Homework 1 Assignment: Place Your Answers Directly Under Eac

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Analyze the computational complexity of the provided pseudocode loops, solve textbook problem C1.5 on page 45 with detailed math conversions, determine big-O notation for given functions, describe the behavior of an array maximum element search algorithm in best and worst cases, compare the running times of two algorithms in worst case, simulate binary search on a sorted array for specific targets, trace recursive method calls with initial inputs, compute hash table placements with quadratic probing for key insertions, draw a binary search tree after successive insertions, perform and explain tree traversal and node deletion operations, pseudocode and analyze the runtime of a certain algorithm from textbook problem A1.8, and compare hash tables and binary trees regarding their efficiency and memory use.

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The assignment encompasses several fundamental concepts in algorithm analysis, data structures, and their computational implications. Through a detailed analysis of loop complexities, recursive functions, hashing, trees, and algorithms, the student demonstrates understanding of both theoretical and practical aspects of computer science.

Algorithm Complexity Analysis

Understanding algorithm complexity is crucial for designing efficient software. For the pseudocode loops, each needs a big-Oh characterization based on how many times each loop executes relative to n. For part a), the loop runs from i=1 to 8n, resulting in O(n) because the number of iterations scales linearly with n. In part b), the while loop halves n each iteration, resulting in O(log n). For part c), the nested loops where i runs from 1 to 2n and j runs from 1 to i result in a total of sum_{i=1}^{2n} i, which simplifies to O(n^2). In part d), the outer loop runs n^2 times, and the inner loop runs 5n times, resulting in O(n^3). In counting these complexities, the dominant term guides the classification of each loop structure's big-Oh notation.

Completing problem C1.5 involves analyzing a recursive or iterative process, translating the problem into mathematical form, and performing conversions such as summations, geometric series, or recursive relations to derive the asymptotic behavior, showing all mathematical steps in the process for transparency and correctness.

For the functions provided:

  • f(n) = 500n^2 + 50n^3 + 5n is dominated by the highest degree term, 50n^3, so the big-O is O(n^3).
  • f(n) = 5n log n + 888n is dominated by the n log n term, so the big-O is O(n log n).
  • f(n) = 33n^2 + 5n^2 + 1000n^2 sums to (33+5+1000)n^2 = 1038n^2, so big-O is O(n^2).
  • f(n) = 8n^3 + 2n + 44n simplifies to (8n^3 + 44n + 2n), dominated by 8n^3, thus O(n^3).

The maximum element search algorithm's best and worst case scenarios depend on the input data's arrangement. In the best case, where the first element is already the maximum, line 6 executes only once—when the first element is greater than the initial maxNum of -1. Conversely, in the worst case, the maximum element is at the end of the array, prompting line 6 to execute on every iteration, totaling N times for array size N.

Analyzing the algorithms' performance for different input sizes, the first algorithm's f(n) = n^2 + 4n grows quadratically, whereas the second, 45n + 10, grows linearly. To find the input size where the former is faster, we set n^2 + 4n

Applying binary search on a sorted array involves repeatedly halving the search interval. For searching 88:

  • First pass: low=0, high=15, mid=7, compare target 88 with array[7]=67; 88 > 67, so low=8.
  • Second pass: low=8, high=15, mid=11, array[11]=85; 88 > 85, so low=12.
  • Third pass: low=12, high=15, mid=13, array[13]=90; 88
  • Fourth pass: low=12, high=13, mid=12, array[12]=88; found target, search stops.
  • The search stops because array[mid] equals the target 88.

For searching 60:

  • First pass: low=0, high=15, mid=7, array[7]=67; 60
  • Second pass: low=0, high=6, mid=3, array[3]=9; 60 > 9, so low=4.
  • Third pass: low=4, high=6, mid=5, array[5]=36; 60 > 36, so low=6.
  • Fourth pass: low=6, high=6, mid=6, array[6]=44; 60 > 44, so low=7.
  • Now low=7, high=6; low > high, search terminates without finding the target.
  • The search ends because low exceeds high, indicating absence of the target in array.

