Consider an n by n matrix, where each of the n² entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n² entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 1 47 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 18 21 23 26 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no linear time algorithm exists.

Consider an n by n matrix, where each of the n² entries is a positive integer. If the entries in this matrix are unsorted, then determining whether a target number t appears in the matrix can only be done by searching through each of the n² entries. Thus, any search algorithm has a running time of O(n²). However, suppose you know that this n by n matrix satisfies the following properties: • Integers in each row increase from left to right. • Integers in each column increase from top to bottom. An example of such a matrix is presented below, for n=5. 1 47 11 15 2 5 8 12 19 3 6 9 16 22 10 13 14 17 24 18 21 23 26 30 Here is a bold claim: if the n by n matrix satisfies these two properties, then there exists an O(n) algorithm to determine whether a target number t appears in this matrix. Determine whether this statement is TRUE or FALSE. If the statement is TRUE, describe your algorithm and explain why your algorithm runs in O(n) time. If the statement is FALSE, clearly explain why no linear time algorithm exists.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Concept explainers

Counting sort

The counting sort algorithm sorts an array’s contents by counting the repetition of each element that occurs in the array. A counting sort algorithm is a stable algorithm that works the best when the range of elements in the array is small, say for example from 0 to 9. The count which is the number of repetitions of an element in the array is mapped to an index of an auxiliary array, which is then saved in an auxiliary array. The counting sort algorithm has a stable time complexity of the order of O(size of array + size of auxiliary array). Let’s discuss the algorithm in detail as follows.

Question

Consider an n by n matrix, where each of the n2 entries is a
positive integer.
If the entries in this matrix are unsorted, then determining
whether a target number t appears in the matrix can only be done
by searching through each of the n2 entries. Thus, any search
algorithm has a running time of O(n²).
However, suppose you know that this n by n matrix satisfies the
following properties:
• Integers in each row increase from left to right.
• Integers in each column increase from top to bottom.
An example of such a matrix is presented below, for n=5.
4 7 11 15
2 5 8 12 19
3 6 9 16 22
10 13 14 17 24
1
18 21 23 | 26 | 30
Here is a bold claim: if the n by n matrix satisfies these two
properties, then there exists an O(n) algorithm to determine
whether a target number t appears in this matrix.
Determine whether this statement is TRUE or FALSE. If the
statement is TRUE, describe your algorithm and explain why your
algorithm runs in O(n) time. If the statement is FALSE, clearly
explain why no linear time algorithm exists.

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