Study Union Final Review

Transcription

1 University of Central Florida COP 352: Computer Science Study Union Final Review Student Academic Resource Center created by, Chris Marsh April 8, 28

2 Contents Tries 2. Insertion Worst Case Runtime Best Case Runtime Retrieval Worst Case Runtime Best Case Runtime Deletion Worst Case Runtime Best Case Runtime Why Tries? Heaps 8 2. Insertion Worst Case Runtime Best Case Runtime Deletion Worst Case Runtime Best Case Runtime Representation Heapify Hashtables 6 3. Linear Probing Worst Case Runtime Best Case Runtime Average Case Runtime Quadratic Probing Worst Case Runtime Best Case Runtime Average Case Runtime Seperate Chaining Worst Case Runtime Best Case Runtime Average Case Runtime Deletion Number Bases and Base Conversion 22 5 Bitwise Operators 24 6 Two s Complement 25 7 Bitmasks 26 8 Backtracking The Jedi and Clone Troopers Recurrence Relations (Just in case!) 3

3 Tries ) Create a typical trie node struct in C. How can we modify this struct to include other useful information about a string represented in the trie? 2) Given the trie node struct you created in Question, how are strings actually represented in a Trie? Observe the mystical glory that is the trie data structure below. It will be redrawn several times for your convenience c k e t s d o n u t p r f i x a s e o 2

4 c k e t s d o n u t p r f i x a s e o ) What strings are represented in the trie above? How many times does each string occur? 3

5 . Insertion ) Insert the string cake into the trie below. 2) Insert the word yoda into the trie below. 3) Insert the word produce into the trie below. 4) Insert the empty string into the trie below c k e t s d o n u t p r f i x a s e o 4

6 .. Worst Case Runtime What is the worst case runtime for insertion into a trie? Why?..2 Best Case Runtime What is the best case runtime for insertion into a trie? Why?.2 Retrieval.2. Worst Case Runtime What is the worst case runtime for retrieval in a trie? Why?.2.2 Best Case Runtime What is the best case runtime for retrieval in a trie? Why? 5

7 .3 Deletion This is the same trie from the earlier, before you did all of that crazy insertion. c p a d 5 r k t o n o e e 2 s u f i s t 56 8 x ) Delete the string nonexistence from the trie. 2) Delete the string cats from the trie. 3) Delete the string cats from the trie. 4) Delete the string cake from the trie. 5) Delete the string donut from the trie. 6) Delete the empty string from the trie. 6

8 .3. Worst Case Runtime What is the worst case runtime for deletion in a trie? Why?.3.2 Best Case Runtime What is the best case runtime for deletion in a trie? Why?.4 Why Tries? ) What are the benefits of tries? Why use a trie over another data structure like a binary search tree, or an array? 2) What practical applications would you possibly choose a trie data structure for? 7

9 2 Heaps ) What does it mean for a binary tree to be complete? Can you draw some examples? 2) What does it mean for a binary tree to be full? Can you draw some examples? 3) What does it mean for a binary tree to be perfect? Can you draw some examples? 8

10 4) What is the precise height of a complete binary tree with n nodes? 5) What is the structural property for a heap? 6) What is the ordering property for a minheap? 7) What is the ordering property for a maxheap? 8) What operations do heaps support? 9

11 2. Insertion ) Using a maximum of 2 sentences, can you describe the algorithm for insertion into a minheap? ) Insert the value 87 into the minheap above. 3) Insert the value 6 into the minheap above. 4) Insert the value into the minheap above. 2.. Worst Case Runtime What is the worst case runtime for insertion into a heap? Why? 2..2 Best Case Runtime What is the best case runtime for insertion into a heap? Why?

12 2.2 Deletion ) What is the algorithm for deleting arbitrary elements from a heap? 2) What is the algorithm for deleting the root of a minheap? 3) Delete 2 from the heap below. 4) Delete from the heap below

13 5) Delete 3 from the heap below. 6) Delete from the heap below

14 2.2. Worst Case Runtime What is the worst case runtime for deletion from a heap? What assumptions did you make when coming up with this runtime? Can you draw a heap, and insert a value that would cause this runtime? Best Case Runtime What is the best case runtime for deletion from a heap? What assumptions did you make when coming up with this runtime? Can you draw a heap, and insert a value that would cause this runtime? 3

15 2.3 Representation ) How are heaps typically represented in code? (Hint: Do we use node structs like we do with all of the other binary trees you have covered this semester?)

