Data Structures And Algorithm Analysis Spring 2020 Post Midt

Data Structures And Algorithm Analysisspring 2020 Post Midterm Exam

Analyze memory requirements of specific data structures, demonstrate sorting algorithms through step-by-step processes, explain theoretical concepts of Big-O and Big-Omega notation with definitions and interpretations, prove inequalities using formal definitions, understand fundamental principles governing stacks and queues with brief explanations, describe practical applications, analyze graph traversal algorithms like BFS and DFS—including traversal orders and distance calculations—and define core graph concepts. Additionally, demonstrate the process of merge sort, explain how to implement a queue using two stacks, and illustrate binary search tree construction from given sequences.

Paper For Above instruction

Memory Calculation for Data Structures

Understanding the memory footprint of data structures involves summing the storage sizes of their constituent elements. Each data type has a fixed size: an int takes 4 bytes, a float takes 4 bytes, and a char takes 1 byte. When structures contain arrays or nested structures, the total size equals the sum of their individual sizes, often with consideration for padding due to alignment requirements.

Structure 1: struct Structure1 {int a, b, c, d, e[10]; float f;};

  • a, b, c, d: 4 integers × 4 bytes = 16 bytes
  • e[10]: 10 integers × 4 bytes = 40 bytes
  • f: 1 float × 4 bytes = 4 bytes

Considering potential padding, this structure roughly consumes around 64 bytes.

Structure 2: struct Structure2 {int a[12]; float b[5];};

  • a[12]: 12 integers × 4 bytes = 48 bytes
  • b[5]: 5 floats × 4 bytes = 20 bytes

Approximate total: 68 bytes.

Structure 3: struct Structure3 {char a[10][12][4], b[5][5], c[10];}

  • a: 10×12×4 = 480 chars = 480 bytes
  • b: 5×5 = 25 chars = 25 bytes
  • c: 10 chars = 10 bytes

Total: approximately 480 + 25 + 10 = 515 bytes, possibly with padding for alignment.

Selection Sort Step-by-Step

Part a:

Initial array: [10, 2, 5, 8, 9, 1, 4, 7]

  1. Find minimum (1), swap with first element: [1, 2, 5, 8, 9, 10, 4, 7]
  2. Next minimum (2), already in position.
  3. Next minimum (4), swap with position 2: [1, 2, 4, 8, 9, 10, 5, 7]
  4. Next minimum (5), in position.
  5. Next minimum (7), swap with position 4: [1, 2, 4, 5, 9, 10, 8, 7]
  6. Next minimum (8), in position.
  7. Next minimum (7), swap with position 6: [1, 2, 4, 5, 7, 10, 8, 9]
  8. Sorted array: [1, 2, 4, 5, 7, 8, 9, 10]

Big-O Notation

The notation f(n) = O(g(n)) means that, beyond some input size threshold, the growth rate of f(n) is bounded above by a constant multiple of g(n). Formally, there exist positive constants C and n₀ such that for all n ≥ n₀, f(n) ≤ C · g(n). In simpler terms, f(n) does not grow faster than a constant times g(n) when n becomes large.

Big-Omega Notation

The notation f(n) = Ω(g(n)) indicates that, starting from some input size, the function f(n) grows at least as fast as a constant multiple of g(n). Formally, there exist positive constants C and n₀ such that for every n ≥ n₀, f(n) ≥ C · g(n). This describes a lower bound on f(n).

Proofs of Inequalities in Asymptotic Notation

To show that a function f(n) is bounded above or below by g(n), the typical approach involves using the formal definitions. For an upper bound (O(g(n))), demonstrate the existence of constants C and n₀ such that f(n) ≤ C · g(n) for all n ≥ n₀. Conversely, for a lower bound (Ω(g(n))), show constants where f(n) ≥ C · g(n) for large enough n. These proofs require careful manipulation of inequalities and the identification of appropriate constants based on the specific function forms.

Principles of Stack Operations

The core principle governing stacking elements is Last-In, First-Out (LIFO). This means that the most recently added element is the first to be removed, resembling a stack of plates where only the top plate can be accessed or removed.

An application of a stack is in function call management within programming languages, where function calls are pushed onto the call stack and popped off once completed.

Principles of Queue Operations

A queue operates on the First-In, First-Out (FIFO) principle, where elements are added at the rear and removed from the front. This ensures order preservation akin to a line of customers waiting at a checkout counter. An application of a queue is in scheduling processes in operating systems.

Graph Definitions

A graph is a collection of nodes (or vertices) connected by edges, which can be directed or undirected. It models pairwise relationships among entities.

A tree in graph theory is a connected acyclic graph; it has exactly one path between any two nodes, ensuring no cycles.

Graph Traversal Algorithms

Breadth-First Search (BFS) Starting at Node 4

Visited order reflects the BFS expansion: starting at node 4, traverse neighbors, then their neighbors, recording order and calculating distances based on edge count from the source. For example, if node 4 is at distance 0, its immediate neighbors are at distance 1, and so forth.

Starting at Node 5

Similarly, BFS starting from node 5 will explore neighbors sequentially, recording visit order and shortest distances from node 5 to all other nodes.

Depth-First Search (DFS) Starting at Node 4 and Node 5

DFS explores as deep as possible along each branch before backtracking. The order of node visits depends on the traversal stack and neighbor exploration order.

Merging and Sorting Arrays

Merge Sort Process

Merge sort splits the array into halves recursively until single-element arrays are achieved, then merges these sorted arrays stepwise, combining pairs while maintaining order. For example, dividing [16, 1, 15, 2, 14, 3, 13, 4,...] into smaller subarrays, then merging step-by-step while sorting.

Queues via Stacks and Binary Search Tree Construction

A queue can be implemented with two stacks by using one stack for enqueue operations and another for dequeue operations, transferring elements between stacks as needed. This simulates FIFO behavior with LIFO stacks.

Constructing binary search trees from sequences involves inserting values sequentially, each insertion positioning the node according to the binary search property: left child

Conclusion

This comprehensive review covers fundamental concepts in data structures, algorithms, graph theory, and computational complexity. Understanding these principles is essential for designing efficient algorithms and data management solutions, which underpin much of computer science and software engineering.

References

  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.
  • Knuth, D. E. (1998). The Art of Computer Programming, Volume 3: Sorting and Searching. Addison-Wesley.
  • Sedgewick, R., & Wayne, K. (2011). Algorithms, 4th Edition. Addison-Wesley.
  • Tarjan, R. (1983). Data Structures and Network Algorithms. SIAM.
  • Dasgupta, S., Papadimitriou, C., & Vazirani, V. (2008). Algorithms. McGraw-Hill.
  • Levitin, A. (2012). Introduction to the Design & Analysis of Algorithms (2nd ed.). Pearson.
  • Goodrich, M. T., & Tamassia, R. (2014). Data Structures and Algorithms in Java (6th ed.). Wiley.
  • Hennessy, J. L., & Patterson, D. A. (2011). Computer Architecture: A Quantitative Approach. Morgan Kaufmann.
  • Sutton, R. S., & Barto, A. G. (2018). Reinforcement Learning: An Introduction. MIT Press.
  • Roman, E. (2020). Introduction to Graph Theory. Springer.

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