Prof Pitts 557a: Algorithms Fall 2019 Study Guide And Practi

Prof Pitts 557a: Algorithms Fall 2019 Study Guide and Practice Problems

The final exam will not be comprehensive. It will cover only material after the midterm. I will not have questions on the exam that deal with text searching algorithms. The material dealing with P, NP, NP-Hard, NP-Complete, problem reductions, etc., will not be covered on the exam.

See the questions below for examples of questions similar to those that may appear on the final exam. As with the midterm exam, the final exam will be closed book and closed note, but you may bring two sheets of paper with prepared notes to use during the final exam.

Paper For Above instruction

Add this list of values to the open hash table with 17 slots: 52, 85, 3, 21, 49, 34, 15, 20, 48, 11, 89, 78, 2, 33. What is the maximum number of key comparisons for a successful search? What is the average number of key comparisons for a successful search? For this hash table, the number of key comparisons for successful searches are: Value # Key Comps So, the maximum number of key comparisons for a successful search is 2 and the average is 1.3 key comparisons for a successful search. Repeat the previous question using closed hashing and linear probing. For this hash table, the number of key comparisons for successful searches are: Value # Key Comps So, the maximum number of key comparisons for a successful search is 10, while the average is 2.5 key comparisons per successful search. How has clustering affected the performance between the open and closed hashing tables? Clustering had a significant effect on maximum key comparisons; in the open hash table, 33 required 10 comparisons while only one was required in the closed hash table.

Add the following sequence of values to a red/black tree, showing all rotations/recolorings: 1, 2, 3, 4, 5, 10, 9, 8, 7, 6. Add these numbers to a regular binary search tree. How does the height of this tree compare with the height of the red/black tree? The height of the binary search tree in the worst case is linear, n-1=9 for 10 nodes. The red/black tree height is 4, which is approximately log₂n + 1. Red/black trees do not always exhibit minimal height, but they remain close to balanced.

Solve the coin selection problem for the set: 5, 7, 3, 2, 1, 6, 1. Using the dynamic programming method, from the provided F array, the selected coins are 1, 3, 5, and 7, summing to 17. Solve the robot coin collector problem by following the path from bottom right up to row 1, then left to column 1.

Using the given undirected graph, apply Prim's algorithm to find the minimum spanning tree. Repeat with Kruskal's algorithm, processing edges in order of increasing weight: (4, 7): 2; (5,8): 2; (2,7): 3; (3,7): 3; (4,5): 3; (2,4): 4; (1, 2): 5; (3,6): 5; (6, 8): 5; (6, 7): 6; (1, 7): 7; (4, 8): 7. Edges are added only if they do not form cycles in the spanning tree.

Use Dijkstra's algorithm from vertex 1 to find shortest paths to all other vertices in the graph.

Scenario analysis involves creating different scenarios in Excel to forecast results, such as varying shares purchased in a portfolio. For example, creating scenarios with 0, 200, and 300 shares for certain companies, and plotting total return vs. shares to analyze impacts.

Create a custom Excel function called Commisn with three inputs to calculate broker commissions based on purchase price and shares, with specified flat rates for larger commissions.

Develop two macros in Excel: Macro 1 creates a scatter plot of last prices vs. purchase prices, adds a regression line and equation, and sorts stocks by highest last price; Macro 2 plots last prices vs. purchase prices but with axes reversed, adds regression and sorts by highest purchase price.

Paper For Above instruction

The final examination in a course on algorithms, specifically for Prof. Pitts's 557A class in Fall 2019, emphasizes material covered after the midterm, explicitly excluding topics such as text searching algorithms, P versus NP, NP-hard, NP-complete problems, and problem reductions. The exam format is closed book and closed notes, with the allowance of two sheets of prepared notes. This specificity guides students to focus their preparation on the covered topics, including hashing techniques, data structures, algorithms, and their analysis.

Hashing and Clustering Impact Analysis

Open hashing with separate chaining involves storing keys in linked lists within each hash table slot. When inserting a series of values such as 52, 85, 3, 21, 49, 34, 15, 20, 48, 11, 89, 78, 2, 33 into a table with 17 slots, key comparisons during successful searches depend heavily on the length of chains. The maximum comparisons for success are 2, with an average around 1.3 comparisons, reflecting relatively efficient lookups due to minimal clustering.

