Write a brief justification to your answer to Q1-c1.

The following figure shows red-black tree (RBT) in which a square denotes a black node, a circle denotes a red node, and the NIL nodes are omitted. The number inside a circle/square is the key value of the corresponding node. The label (upper-case letter) next to a node is a pointer pointing to the memory location of the corresponding node. You should use the label when referring to a node.   (a) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. After BST insertion (before RBT insertion fixup), the parent of O is [a]   (b) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. After BST insertion (before RBT insertion fixup), is O the left child of its parent or the right child of its parent? Write LEFT or RIGHT. [b]   (c) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. After BST insertion (before RBT insertion fixup), which property of the RBT is violated? Select 0 if none of the properties is violated. [c]   (d) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. Then we perform insertion fixup if necessary. In the resulting RBT, what is the parent of node E? [d]   (e) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. Then we perform insertion fixup if necessary. In the resulting RBT, what is the color of node J? [e]   (f) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. Then we perform insertion fixup if necessary. In the resulting RBT, what is the left child of node O? [f]   (g) Suppose that we want to insert 33 into the RBT in the figure. We first allocate memory for a tree node O and set its color to red and its key to 33. Then we insert it into tree T as if inserting into a binary search tree. Then we perform insertion fixup if necessary. In the resulting RBT, what is the right child of node O? [g]     (h) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the right child of node A? [h]   (i) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the color of node M? Write either BLACK or RED. [i]   (j) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the left child of node M? [j]   (k) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the right child of node M? [k]   (l) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the color of node F? Write either BLACK or RED. [l]   (m) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the color of node N? Write either BLACK or RED. [m]   (n) Suppose that we want to delete node M (with key=60) from the RBT  in the figure. In the resulting RBT, what is the color of node G? Write either BLACK or RED. [n]

Given an unsorted array A of n distinct integers and an integer k, you need to return the k smallest integers in the array in sorted order, where k may be any integer between 1 and n. Suppose that you have the following three algorithms to solve this problem. A1: Sort the array in increasing order, then list the first k integers after sorting. A2: Build a min-heap from these n integers, then call Extract-Min k times. A3: Use the linear time selection algorithm to find the k-th smallest integer in the array, then partition the array about that number to obtain the k smallest numbers in the array, and finally sort the k smallest numbers. Assume that you are using mergesort as your sorting algorithm, and use the linear time build-heap algorithm to build the heap. Let T1(n, k) denote the worst-case running time of Algorithm A1. Let T2(n, k) denote the worst-case running time of Algorithm A2. Let T3(n, k) denote the worst-case running time of Algorithm A3. Analyze the worst-case running times of the algorithms. What is the asymptotic notation for T3(n, k)? Use the most accurate big-O notation in your answer. Note that k is between 1 and n. Hence k is nominated by n.

Write a brief justification to your answer to Q1-b1.

Match each thesis statement to its description: bad in what way or good.

Why is water a good solvent for organic chemistry and life on Earth? **Note: No partial credit – no exceptions** 

Repeated encounters with a musical work will help one to

Which relatively new instrument did Mozart revise Symphony No. 40 to include?

Which of the following best describes the melodies of the Classical period?

The deliberate upsetting of the normal pattern of accents is called