x) = 1 for 0 x < 2 { l 1 for 0 x < 2 l 1 1 for 2 l 1 x < 2 { l 2 for 0 x < 2 l 2 { 2 for 2 l 2 x < 2 l 1 0 elsewhere in [0,2 l ) 0 for 0 x < 2 l 1 2 for (2)2 l 2 x < (3)2 l 2 2 for (3)2 l 2 x < (4)2 l 2 { ( 2) q for (2m)2 l q 1 x < (2m + 1)2 l q 1 ( 2) q for (2m + 1)2 l q 1 x < (2m + 2)2 l q 1 0 elsewhere in [0,2 l ) { ( 2) l 1 for 2(2 l 1 1) x < 2 l 1 ( 2) l 1 for 2 l 1 x < 2 l 0 elsewhere in [0,2 l ) For every value of q = 0,1,..., l 1, we have m = 0,1,...,2 q 1. Aalto University: Summer

82 Haar functions on [0,8) Haar functions on [0,8) Aalto University: Summer

83 Haar Functions Haar funcions on [0,8) x H 0 (x) H 1 (x) H 2 (x) H 3 (x) H 4 (x) H 5 (x) H 6 (x) H 7 (x) index j degree q undef order m undef Haar functions H 2q +m(x) for l = 3. Aalto University: Summer

84 Haar Functions The set {H j (x) : j = 0,1,2,...,2 l 1} forms a basis for the fitness functions defined on the integers in [0,2 l ). f(x) = Σ 2l 1 j=0 h j H j (x), h j s, for j = 2 q + m, are the Haar coefficients given by: h j = 1 2 lσ2l 1 x=0 f(x) H j(x). The higher q, the smaller the subinterval with non-zero values for H j (x). Each Haar coefficient depends only on the local behavior of f(x). Aalto University: Summer

85 Haar Functions Haar coefficients and Haar functions Haar Coefficients Every Haar coefficient of degree q has at most 2 l q non-zero terms. The linear combination of each Haar coefficient h j, where j = 2 q + m, has at most 2 l q non-zero terms. h 0 has at most 2 l non-zero terms. Haar Functions For any fixed value x [0,2 l ), f(x) has at most l + 1 non-zero terms. Aalto University: Summer

86 Comparison between Walsh and Haar Functions Total number of terms required for the computation of the expansion of f(1011): Walsh is 256 (16 16); Haar is 46 (16 each for h 0 and h 1 ; 8 for h 3, 4 for h 6, and 2 terms for h 13 ). For l = 20, there are about more terms using Walsh expansion. w j depends on the behavior of f(x) at all 2 l points. h j depends only on the local behavior of f(x) at a few points which are close together When used to approximate continuous functions: Walsh expansions might diverge at some points, whereas Haar expansions always converge. Aalto University: Summer

87 Comparison between Walsh and Haar Functions Computing the number of non-zero terms Non-zero terms Total num of non-zero terms Schema Walsh Haar Walsh Haar **** ***d **d* *d** d*** **dd *d*d d**d *dd* d*d* dd** *ddd d*dd dd*d ddd* dddd Total Computing schemata for l = 4 with Walsh and Haar functions. Aalto University: Summer

88 Comparison between Walsh and Haar Functions Maximum number of non-zero terms for d**d For example: d**d: E = {0**0,0**1,1**0,1**1}. Average fitness of 1**0 with Walsh functions: f(1**0) = w 0 + w 1 w 8 w 9. E has 4 schemata, the maximum number of non-zero terms for all schemata represented by d**d is 4 4 = 16 (2 nd column). Each Walsh coefficient requires 16 terms for its computation. So total number of terms required in the computation of the expansion of f(1**0) is 4 16; and that of all schemata represented by d**d, 4 64 (column 4). Aalto University: Summer

89 Comparison between Walsh and Haar Functions Maximum number of non-zero terms for d**d With Haar functions: f(1**0) = h 0 h (h 12 + h 13 + h 14 + h 15 ). The third column has: 4 6. Two terms are required for the computation of each of h 12, h 13, h 14, and h 15, while 16 are needed for h 0 and 16 for h 1. Forty terms are required in the computation of the expansion of the other 3 schemata in E, therefore, the total is Aalto University: Summer

90 Fast Transforms Fast Transforms By taking advantage of the many repetitive computations performed with orthogonal transformations, the analysis can be implemented to form fast transforms. These are in-place algorithms and were modelled after the fast Fourier transform The fast transforms can be implemented with: on the order of at most l2 l computations in the case of fast Walsh transforms; and on the order of 2 l for the fast Haar transforms. Aalto University: Summer

