Answer
See the explanation.
Work Step by Step
a.To determine which sequence does a better job of sorting a list of length three, let's analyze each sequence:
1. Sequence \( (1,3)(3,2) \):
- First, it compares the first and third elements, potentially swapping them if necessary.
- Then, it compares the third and second elements, potentially swapping them if necessary.
2. Sequence \( (1,2)(2,3)(1,2) \):
- It first compares the first and second elements, potentially swapping them if necessary.
- Then, it compares the second and third elements, potentially swapping them if necessary.
- Finally, it compares the first and second elements again.
Comparing the two sequences, the second sequence \( (1,2)(2,3)(1,2) \) does a better job of sorting a list of length three because it ensures that the list is fully sorted by performing additional comparisons and swaps.
b.To describe a genetic algorithm for developing a program that sorts lists of length ten, we can follow these steps:
1. **Initialization**: Generate an initial population of sorting programs (genomes). Each sorting program represents a potential solution to the sorting problem.
2. **Evaluation**: Evaluate the fitness of each sorting program in the population. Fitness can be determined by how well the program sorts lists of length ten compared to a predefined target sorting criterion (e.g., minimizing the number of incorrect elements or maximizing the number of correctly sorted elements).
3. **Selection**: Select sorting programs from the current population to serve as parents for the next generation. Programs with higher fitness values are more likely to be selected, but some degree of randomness is often introduced to maintain diversity.
4. **Crossover**: Perform crossover (recombination) between pairs of selected sorting programs to create offspring. This involves exchanging genetic material (sorting algorithm segments) between the parent programs to generate new sorting algorithms.
5. **Mutation**: Introduce random changes (mutations) to the offspring sorting programs to explore new regions of the search space. Mutations can involve altering individual sorting algorithm components or parameters.
6. **Replacement**: Replace the least fit individuals in the current population with the newly created offspring. This ensures that the population size remains constant across generations.
7. **Termination**: Repeat steps 2-6 for a fixed number of generations or until a termination condition is met (e.g., reaching a satisfactory solution or exhausting computational resources).
By representing sorting programs as genomes and applying selection, crossover, and mutation operators inspired by natural selection and genetic inheritance, genetic algorithms iteratively evolve and improve solutions to the sorting problem over successive generations.
