COP Study Union - Fall 2017

Transcription

1 COP Study Union - Fall 2017 Chris Marsh Contents 1 Tries Insertion Worst Case Runtime Best Case Runtime Retrieval Worst Case Runtime Best Case Runtime Deletion Worst Case Runtime Best Case Runtime Why Tries? Heaps Insertion Worst Case Runtime Best Case Runtime Deletion Worst Case Runtime Best Case Runtime Representation Heapify Hashtables Linear Probing Worst Case Runtime Best Case Runtime Average Case Runtime Quadratic Probing Worst Case Runtime Best Case Runtime

2 3.2.3 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!) 31 2

3 1 Tries 1) How are strings represented in a trie? 2) 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? Observe the mystical glory that is the trie data structure below. 3

4 3) What strings are represented in the trie above? How many times does each string occur? 1.1 Insertion 1) Insert the string cake into the trie above. 2) Insert the word in into the trie above. 3) Insert the word integrate into the trie above. 4) Insert the empty string into the trie above Worst Case Runtime What is the worst case runtime for insertion into a trie? Why? Best Case Runtime What is the best case runtime for insertion into a trie? Why? 4

5 1.2 Retrieval Worst Case Runtime What is the worst case runtime for retrieval in a trie? Why? Best Case Runtime What is the best case runtime for retrieval in a trie? Why? 1.3 Deletion This is the same trie from above, before you did all of that crazy insertion. 5

6 1) Delete the string nonexistance from the trie. 2) Delete the string cats from the trie. 3) Delete the string cat 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

7 1.3.1 Worst Case Runtime What is the worst case runtime for deletion in a trie? Why? Best Case Runtime What is the best case runtime for deletion in a trie? Why? 1.4 Why Tries? 1) 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

8 2 Heaps 3) What does it mean for a binary tree to be complete? Can you draw some examples? 4) What does it mean for a binary tree to be full? Can you draw some examples? 5) What does it mean for a binary tree to be perfect? Can you draw some examples? 8

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

10 2.1 Insertion 1) What is the algorithm for insertion into a minheap? 2) Is the binary tree below a minheap? 2) Insert the value 87 into the heap above. 3) Insert the value 6 into the heap above. 4) Insert the value 1 into the heap above Worst Case Runtime What is the worst case runtime for insertion into a heap? Why? 10

11 2.1.2 Best Case Runtime What is the best case runtime for insertion into a heap? Why? 2.2 Deletion 1) 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 12 from the heap below. 4) Delete 1 from the heap below. 11

12 3) Delete 30 from the heap below. 5) Delete 1 from the heap below. 12

13 2.2.1 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 that would cause this runtime? 13

14 2.2.2 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 that would cause this runtime? 2.3 Representation 1) 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?) 14

15 2.4 Heapify 1) How does heapify work? What is its runtime? Why? 15

16 3 Hashtables 1) Obtain the hash values for the keys using the hash function below. Keys: 64, 89, 77, 50, 99, 159 int hash(int key, int table_size) { return ((key - 47) * 3) % table_size; } hash(64) = hash(89) = hash(77) = hash(50) = hash(99) = hash(159) = 3.1 Linear Probing 1) Insert the keys above, using the hashtable obtained above using linear probing as your collision resolution policy Worst Case Runtime What is the worst case runtime for for insertion and search using linear probing? Why? 16

17 3.1.2 Best Case Runtime What is the best case runtime for for insertion and search using linear probing? Why? Average Case Runtime What is the average case runtime for for insertion and search using linear probing? Why? 3.2 Quadratic Probing 1) Insert the keys above, using the hashtable obtained above using quadratic probing as your collision resolution policy. 17

18 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? 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? 18

19 3.2.3 Average Case Runtime What is the average case runtime for for insertion and search using quadratic probing? Why? 3.3 Seperate Chaining 1) Insert the keys above, using the hashtable obtained above using seperate chaining as your collision resolution policy. 19

20 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? Worst Case Runtime What is the worst case runtime for for insertion and search using seperate chaining? Why? 20

21 3.3.2 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 ways that you can delete from a hash table? 21

22 4 Number Bases and Base Conversion 1) Convert ( ) 2 to base 10. 2) Convert (63526) 7 to base 10. 3) Convert (ef45) 16 to base 3. Do the following converstions without using decimal as an intermediary base. Why is this possible? 4) Convert (42763) 8 to base 2. 5) Convert (5362) 9 to base 3. 22

23 6) Convert (211021) 3 to base 9. Fun questions about number bases. 7) How many different integers can we represent with 5 digits in base 2? 8) How many different integers can we represent with 3 digits in base 4? 9) How many different integers can we represent with m digits in base n? 23

24 5 Bitwise Operators 1) 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 << 31); printf("%d\n", c = (~c) + 1); printf("%u\n", ~c); printf("%d\n", 0xfe1); printf("%d\n", 011); 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)); 24

25 } return 0; 6 Two s Complement 1) What is decimal value of the following signed 32-bit numbers, using twos complement? a) b) c) d) e) f) ) What is the range of positive values including the integer 0 than you can make with 32-bit numbers, using two s complement? 3) What is the range of negative values that you can make with 32-bit numbers, using two s complement? 25

26 7 Bitmasks Observe the code segment below. #define LIGHT_SIDE 0x0001 #define FORCE_SENSITIVE 0x0002 #define DARTH_VADER 0x0004 #define LIKES_SAND 0x0008 typedef struct StarWarsCharacter { unsigned int attributes; char *name; } StarWarsCharacter; void panic(char *s) { fprintf(stderr, "%s\n", s); exit(1); } Character *createcharacter(char *name) { Character *c = NULL; if ((c = malloc(sizeof(character))) == NULL) panic("failed to allocate enough memory in createcharacter()!\n"); c->attributes = -1; c->name = malloc(sizeof(char) * (strlen(name) + 1)); strcpy(c->name, name); } return c; int main(void) { StarWarsCharacter *anakin = createcharacter("anakin"); // Assume there is code here that sets the least significant // binary digits of anakin s attributes member to unknown 26

27 // values, and the rest of the digits are zeros // (1) // (2) // (3) // (4) } return 0; Let the integer with the identifier attributes, use the bitmasks as follows: a) If the first bit is 1, the StarWarsCharacter is on the light side, and not on the lightside otherwise. b) If the second bit is 1, the StarWarsCharacter is force sensitive, and not force sensitive otherwise. c) If the third bit is 1, the StarWarsCharacter is Darth Vader, and not Darth Vader otherwise. 1) Beneath the comment labeled (1) 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. 27

28 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. 28

29 8 Backtracking 8.1 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. 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. 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 take the Jedi to Emperor Palpatine. 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 no Jedi gets taken to Emperor Palpatine. 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! 29

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31 9 Recurrence Relations (Just in case!) Derive a closed form for the following recurrence relation taken Dr. More Recursion; Recurrence Relations. Szumlanski s page T (0) = T (1) = c 1 T (n) = n + T (n/2) for n 2 31