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EECS 281: Data Structures and Algorithms

Fall 2024

Lab 3: Complexity Analysis, Amortization, and Strings

1 Logistics

1. What is the due date of lab 2 quiz and AG?

A. 9/21/2024

B. 9/23/2024

C. 9/19/2024

D. 9/25/2024

2. What is the due date of lab 3 handwritten?

A. 9/24/2024

B. 9/27/2024

C. 9/23/2024

D. 9/26/2024

3. What is the due date of lab 3 quiz and AG?

A. 9/28/2024

B. 9/26/2024

C. 10/04/2024

D. 9/30/2024

4. What is the due date of Project 2?

A. 10/6/2024

B. 10/10/2024

C. 10/12/2024

D. 10/14/2024

2 Recurrence Relations and Complexity Analysis

5. What is the best approach for deducing the following relation, given T(1) = 1?

T(n) = 8T(2/n) + 3n

A. The Substitution Method

B. The Master Theorem

6. What is the best approach for deducing the following relation, given T(1) = 1?

T(n) = 2T(n − 1) + 2

A. The Substitution Method

B. The Master Theorem

7. Given the function below, calculate the recurrence relation. Assume bar(n) runs in Θ(n) time.

A. T(n) = T(4/n) + n3log n + T(2/n) + n2

B. T(n) = T(4/n) + n3log n + nT(2/n) + n2

C. T(n) = T(4/n) + n4 + nT(2/n) + n2

D. T(n) = T(4/n) + n4log n + nT(2/n) + n2

8. Which ordering lists Θ(n 100), Θ(100n ), Θ(n!), Θ(log(n!)) in order of increasing complexity?

Hint: Θ(log(n!)) can be simplified to a complexity class that you’ve likely seen before.

Think about any mathematical identities that may help you perform this simplification.

A. Θ(n 100), Θ(100n ), Θ(log(n!)), Θ(n!)

B. Θ(100n), Θ(n 100), Θ(log(n!)), Θ(n!)

C. Θ(log(n!)), Θ(n 100), Θ(n!), Θ(100n)

D. Θ(log(n!)), Θ(n 100), Θ(100n ), Θ(n!)

E. Θ(log(n!)), Θ(100n ), Θ(n 100), Θ(n!)

9. Solve the following recurrence relation:

A. 281n + 370

B. 281n + 651

C. 370n + 281

D. 370n - 89

E. 370n - 459

10. Consider the following recurrence relation:

What is the tightest complexity class that you can attribute to this recurrence relation?

A. T(n) = Θ(√n)

B. T(n) = O( √n)

C. T(n) = Ω(log2n)

D. T(n) = Θ(n log2n)

E. T(n) = Θ(n2)

11. What is the worst-case complexity of this function?

void foo ( vector & v ) {

int n = static_cast ( v . size ());

int rt = static_cast ( floor ( sqrt ( n )));

for (int i = 0; i < rt ; ++ i ) {

for (int j = 0; j < rt * rt ; j += rt ) {

cout << v [ j + i ] << " ";

}

cout << endl ;

}

}

A. Θ(√n)

B. Θ(n)

C. Θ(n√n)

D. n log n

E. n 2

12. What is the worst-case time complexity of this function?

void potato ( int n ) {

for (int i = 1; i < n ; i *= 2) {

for (int j = 1; j < n ; ++ j ) {

for (int k = 1; k < j ; ++ k ) {

cout << "I'm a smart potato ! o(^ -^)o \n";

}

A. Θ(n2)

B. Θ(n2log n)

C. Θ(n log2 n)

D. Θ(n 2 log2 n)

E. Θ(n 3 )

3 Vectors and Strings

13. What is the worst-case time complexity of the push back operation discussed in the lab slides, given a vector of size n?

A. Θ(1)

B. Θ(log n)

C. Θ(n)

D. Θ(n2)

14. What is the amortized time complexity of the push back operation discussed in the lab slides, given a vector of size n?

