Data Structures: Fall 2004

Finals Review

Draw figures, etc.

• Given a binary search tree, show the order in which nodes are visited in the following traversals (i) preorder, (ii) postorder, (iii) inorder, and (iv) level-order.
• Given a sequence of insert and delete operations on (i) a BST, (ii) an AVL tree, or (iii) a hash table (open addressing or chaining), draw a figure of the appropriate data structure.
• Given a sequence of insert and search operations on a self-adjusting BST of a (single rotations or move to root), draw the tree.
• Given a sequence of insert (push) and DeleteMax/DeleteMin (pop) on a Max/Min heap, draw the heap, using pointers and using an array.
• Given an array, draw figures to show the steps in the heap initialization algorithm.

Write code fragments

• Write code for the following: (i) tree traversals, (ii) insert, delete, and search on BST, AVL tree, self-adjusting BST, and hash tables, (iii) push, pop, and top on max and min heaps, (iv) heap initialization, (v) rotations on different types of BSTs, (vi) largest node, smallest node, etc, of a BST.
• Write an iterator class for a given class.
• Learn the three basic features of an STL priority_queue class (push, pop, and top -- these correspond to certain operations on a heap).

Times complexities

• Give the time complexities for performing various operations on all the data structures that we have studied in this class.
• Given an application, choose a suitable data structure for it, giving a rational justification for your choice.

Proofs and analysis

• Given a recursive function, give the time complexity of executing it as a function of its arguments.
• Derive the expected and worst case time complexities for insertion, deletion, and search on BST, AVL trees, self-adjusting BST, and hash tables.
• Derive bounds on the height, number of nodes, and number of leaves of binary trees, BST, AVL trees, and heaps.
• Derive the time complexity of push, pop, top, and initialization of a heap.
• Prove properties for different types of trees.
• Prove that algorithms with certain time complexities are impossible, using the fact that comparison base sorting cannot be performed in time O(n).
• Disprove the correctness of some statement, using a counterexample.
• Determine the index of the children and the parent of a node, in the array representation of a complete tree, which need not be a binary tree, and prove that our answer is correct, using induction.