In contrast, closed hashing with linear probing manages collisions differently, storing keys in contiguous slots. Insertions of the same values lead to longer probe sequences; maximum comparisons rise to 10, with an average of 2.5. Clustering significantly affects performance, especially in open hashing, where similar values tend to cluster, affecting maximum comparison counts. The open hash table experienced a chain requiring 10 comparisons for value 33, whereas closed hashing with linear probing kept most searches shorter, but with increased average comparisons overall.

The clustering phenomenon notably degrades performance in open hashing by creating long chains that necessitate many comparisons during successful searches. Closed hashing's strategy of probing sequential slots disperses keys more evenly, resulting in generally shorter search lengths despite occasional long probes under certain distributions. Hence, clustering influences maximum search comparisons more profoundly in open hashing, emphasizing the importance of collision management techniques in hash table design.

Red-Black Trees versus Binary Search Trees

When inserting the sequence 1, 2, 3, 4, 5, 10, 9, 8, 7, 6 into a red/black tree, rotations and recoloring operations preserve balance. Red/black trees maintain a height approximately proportional to log₂n + 1, which for 10 nodes results in a height of 4. Regular binary search trees, especially in unbalanced insertions, can degenerate into a linear chain with height 9, representing the worst-case scenario. This difference illustrates the efficiency of self-balancing trees like red/black trees in maintaining near-minimal height, which is crucial for optimizing search, insertion, and deletion operations.

Dynamic Programming for Coin Selection

The coin selection problem employs dynamic programming to determine the minimum number of coins needed to reach a target value. In the provided set {5, 7, 3, 2, 1, 6, 1}, the method involves constructing a table that iteratively finds optimal solutions for smaller amounts, culminating in the selection of coins {1, 3, 5, 7} totaling 17. This approach minimizes computation time compared to brute-force enumeration, particularly when extended to larger coin sets or target values, demonstrating the practicality of dynamic programming in optimal coin change problems.

Graph Algorithms for Minimum Spanning Trees and Shortest Paths

Prim’s and Kruskal’s algorithms serve to find minimum spanning trees (MSTs) in undirected weighted graphs. Prim's algorithm grows the MST by adding the smallest edge connecting the tree to a new vertex iteratively, ensuring no cycles. Kruskal's algorithm sorts edges by weight and adds them sequentially if they do not form cycles, utilizing a union-find structure. Applying these algorithms to the specified graph yields MSTs with minimal total edge weights, essential for network design and optimization.

Dijkstra's algorithm calculates shortest paths from a source vertex to all others, updating path lengths based on current estimates and exploring vertices in order of increasing distance. Its application from vertex 1 enables the derivation of shortest paths to each node, facilitating efficient route planning and network analysis.

Scenario Analysis and Excel Custom Functions

Scenario analysis, as used in financial modeling, involves creating multiple "what-if" scenarios to forecast outcomes under varying assumptions—such as changes in shares purchased or stock prices. For instance, varying the number of shares in a portfolio impacts total returns, which can be graphically represented, aiding decision-making. Such tools are invaluable in financial analysis to assess risk and potential gains.

Excel's custom functions extend its capabilities by automating repetitive calculations. The Commisn function computes broker commissions based on purchase price and shares, applying flat rates for larger commissions. This function streamlines calculations, reduces errors, and enhances productivity in portfolio management tasks.

Macros in Excel automate complex or repetitive tasks such as plotting data points, fitting regression lines, and sorting datasets. Creating macro buttons for these functions enhances user efficiency, supports data visualization, and ensures reproducibility—key qualities in data analysis and reporting workflows.

Conclusion

This comprehensive review of algorithmic strategies, data structures, and analytical tools reflects core competencies required for the final examination for Prof. Pitts's course. Emphasizing hashing techniques, balanced trees, dynamic programming, graph algorithms, and practical Excel applications, the material prepares students to analyze, implement, and optimize algorithms in diverse computational contexts, reinforcing both theoretical understanding and practical skills essential for advanced computer science studies and professional applications.

References

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• Sedgewick, R., & Wayne, K. (2011). Algorithms (4th ed.). Addison-Wesley.
• Knuth, D. E. (1998). The Art of Computer Programming, Volume 3: Sorting and Searching. Addison-Wesley.
• Tarjan, R. E. (1983). Data structures and network algorithms. SIAM.
• Goodrich, M. T., & Tamassia, R. (2014). Data Structures and Algorithms in Java (6th ed.). Wiley.
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• Ballard, G., & Madabhushi, K. (Eds.). (2018). Graph Algorithms in Computer Science. Springer.
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