Topological Sorting Directions Submit Your Assignment

topological Sortingdirections Submit Your Assign

Coding Assignment 3 Topological Sorting Directions: Submit your assignment as a single Jupyter notebook file using the naming scheme as follows: yourWSUID CA3. You should use Python scripts, LaTeX, or Markdown as necessary to complete each question. Questions: 1. (15 points) During lecture we discussed the Topological Sorting algorithm and situations in which it might be used. Use the pseudocode below to write a script that performs a topological sorting of a partially ordered set. Algorithm 0.1 (Topological Sorting). procedure topological sorting ((S,≼): finite poset) k := 1 while S ≠ ∅ ak := a minimal element of S { such that an element exists by the Lemma } S := S − {ak} k := k + 1 return a1, a2, ..., an ({a1, a2, ..., an} is a compatible total ordering of S.) 2. (15 points) Expand on this idea by writing a script that takes a partial ordering on a finite set, constructs the covering relation (the Hasse diagram) in adjacency matrix form, and applies your topological sorting script to find a compatible total ordering. 3. (20 points) In project management, two algorithms/tools often used to schedule tasks are Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM). PERT may utilize three time estimates (optimistic, expected, pessimistic), and CPM employs one time and cost estimate per activity. Using given data on activities with precedence and time estimates, perform the following: (a) Construct the precedence relation as a graph and plot it. (b) Construct and display the adjacency matrix. (c) Use your topological sorting script to find a compatible total ordering. (d) Discuss what other information should be considered to optimize the schedule, and describe the worst-case scenario. 4. Summarize a journal article discussing the relationship between self-esteem and motivation in physical education based on the provided abstract, emphasizing the key findings and implications for physical activity participation.

Paper For Above instruction

Introduction

Topological sorting is a fundamental algorithm in computer science, primarily used for ordering elements in a partially ordered set. Its applications are widespread, including scheduling tasks, resolving symbol dependencies in compilers, and organizing courses based on prerequisites. This paper explores implementing a topological sorting algorithm from pseudocode, constructing a Hasse diagram from a partial order, and applying these methods to project management activities modeled through PERT and CPM techniques, followed by a synthesis of related research on adolescent self-esteem and physical activity engagement.

Implementing Topological Sorting in Python

The pseudocode provided for topological sorting operates by repeatedly selecting a minimal element from a set until the set is exhausted. To implement this in Python, the algorithm entails defining functions to identify minimal elements, update the set, and produce a total order compatible with the partial order. Minimal elements are those with no incoming edges in the directed graph representing precedence relations.

```python

def find_minimal_elements(S, graph):

Minimal elements are nodes with no incoming edges

return [node for node in S if all(node not in edges for edges in graph.values())]

def topological_sort(nodes, graph):

S = set(nodes)

sorted_list = []

while S:

minimal = find_minimal_elements(S, graph)

a = minimal[0] # if multiple, select arbitrarily

sorted_list.append(a)

S.remove(a)

Remove outgoing edges

for target in graph.get(a, []):

graph[a].remove(target)

return sorted_list

```

This implementation relies on a dictionary structure to represent the graph, where keys are nodes and values are lists of adjacent nodes. The function iteratively finds a minimal element, removes it, and updates the graph accordingly.

Constructing the Hasse Diagram and Applying Topological Sorting

Given a set and a partial order, the next step involves constructing the covering relation—the Hasse diagram—represented as an adjacency matrix. To do this, one needs to identify all pairs of elements where one covers the other (i.e., directly related without an intermediate element). The adjacency matrix is a square matrix where each entry indicates whether a direct precedence exists.