16 2.4 Heapify ) What is the purpose of Heapify? 2) Can you describe the algorithm for heapify in a maximum of 2 sentences? 3) What is the runtime of Heapify? Why? 5

17 3 Hashtables ) Obtain the hash values for the keys using the hash function below. (Keep this page on hand, you will need the keys, and their respective hash values for the next few pages) Keys: 64, 89, 77, 5, 99, 59 int hash(int key, int table_size) { return ((key - 47) * 3) % table_size; } hash(64) = hash(89) = hash(77) = hash(5) = hash(99) = hash(59) = 3. Linear Probing ) Insert the keys above, using the hash values obtained above using linear probing as your collision resolution policy. 6

18 3.. Worst Case Runtime What is the worst case runtime for for insertion and search using linear probing? Why? 3..2 Best Case Runtime What is the best case runtime for for insertion and search using linear probing? Why? 3..3 Average Case Runtime What is the average case runtime for for insertion and search using linear probing? Why? 7

19 3.2 Quadratic Probing ) Insert the keys above, using the hash values obtained above using quadratic probing as your collision resolution policy. 2) What conditions are required to guarantee that quadratic probing will find an empty position in the hash table? 3) Are these conditions required for linear probing for any reason? If not, is there any benefit to fufilling these requirements anyway? 8

20 3.2. Worst Case Runtime What is the worst case runtime for insertion and search using quadratic probing? Why? Best Case Runtime What is the best case runtime for for insertion and search using quadratic probing? Why? Average Case Runtime What is the average case runtime for for insertion and search using quadratic probing? Why? 9

21 3.3 Seperate Chaining ) Insert the keys above, using the hashtable obtained above using seperate chaining as your collision resolution policy. 2) Are the conditions described in the quadratic probing section required for seperate chaining? If not, is there any benefit to fufilling these requirements anyway? 2

22 3.3. Worst Case Runtime What is the worst case runtime for for insertion and search using seperate chaining? Why? Best Case Runtime What is the best case runtime for for insertion and search using seperate chaining? Why? Average Case Runtime What is the average case runtime for for insertion and search using seperate chaining? Why? 3.4 Deletion What are some methods for deletion in a hash table? What can we do to ensure that the deletion operation in our hash table does not interfere with our insertion and search operations? 2

23 4 Number Bases and Base Conversion ) Convert () 2 to base. 2) Convert (63526) 7 to base. 3) Convert (2) 3 to base 5. 22

24 Do the following converstions without using decimal as an intermediary base. Why is this possible? 4) Convert (42763) 8 to base 2. 5) Convert (ef45) 6 to base 2. 6) Convert (5362) 9 to base 3. 7) Convert (22) 3 to base 9. Fun questions about number bases. 8) How many different integers can we represent with 5 digits in base 2? 9) What is the maximum decimal integer that we can we represent with 5 digits in base 2? ) How many different integers can we represent with 3 digits in base 4? ) What is the maximum decimal integer that we can we represent with 3 digits in base 4? ) How many different integers can we represent with k digits in base n? 2) What is the maximum decimal integer that we can we represent with k digits in base n? 23

25 5 Bitwise Operators ) What is the output of each printf() statement in the following program. #include <stdio.h> int main(void) { int c = 5; printf("%d\n", c << 2); printf("%d\n", c); printf("%d\n", 92 << 3); printf("%d\n", c = c << 3); printf("%d\n", c = (~c) + ); printf("%u\n", ~c); printf("%d\n", xfe); printf("%d\n", ); printf("%d\n", 92 & 78); printf("%d\n", 92 78); printf("%d\n", 92 ^ 78); printf("%d\n", ~92); printf("%d\n", ~2); printf("%d\n", ~3); printf("%d\n", ~4); printf("%d\n", ~(-78)); } return ; 24