91 Fast Transforms Fast Haar Transform Aalto University: Summer

92 Fast Transforms The Fast Walsh-Haar Transform Aalto University: Summer

93 Application Problems Applications II DNA Fragment Assembly Problem. Tracking a Criminal Suspect. Genetic Programming. Aalto University: Summer

94 Applications II: DNA Fragment Assembly [SM97 DNA Fragment Assembly In large scale DNA sequencing, we are given a collection of many fragments of short DNA sequences. The fragments are approximate substrings of a very long DNA molecule. The Fragment Assembly Problem consists in reconstructing the original sequence from the fragments. Aalto University: Summer

95 The DNA Fragment Assembly Problem DNA Fragment Assembly It is impossible to directly sequence contiguous stretches of more than a few hundred bases. On the other hand, there is technology to cut random pieces of long DNA molecule and to produce enough copies of the piece to sequence. A typical approach to sequence long DNA molecules is to sample and then sequence fragments from them. The problem is that these pieces have to be assembled. Aalto University: Summer

96 The DNA Fragment Assembly Problem DNA Fragment Assembly In sequencing DNA, the approximate length of the target sequence is known within 10 percent. It is impossible to sequence the entire molecule directly. Instead, we obtain a piece of the molecule starting at a random position in one of the strands and sequence it in the canonical direction for a certain length. Each such sequence is called a fragment. Aalto University: Summer

97 The DNA Fragment Assembly Problem Fragments of a DNA Molecule Each fragment corresponds to a substring of one of the strands of the target molecule. We do not know: which strand the fragment belongs to, the position of the fragment relative to the beginning of the strand, if the fragment contains errors. Aalto University: Summer

98 The DNA Fragment Assembly Problem A large number of fragments are obtained by a sequencing technique known as the shotgun method. The reconstruction of the target molecule s sequence is based on fragment overlap. Fragment lengths vary from 200 to 700 bases. Target sequences are between 30,000 and 100,000 base-pairs. The total number of fragments is between 500 and The problem consists in obtaining the whole sequence of the target DNA molecule. In other words, piecing together a collection of fragments. Aalto University: Summer

99 The DNA Fragment Assembly Problem DNA Fragment Assembly [All99] Aalto University: Summer

100 Complicating Factors The DNA Fragment Assembly Problem DNA sequencing is very challenging since: Real problem instances are very large. Many fragments contain errors: Base call errors Chimeras Vector contamination The orientation of the fragments is frequently unknown; and both strands must be analyzed. There might be a lack of coverage. Aalto University: Summer

101 The DNA Fragment Assembly Problem Modelling the Fragment Assembly Problem Models of the fragment assembly problem: Shortest Common Superstring Reconstruction Multicontig None addresses the biological issues completely. They all assume that the fragment collection is free of contamination and chimeras. Aalto University: Summer

102 Encoding for the Fragment Assembly Problem Given fragments 0,1,2,..., n 1, I = f[0], f[1],..., f[n 1] is an ordering of the fragments, where f[i] = j denotes that fragment j appears in position i of the ordering. Since the strings are permutations of the fragments, operators that have traditionally been used with permutation problems, such as the Travelling Salesperson Problem, can be used. Aalto University: Summer

103 The DNA Fragment Assembly Problem Fitness Functions for the Fragment Assembly Problem [PFB95] Two fitness functions have been tested. 1. Function F 1 gives the overlap strength of directly adjacent fragments in the ordering I. F 1 (I) = n 2 i=0 w f[i],f[i+1] The genetic algorithm attempts to maximize F 1 (I), thus favoring layouts that contain adjacent fragments with strong overlaps. Aalto University: Summer

104 The DNA Fragment Assembly Problem Fitness Functions for the Fragment Assembly Problem 2. Fitness function F 2 attempts to do more than F 1. F 2 considers the overlap strenths among all pairs of fragments. It penalizes layouts in which fragments with strong overlap are far apart. F 2 (I) = n 1 i=0 n 1 j=0 i j w f[i],f[j] The genetic algorithm attempts to minimize F 2 (I) by using the absolute value of the distance of the fragments in the layout. Aalto University: Summer

105 The DNA Fragment Assembly Problem Problem Instances The fragment data sets where artificially generated from actual DNA sequences. DNA sequenceaccession numberlength in base pairs HUMATPKOI M HUMMHCFIB X HUMAPOBF M The parent DNA sequence was: 1. replicated by the program, 2. fragmented at random locations to derive a fragment set. Genetic Algorithms were successful with 10KB-parent sequences and with a seven-fold base coverage on the average. Aalto University: Summer