A. Θ(1)

B. Θ(log n)

C. Θ(n)

D. Θ(n 2)

15. Consider the following snippet of code:

char str1 [] = " BING ";

cout << strlen ( str1 );

cout << sizeof ( str1 );

string str2 (" BING ");

cout << str2 . length ();

cout << str2 . size ();

What is the output?

A. 4444

B. 4455

C. 4544

D. 5544

E. None of the above is correct.

16. Consider the following snippet of code:

const char * s = " hello ";

char ss [20];

int length = strlen ( s );

for (int i = 0; i < length ; ++ i )

ss [ i ] = s [ length - i ];

printf ("%s", ss );

What is the output?

A. hello

B. olleh

C. olle

D. segmentation fault

E. no output is printed

4 Handwritten Problem

This problem is to be submitted independently. We recommend trying it on your own, checking your answer with a group and discussing solutions, and then submitting it in lab. These will be graded on completion, not by correctness. However, we want to see that you were thinking about the problem. Please implement your solution on the lab assignment template, and you may test the solution using anagram.h. The starter files can be found on Canvas.

17. Write a function that takes in two strings and returns whether they are anagrams of each other (words that contain the same letters). The only characters will be spaces and lowercase letters. Do this in Θ(n) time.

• Example 1: Given s1 = “anagram” and s2 = “nagaram” , return true.

• Example 2: Given s1 = “i love eecs” and s2 = “i scole ve e” , return true.

• Example 3: Given s1 = “anagrams” and s2 = “anagrams anagrams” , return false.

• Example 4: Given s1 = “cats” and s2 = “cat” , return false.

// check if two strings are anagrams

bool isAnagram ( const string & s1 , const string & s2 );

5 Coding Assignment

You can work on these problems by yourself or with your group, but a solution must be submitted to the autograder for each individual.

18. String Library Implementation

The following specs contains every detail about the problem - if you want a quick TL;DR summary, see appendix A.

In lab and lecture, you learned about C++ strings and the C++ string library. The string class provides many helpful member functions that simplify the usage of string objects (such as find, insert, erase, just to name a few!). While you do not have to fully understand how a string object works under the hood, knowing these string operations will be helpful for future projects and interviews. In this lab assignment, you will implement five of these operations within a String class that we have provided for you. You will be given three submissions to the autograder per day (but we will give you the autograder’s test cases; this will be covered later).

Download the starter files from either Canvas or Github (https://github.com/eecs281staff/l3-string-library). Please complete the assignment in the String.cpp file. Do not include any libraries beyond the ones given to you or modify anything outside the parts labeled with TODO. Doing so may cause your program to fail on the AG. To clone the directory on to your local machine, go to your terminal and type the following command: git clone https://github.com/eecs281staff/l03-string-library.git

You will be implementing a portion of this String class so that its behavior emulates that of a standard C++ string. Before you begin, there are a few things you should know. Please read the following items carefully:

• On lines 162 and 163 of the original String.h file, you will see two member variables: cstr and sz. cstr is a c-string member variable that stores the underlying contents of the String object. Recall that c-strings are arrays of characters that are terminated by a null-character, or sentinel (’\0’). The variable sz stores the size of the String object, not including the null-terminating character. Make sure that both variables are updated correctly after each operation.

• The null-terminating character (’\0’) is defined as the member variable a null byte (on line 29 of the StringVerifier.cpp file). You may compare a character with a null byte or assign a character to a null byte. Remember that this character must be found at the end of the c-string representing the contents of the String object, as it denotes the end of the c-string.

• Strings have a constant value called npos that represents the largest possible value for an element of type size t (or 18,446,744,073,709,551,615). This is defined on line 28 of the StringVerifier.cpp file. This constant is often returned by find operations if the target is not found in the string.

• You won’t have to worry about illegal operations (e.g. indexing out of bounds, etc.) or growing the c-string array if an operation causes its contents to exceed its capacity, although it is important to note that such situations can exist and must be accounted for.