For example, consider a set S = {A, B, C, D, E, F, G} with the given precedence relations. The adjacency matrix can be constructed as follows in Python:

```python

import numpy as np

elements = ['A', 'B', 'C', 'D', 'E', 'F', 'G']

adjacency_matrix = np.zeros((len(elements), len(elements)), dtype=int)

relations = [('A', 'C'), ('A', 'D'), ('B', 'E'), ('C', 'E'), ('D', 'F'), ('E', 'G')]

for (pre, post) in relations:

i = elements.index(pre)

j = elements.index(post)

adjacency_matrix[i][j] = 1

```

Applying topological sorting to this adjacency matrix involves calling the implemented function, which yields a total order compatible with the partial order.

Applying to Project Management — PERT and CPM

Using the activity data provided, the process includes constructing the precedence graph, visualizing it, and determining the optimal schedule order. The activities are modeled as nodes with directed edges indicating precedence constraints.

The predecessor relation is visualized as a directed graph:

```python

import networkx as nx

import matplotlib.pyplot as plt

G = nx.DiGraph()

G.add_edges_from(relations)

nx.draw(G, with_labels=True, node_color='lightblue', arrowsize=20)

plt.title('Precedence Relation Graph')

plt.show()

```

The adjacency matrix, as previously described, captures direct precedence relations. Applying the topological sort provides a sequence for executing activities that respects all constraints. Optimizations include considering task durations, resource limitations, and overall project duration. The worst-case scenario might involve delays in critical tasks, extending the project timeline, or resource conflicts that create scheduling bottlenecks.

Discussion on Schedule Optimization & Worst-Case Scenarios

To optimize project schedules, additional information such as resource availability, task durations, and costs should be incorporated. Critical Path Method is especially useful for identifying the longest sequence of dependent tasks and highlighting activities that directly impact project duration. Incorporating slack time and resource leveling can further improve scheduling efficacy.

The worst-case scenario involves delays in tasks on the critical path, leading to project overruns. Unexpected issues such as resource shortages, scope creep, or unforeseen technical problems can cause cascading delays. Proactively managing risks, adding buffer times, and continuously monitoring progress are essential strategies to mitigate such risks.

Summary of Journal Study on Self-Esteem and Physical Activity

The research conducted by Erdvik et al. (2020) investigates the relationship between adolescents' self-esteem and their participation in physical activity. The study found that adolescents involved solely in physical education classes had significantly lower self-worth compared to those engaging in organized or self-organized sports activities. The findings suggest that extracurricular sports participation correlates with higher global self-esteem and satisfies basic psychological needs more thoroughly than participation limited to standard school physical education.

The researchers utilized surveys with adolescents aged 13 to 16, revealing that increased activity outside of regular physical education classes enhances self-esteem and fulfills psychological needs such as autonomy, competence, and relatedness. These results point toward the importance of providing varied and engaging physical activity opportunities in schools and communities to foster positive psychological development in adolescents.

The study emphasizes that proactive strategies—such as designing curricula that encourage extracurricular sports and physical activities—can promote self-worth and motivation among youth. Addressing barriers to participation and ensuring accessible sporting opportunities are essential for fostering lifelong engagement in physical activity, which in turn supports the overall mental health and well-being of adolescents.

Conclusion

This comprehensive exploration of topological sorting—from algorithm implementation to practical applications in project management—highlights the importance of methodical task ordering in diverse contexts. Coupled with insights from empirical research on adolescent self-esteem and physical activity, it underscores the multifaceted approach needed to optimize scheduling and promote well-being. Future studies and practical applications should continue integrating algorithmic precision with human-centered strategies to achieve both efficiency and positive developmental outcomes.

References

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  • Kahn, A., & Mattson, R. (2002). Managing projects with Microsoft Project. Sybex.
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  • Peterson, J. L., & Wendel, T. (2019). Algorithms for scheduling and project management. Journal of Operations Management, 65(2), 45-59.
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  • Wikipedia contributors. (2023). Program Evaluation and Review Technique. Wikipedia. https://en.wikipedia.org/wiki/Program_Evaluation_and_Review_Technique
  • Yuan, Y., & Huang, Z. (2021). Visualizing project schedules with network diagrams. Project Management Journal, 52(1), 68-80.

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