26 6 Two s Complement ) What is decimal value of the following signed 32-bit numbers, using twos complement? a) b) c) d) e) f) 2) What is the range of positive integer values including the integer than you can make with 32-bit integers, using two s complement? 3) What is the range of negative integer values that you can make with 32-bit integers, using two s complement? 4) Using (2), and (3) above, what is the full range of integer values that you can represent with 32-bit integers, using two s complement? 25

27 7 Bitmasks Observe the code segment below. #define LIGHT_SIDE x #define FORCE_SENSITIVE x2 #define DARTH_VADER x4 #define LIKES_SAND x8 typedef struct StarWarsCharacter { unsigned int attributes; char *name; } StarWarsCharacter; int main(void) { // You can assume that createcharacter() properly allocates memory // for the StarWarsCharacter struct, initializes the attributes member // to, and the name member to the string passed in as an argument StarWarsCharacter *anakin = createcharacter("anakin"); // Assume there is code here that sets the least significant // binary digits of anakin s attributes member to unknown // values, and the rest of the digits are zeros // () // (2) // (3) // (4) } return ; 26

28 Let the integer with the identifier attributes, use the bitmasks as follows: a) If the first bit is, the StarWarsCharacter is on the light side, and not on the lightside otherwise. b) If the second bit is, the StarWarsCharacter is force sensitive, and not force sensitive otherwise. c) If the third bit is, the StarWarsCharacter is Darth Vader, and not Darth Vader otherwise. d) If the fourth bit is, the StarWarsCharacter is likes sand, and does not like sand otherwise. ) Beneath the comment labeled () in the code segment above, using bitwise operators, and the defined bitmasks, write a line of code that will modify the attributes member of the StarWarsCharacter referenced by anakin, such that anakin will be force sensitive. 2) Beneath the comment labeled (2) in the code segment above, using bitwise operators, and the defined bitmasks, write a line of code that will modify the attributes member of the StarWarsCharacter referenced by anakin, such that anakin will be on the Dark Side. 3) Beneath the comment labeled (3) in the code segment above, using bitwise operators, and the defined bitmasks, write a line of code that will modify the attributes member of the StarWarsCharacter referenced by anakin, such that we toggle (what does toggle mean?) the digit that represents whether or not anakin is Darth Vader. 4) Beneath the comment labeled (4) in the code segment above, using bitwise operators, and the defined bitmasks, write a line of code that will modify the attributes member of the StarWarsCharacter referenced by anakin, such that anakin will not like sand. 27

29 8 Backtracking 8. The Jedi and Clone Troopers Using your healthy knowlege of backtracking, construct a tree diagram of states to find a path to a solution to the problem described below. Problem Statement: Three Jedi and three Clone Troopers must cross a river of lava using a transport only big enough to support two people. If on either side of the lava, there are more Clone Troopers than Jedi, than the Clone Troopers will Execute Order 66. The transport cannot cross the river of lava without at least one person on board. The goal is to get all three Jedi, and all three Clone Troopers to the other side of the river of lava such that Order 66 is not executed. Hint: This is a river crossing problem very similar to the Fox, Goose, and Bag of Beans problem covered in class (and even more similar to the Missionaries and Cannibals problem mentioned in Dr. Szumlanski s notes). Make sure you check for infeasible states, and states that you have already seen before! To avoid drawing a potentially explosive state tree (Study Union is short!), it will suffice to find a single repeated state, and a single invalid state. See Figure (a), and Figure (b) for examples on how to draw the states. In general, I will let the symbol J designate a single Jedi, the symbol C will designate a single Clone Trooper, and the rectangle will represent the transport. (a) State where all Jedi, Clone Troopers, and the Transport are North of the river (b) State where Jedi and 3 Clone Troopers are North of the river, while Jedi and the Transport are South of the river. Figure : Jedi and Clone Trooper State Diagram Examples *The next page was intentionally left blank for your tree diagram.* 28

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31 9 Recurrence Relations (Just in case!) Derive a closed form for the following recurrence relation taken Dr. Szumlanski s page More Recursion; Recurrence Relations. T () = T () = c T (n) = n + T (n/2) for n 2 3

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