• For String operations that require returning a reference to a String, you should return *this in the function implementation. this is a pointer to the current String, so *this dereferences the pointer and returns a reference to the current String.

You may notice that the starter file we gave you supports many string operations. For this lab assignment, you will only be responsible for five of these functions.

• erase

• insert

• replace

• find first of

• find last of

Other than the information presented above, you may skip over everything that doesn’t have TODO attached to it. However, feel free to look over these other functions if you are curious! It is recommended that you review the “Arrays and Containers” lecture before starting this assignment. Go to the first TODO in the starter file (search the document for “TODO #1,” which should be located on line 186 of the String.cpp file if the original file was not modified).

TODO #1: Erase Function

The erase() function is defined as follows:

String & String :: erase ( size_t pos , size_t len = npos );

This function can be used to remove a range of characters within a String. When erase() is called using two parameters, pos and len, this function erases the portion of the String that begins at index pos and spans len characters, or until the end of the String, whichever comes first. If len is not defined, it defaults to a value of npos, and the String is erasedfrom pos to the end. A reference to the modified String is then returned. For example:

String str = " darden ";

str . erase (3 , 2);

erasing would begin at index 3 of str, and 2 characters would get erased. In this case, the second ’d’ (the character at index 3) and the following ’e’ would be erased, and the contents of str after the call to erase would be ”darn”. You may assume that the starting position pos will always be valid, and len will be greater than 0. A valid pos can take on any value in the inclusive range 0 ≤ pos ≤ sz.

Make sure that the final c-string you end up with has a sentinel at the end. This can be done by assigning a character in the c-string with a null byte, a member variable that has a value of ’\0’:

cstr [ last_index ] = a_null_byte ;

where last index is the index one past the last character of the modified String.

One thing to note is that the underlying c-string that is used to store the contents of the String is an array of characters, which stores elements contiguously in memory. Thus, once you erase a portion of the array, all the elements after the point of erasure must be shifted over so that all the characters can remain contiguous in memory.

To accomplish this, you may use the following algorithm:

1. First, declare an index that references the index of the first character to be erased. This can be done by accessing the element at position pos of the c-string cstr.

2. Next, declare an index that references the index of the first character NOT to be erased. This can be done by accessing the element at position pos + len of the c-string cstr.

3. After declaring both indices, overwrite each character at the first index with the char-acter at the second index, then increment both pointers, until removal is complete. Removal is complete when the second index reaches the end of the string. Make sure the last index of the final modified string is set to the sentinel character, a null byte (or ’\0’).

4. Adjust the value of sz so that it reflects the new size of the modified String.

This process is shown visually on the next page. Note that the following figure uses the word “pointer” instead of “index” to represent positions in the c-string array. However, this does not mean that you must use pointers to implement the erase operation. It is more than possible to use indices to keep track of the two positions needed to perform the erase. In other words, instead of keeping track of a pointer to the address of cstr[2], for example, you could just keep track of the element’s integer index, or 2.

The erase function, visualized: In the above diagram, after erase is complete, the contents

of the c-string are ’d’,’a’,’r’,’n’,’\0’,’n’,’\0’. Are we done here, or do we have to remove the last ’n’ and sentinel character at the end? Think about what the purpose of the sentinel character is in a c-string. Do the data elements after this character matter? No, it does not! This is because the sentinel tells the program that it has reached the end of the string, so all characters after this sentinel character are ignored.

An alternative solution for those who want a challenge: erase can also be implemented using the copy-swap method. This would require initiating a String object with the updated contents (after the necessary characters are removed) and swapping it with the current String (using the String swap function we defined for you on line 80). To use the copy-swap method to erase characters from a string, you will need to use the String substr function, which is implemented for you on line 165. Note that this copy-swap implementation is completely optional, and the algorithm listed on the previous page is sufficient for getting full credit on the erase function.