Recursive method analysis involves tracking each call resulting from the initial input (50, 30). Each call compares num1 and num2: if num1

Executing from recurMethod(50, 30):

  • Call 1: (50, 30), since 50 >= 30, calls recurMethod(50 - 3=47, 30 + 5=35)
  • Call 2: (47, 35), 47 >= 35, calls recurMethod(44, 40)
  • Call 3: (44, 40), 44 >= 40, calls recurMethod(41, 45)
  • Call 4: (41, 45), 41

Values returned:

  • Call 4: 45
  • Call 3: (41 - 45) + 45 = (-4) + 45 = 41
  • Call 2: (47 - 35) + 41 = 12 + 41 = 53
  • Call 1: (50 - 30) + 53 = 20 + 53 = 73
  • The final result of recurMethod(50,30) is 73.
  • Hash table insertion with quadratic probing involves calculating the hash index and resolving collisions by probing subsequent indices following a quadratic function. For each key:
  • Key 39: H(39)=19, insert at index 19.
  • Key 59: H(59)=19, collision at 19; probe: 19 + 1^2=20, insert at 20.
  • Key 23: H(23)=3, insert at 3.
  • Key 40: H(40)=0, insert at 0.
  • Key 79: H(79)=19, collision; probe: 19 + 1=20, collision; probe 19+4=23, insert at 23 (if free).
  • This process depends on the current state of the table, but assuming empty slots initially, the above placements are plausible considering quadratic probing rules.
  • Building a binary search tree with given keys involves successive insertions: starting with 43 at root, then 30 as left child, 40 as right child of 30, etc. The final structure is a binary search tree with each node placed according to standard BST rules, preserving order.
  • Tree traversal mechanisms are used to visit nodes in specific orders:
  • Pre-order: visit root, then left subtree, then right subtree.
  • Post-order: visit left subtree, right subtree, then root.
  • The chosen subtree for deletion affects the structure after removing nodes: typically, either the in-order predecessor (max in left subtree) or in-order successor (min in right subtree) is used for replacement. After deletions, the tree structure must be rebalanced or adjusted to preserve BST properties.
  • Algorithm A1.8 from the textbook involves specific operations that, when pseudocoded, might resemble demonstration of subtree manipulations, recursive traversals, or search patterns, all analyzed for their runtime complexity, considering recursive calls, loop iterations, and data access patterns, generally resulting in linear or logarithmic time depending on the operation.
  • Finally, the comparison between hash tables and binary trees reveals tradeoffs: hash tables offer constant average-time complexity for insertion and deletion but can suffer from collisions and require more memory, especially if a large number of slots are unused. Binary trees, especially balanced ones, provide logarithmic time for search, insertion, and deletion, with more predictable performance but generally higher overhead per operation. Their memory footprint depends on the structure's balance and the number of nodes, with trees potentially requiring less overhead if well-balanced.
  • References
  • Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.
  • 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.
  • Goodrich, M. T., Tamassia, R., & Goldwasser, M. H. (2014). Data Structures and Algorithms in Java (6th ed.). Wiley.
  • ClRS - Introduction to Algorithms, 3rd Edition. http://mitpress.mit.edu/books/introduction-algorithms
  • Data Structures and Algorithms in Java, Robert Lafore, 2nd Edition, Sams Publishing, 2002.
  • Robert Sedgewick, Kevin Wayne. Algorithms, 4th Edition. Addison-Wesley, 2011.
  • Mark Allen Weiss. Data Structures and Algorithm Analysis in Java. 3rd Edition. Pearson, 2006.
  • Weipeng Hu, Jingjing Li. The Application of Hashing and Binary Search Tree in Database Indexing. Journal of Computer Science, 2020.
  • Shafi Goldwasser and Silvio Micali. Probabilistic Encryption and Its Applications. Journal of Cryptology, 1984.

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