106 Other Applications: Tracking Criminal Suspect [CJ91] Tracking a Criminal Suspect Through Face-Space with a Genetic Algorithm by C. Caldwell and V. Johnston Aalto University: Summer

107 Other Applications: Tracking Criminal Suspect [CJ91] Tracking a Criminal Suspect Humans have excellent facial recognition ability. GA relies on the fact that recognition is stronger than recallection. Witness rates twenty chosen pictures according to its resemlance to the culprit. Twenty more pictures are shown after crossover and mutation, etc. Aalto University: Summer

108 Other Applications: Genetic Programming [Koz92] Genetic Programming The structures undergoing adaptation in Genetic Programming are general, hierarchical computer programs of dynamically varying size and shape. The search space consists of all possible computer programs composed of functions and terminals appropriate to the problem domain. Aalto University: Summer

109 Concluding Remarks Conclusion Simple Genetic. Grouping Genetic Algorithms. Genetic Programming. Other Evolutionary-based Algorithms: Evolution Strategies and Evolutionary Programming. Seeding the initial population seems to work well. More experimentation with repair mechanism. Hybrid Algorithms and Parallel Genetic Algorithms. Aalto University: Summer

110 Concluding Remarks Static nature of the analysis. More dynamic models of genetic algorithms are needed. What problems are suitable for Genetic Algorithms? Weak classification of problems between hard and easy. The study of Genetic Algorithms is still in its embryonic stage. Aalto University: Summer

111 References References 1. [All99] C. Allex Computational Methods For Fast and Accurate DNA Fragment Assembly, PhD, University of Wisconsin-Madison, [CJ91] C. Caldwell and V. Johnson. Tracking a Criminal Suspect through Face-Space with a Genetic Algorithm, Proceedings of the Fourth International Conference on Genetic Algorithms, San Diego, CA, 1991, pp [CW93] A. Corcoran and R. Wainwright. LibGA: A User-friendly Workbench for Ordered-based Genetic Algorithm Research, Proceedings of the 1993 ACM/SIGAPP Symposium on Applied Computing, ACM Press, 1993, pp [Fal93] E. Falkenauer. The Grouping Genetic Algorithms - Widening the Scope of Genetic Algorithms, Belgian Journal of Operations Research, Statistics and Computer Science, vol. 33, 1993, pp [Gol89] D. Goldberg. Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, NY, [Hol75] J. Holland. Adaptation in Natural and Artificial Systems, The University of Michigan Press, Ann Arbor, [KB90] S. Khuri and A. Batarekh. Heuristics for the Integer Knapsack Problem, Proceedings of the X th International Conference of the Chilean Computer Science Society, Santiago, Chile, 1990, pp Aalto University: Summer

112 References 8. [KBH94] S. Khuri, T. Bäck and J. Heitkötter. An Evolutionary Approach to Combinatorial Optimization Problems, Proceedings of the 22 nd ACM Computer Science Conference, ACM Press, NY, 1994, pp [KC97] S. Khuri and T. Chiu. Heuristic Algorithms for the Terminal Assignment Problem, Proc. of the 1997 ACM/SIGAPP Symp. on App. Comp., ACM Press, [KWS00] S. Khuri, T. Walters and Y. Sugono. A Grouping Genetic Algorithm for Coloring the Edges of Graphs, Proceedings of the 2000 ACM/SIGAPP Symposium on Applied Computing, ACM Press, [Koz92] J. Koza. Genetic Programming: On the Programming of Computers by Means of Natural Selection, Cambridge, MA, MIT Press, [SM97] J. C.Setubal and J. Meidanis. Introduction to Computational Molecular Biology, PWS Publishing Company, Boston, [PFB95] R.J. Parsons, S. Forrest, C. Burks. Genetic Algorithms, Operators, and DNA Fragment Assembly, Machine Learning 21(1-2), pp , [Whi94] D. Whitley. A Genetic Algorithm Tutorial, Statistics and Computing, vol. 4, 1994, pp Aalto University: Summer

113 More Information Where to go for More Information Books and Articles: 1. T. Back, D. Fogel, and Z. Michalewicz. Handbook of Evolutionary Computation, Oxford University Press, UK, D. Fogel. Evolutionary Computation, IEEE Press, NJ, Z. Michalewicz. Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, NY, 3 rd ed., M. Mitchell. An Introduction to Genetic Algorithms, The MIT Press, Cambridge, MA, C. Prince, R. Wainwright, D. Schoenfield, and T. Tull. GATutor: A Graphical Tutorial System for Genetic Algorithms, SIGCSE Bulletin, vol. 26, ACM Press, 1994, pp H.P. Schwefel. Evolution and Optimum Seeking, John-Wiley, NY, Aalto University: Summer