TODO #2: Insert Function

Next, we will look at the insert() function, which can be used to insert characters into a String. The insert() function is defined as follows:

String & String :: insert ( size_t pos , const String & str );

When insert() is called using two parameters, pos and str, this function inserts the contents of str BEFORE the character at position pos. A reference to the modified String is returned. For example:

String str = " shelf ";

str . insert (0 , " book ");

the string ”book” would be added before position 0 of str, resulting in a final str value of ”bookshelf”. You may assume that the starting position pos will always be valid.

Since insertion may cause the underlying c-string to exceed its capacity, a check needs to be done to ensure that there is enough memory allocated to hold the new String once everything has been inserted. This check has been provided for you; you do not need to worry about it, but do know that it exists.

How would you begin implementing this insert operation? Recall how an array works when you try to insert an element - all other elements after the insertion point must be shifted. This is true here as well. First, you must shift all characters after the insertion point by a certain distance. Then, you would insert the contents of the new String into the slots you have freed up for insertion. Use the figure below as a reference.

Once again, make sure that your new c-string includes the sentinel at the very end, and that the size is properly updated after the insert is complete.

For those who want to explore copy-swap (optional): insert can also be implemented using the copy-swap method. This would require initiating a String object with the updated contents (after the necessary elements are added) and swapping it with the current String (using the String swap function we defined for you on line 80). Like erase, you would need to use the substr function, which has been implemented for you. As an additional hint, the new String has three “sections” to it, the portion before the point of insertion, the portion after the point of insertion, and the new String that is added, so you will need to combine all three sections before you initiate the copy-swap.

One note that should be made is that you may come across a few test cases where a string is inserted into itself. In these cases, it would not be safe to begin modification of cstr immediately, since the insertion may depend on the original value of the String. There are several ways to bypass this. One way is to check if the address of the String to insert (&str) is equal to the address of the current String (this), and to insert differently if they are. Another method is to simply implement insert in a way that guarantees that str always retains its original value throughout the life of the function (e.g. making a copy of str and using the copy to do the insert).

TODO #3: Replace Function

Next, you will implement the replace() function. The replace() function is defined as follows:

String & String :: replace ( size_t pos , size_t len , const String & str );

When replace() is called using three parameters, pos, len, and str, this function replaces the portion of the String that begins at character pos and spans len characters with the contents of str. For example:

String str = " EECS 281 is hard ";

str . replace (12 , 4 , "fun ");

the substring of length 4 starting at position 12 of str (”hard”) would be replaced with the string ”fun”. The final contents of str after the call to replace would be ”EECS 281 is fun”. You may assume that pos is valid. If the value of len exceeds the end of the String, replace as many characters as possible.

The implementation of replace should not take more than 10 lines of code! This is because you have already implemented erase and insert; a call to replace is simply a combination of these two operations.

If you aren’t sure that your erase or insert are fully functional, an alternative (but longer) approach would be to implement the function from scratch. Shift all the characters after the segment to replace, and move the new String into the space you opened up. If str is shorter than len, the replaced section will be shorter than it was before, and if str is longer than len, the replaced section will be longer than it was before.

Like with insert, you will have to deal with self-replacement. To solve this, use an approach similar to the one you used to overcome self-insertion; either check for the case where &str == this (the String being passed in is the same as the String you are modifying) or implement your function in a way that guarantees that the value of str stays valid throughout the entire replace call.

TODO #4: Find-First-Of Function

Now, we will discuss functions that can be used to find specific characters in a String object. The find first of() function is defined as follows:

size_t String :: find_first_of ( const String & str , size_t pos = 0);

This function checks if ANY characters of str can be found in the String. In other words, given two parameters str and pos, this function searches the String for the first character that matches ANY of the characters in str, starting from position pos of the String. Characters before pos are ignored. It is enough for a single character of str to match for the search to be successful. If a match is found, the function returns a size t that represents the position of the first character that matches. Otherwise, the function returns npos. If pos exceeds the length of the string, the function will never find a match. If pos is not specified, the value of pos is assumed to be 0.

