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Chapter 12: Problem 2

Apply the merge sort to each of the following lists. Draw the splitting and merging trees for each application of the procedure. a) \(-1,0,2,-2,3,6,-3,5,1,4\) b) \(-1,7,4,11,5,-8,15,-3,-2,6,10,3\)

Short Answer

Expert verified
The detailed solution will vary based on visualization used to depict trees. Generally, the merge sort involves repetitively splitting the initial list into smaller parts until only individual elements are left, then merging these elements in a sorted manner. This entire process can be illustrated through trees for a more thorough understanding.

Step by step solution

01

Splitting

For both list a) and b), start by splitting the lists into two halves. Continue splitting each half into two until we are left with lists containing one element each. We can illustrate this process through a tree-like structure, where the root of the tree will be the original list, and each leaf will be one-element lists. Pay close attention to the order in which elements are split.
02

Merging and sorting

Once we have our one-element lists, we start merging them back together in pairs, while also sorting them. This process can also be illustrated using a tree. Starting from each leaf (one-element list), we merge pairs of leaves to form the parent nodes. For each merge operation, the order of elements in the merged list should be sorted. This process is repeated until only one list remains at the root, which is the sorted list.
03

Draw the merge sort trees

For comprehensions, draw out the splitting and merging trees for each application of the merge-sort procedure on both list a) and b). Drawing out these trees can help visualize the sorting process and confirm that it's done correctly. Each leaf of the tree should be the single element list after splitting and each node in the tree should represent a merge operation resulting in a sorted list of its child nodes.

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Most popular questions from this chapter

Related to the merge sort is a somewhat more efficient procedure called the quick sort. Here we start with a list \(L: a_{1}, a_{2}, \ldots, a_{n}\), and use \(a_{1}\) as a pivot to develop two sublists \(L_{1}\) and \(L_{2}\) as follows. For \(i>1\), if \(a_{i}1\), have been processed, place \(a_{1}\) at the end of the first list. Now apply quick sort recursively to each of the lists \(L_{1}\) and \(L_{2}\) to obtain sublists \(L_{11}, L_{12}, L_{21}\), and \(L_{22}\). Continue the process until each of the resulting sublists contains one element. The sublists are then ordered, and their concatenation gives the ordering sought for the original list \(L\).

Let \(T=(V, E)\) be a tree with \(|V|=n \geq 2\). How many distinct paths are there (as subgraphs) in \(T\) ?

Let \(T=(V, E)\) be a tree where \(|V|=v\) and \(|E|=e\). The tree \(T\) is called graceful if it is possible to assign the labels \(\\{1,2,3, \ldots\), v the vertices of \(T\) in such a manner that the induced edge labeling - where each edge \(\\{i, j\\}\) is assigned the label \(|i-j|\), for \(i, j \in\\{1,2,3, \ldots, v\\}\), \(i \neq j\)-results in the e edges being labeled by 1,2 , \(3, \ldots, e\). a) Prove that every path on \(n\) vertices, \(n \geq 2\), is graceful. b) For \(n \in \mathbf{Z}^{+}, n \geq 2\), show that \(K_{1, n}\) is graceful.c) If \(T=(V, E)\) is a tree with \(4 \leq|V| \leq 6\), show that \(T\) is graceful. (It has been conjectured that every tree is graceful.)

Let \(T=(V, E)\) be a balanced complete \(m\)-ary tree of height \(h \geq 2\). If \(T\) has \(l\) leaves and \(b_{h-1}\) branch nodes at level \(h-1\), explain why \(l=m^{h-1}+(m-1) b_{h-1}\).

On the first Sunday of 1993 Rizzo and Frenchie start a chain letter, each of them sending five letters (to ten different friends between them). Each person receiving the letter is to send five copies to five new people on the Sunday following the letter's arrival. After the first seven Sundays have passed, what is the total number of chain letters that have been mailed? How many were mailed on the last three Sundays?
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