As long as a single character in str can be found in the String, the search is successful and returns the position of the match, as demonstrated in the examples below:

String str = " EECS 281 is fun";

size_t found = 0;

found = str . find_first_of ("281", 0);

// found is 5 since the char '2' can be found at index 5

found = str . find_first_of ("280", 0);

// found is 5 since the char '2' can be found at index 5

found = str . find_first_of ("281", 6);

// found is 6 since the char '8' can be found at index 6

The implementation of find first of() only checks for the existence of one character match rather than an entire String match.

TODO #5: Find-Last-Of Function

The find last of() function is defined as follows:

size_t String :: find_last_of ( const String & str , size_t pos = npos );

The find last of() function is very similar to the find first of() function. However, this function looks for the last character in a String that matches any of the characters in str, rather than the first. The search begins at position pos of the String and moves toward position 0, searching for a match along the way. Characters after position pos are ignored. If a match is found, the function returns a size t that represents the position of the last character that matches. Otherwise, the function returns npos. If pos exceeds the length of the String, the entire String is searched. For example:

String str = " EECS 281 is fun";

size_t found = 0;

found = str . find_last_of (" Eggs ", 14);

// found is 10 since an 's' can be found at index 10

Unlike find first of(), where pos defaults to zero, it is possible for find last of() to take in a value of pos that exceeds the size of the String. Make sure you take this into consideration.

One thing you have to watch out for when implementing find last of() is the concept of overflow. Overflow occurs when the value of a data type exceeds the maximum or goes below the minimum value that a data type can hold. When this happens, the value “wraps around” - that is, if you increment a data type past its maximum value, the new value wraps around to the minimum value, and vice versa. Consider the following code:

int main () {

int counter = 10 , val = 2147483643;

while ( - - counter >= 0) {

cout << val << "\n";

++ val ;

}

}

The largest possible value of an int is 2,147,483,647, so incrementing 2,147,483,647 does not give you 2,147,483,648. Instead, val would wrap around to the minimum possible value an int can hold (-2,147,483,648), so incrementing 2,147,483,647 by one instead gives you -2,147,483,648.

The above code prints the following. Notice that overflow occurs when 2,147,483,647 is incremented.

2147483643

2147483644

2147483645

2147483646

2147483647

-2147483648

-2147483647

-2147483646

-2147483645

-2147483644

The same thing applies to values of type size t. A size t is unsigned, so the minimum value a size t can take on is 0. If you attempt to decrement a size t below 0, the value of size t ends up wrapping around to npos, which is the largest value a size t can take on! As a result, this for loop will never exit:

for ( size_t i = std :: min ( pos , sz - 1); i >= 0; --i ) { /* do stuff */ }

This is because the expression i >= 0 is always true for a size t. You will need to find another way to break out of the loop when implementing the function.

Testing Your Code

To test your code, you may use the provided StringVerifier.cpp test file, which includes all the tests that will be used to judge your implementation on the autograder. To begin testing, run make using the Makefile we provide you in the starter files. This will generate an executable file called strlib, which can be used to test your solution.

Submitting to the Autograder

You may work with a partner on this lab. To submit to the autograder, create a .tar.gz file containing just String.cpp, as shown below. Do not modify the String.cpp file name, or your submission won’t run! Make sure the capitalization of your command matches.

tar - czvf lab3 . tar . gz String . cpp

If you are using the Makefile we provide you, you can type make fullsubmit, which will also run the command above.

If you are working with a partner, both partners must submit to the autograder. Only students who submit code to the autograder will receive points. It’s perfectly fine for both partners to submit identical code, as long as the code was written by both of the partners. You will be able to make three submissions to the autograder per day, up until the due date. Make sure the assignment identifier is on all code files you submit. The autograder will be very lenient with time and memory; the score you see is the score you will get, even if you have some test cases that are blue.

The test cases in the provided starter files are separated on the autograder, so if you are failing only 1 of the insert tests, you can get points for the other ones you are passing. The first letter of the test case denotes the operation that is being tested, and the number represents the test number (the same as in your starter files). Thus, test case E05 is the 5th erase test in StringVerifier.cpp.