CS210 Study Guide Swami Iyer. 1 Programming Model (Algorithms 1.1) 2. 2 Data Abstraction (Algorithms 1.2) 4

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1 Contents 1 Programming Model (Algorithms 1.1) 2 2 Data Abstraction (Algorithms 1.2) 4 3 Analysis of Algorithms (Algorithms 1.4) 7 4 Basic Data Structures (Algorithms 1.3) 9 5 Union-find (Algorithms 1.5) 11 6 Elementary Sorts (Algorithms 2.1) 12 7 Merge Sort (Algorithms 2.2) 13 8 Quick Sort (Algorithms 2.3) 13 9 Priority Queues (Algorithms 2.4) Sorting Applications (Algorithms 2.5) Symbol Tables (Algorithms 3.1) Binary Search Trees (Algorithms 3.2) Balanced Search Trees (Algorithms 3.3) Hash Tables (Algorithms 3.4) Searching Applications (Algorithms 3.5) Undirected Graphs (Algorithms 4.1) Directed Graphs (Algorithms 4.2) Minimum Spanning Trees (Algorithms 4.3) Shortest Paths (Algorithms 4.4) String Sorts (Algorithms 5.1) Tries (Algorithms 5.2) Substring Search (Algorithms 5.3) Regular Expressions (Algorithms 5.4) Data Compression (Algorithms 5.5) 29 1 of 29

2 1 Programming Model (Algorithms 1.1) Problem 1. Consider the following method: public static int mystery ( int [] a) { int x = 0; for ( int i = 0; i < a. length ; i ++) { x += a[i] * a[i]; return x; a. What does mystery() compute and return in general? b. What will mystery() return if the argument a is an integer array containing the values 1, 2, 3, 4, and 5? Problem 2. Consider the following method: public static int [][] mystery ( int n, int k) { int [][] a = new int [n][n]; for ( int i = 0; i < n; i ++) { for ( int j = 0; j < n; j ++) { a[i][j] = (i > j)? 0 : k; return a; a. What does mystery() compute and return in general? b. What will mystery() return if the arguments are n = 4 and k = 5? Problem 3. Consider the following recursive method: public static int mystery ( int a, int b) { if (b == 0) { return 0; if ( a == 0) { return mystery ( b - 1, a); return b + mystery (b, a - 1); a. What will mystery(0, 10) return? b. What will mystery(10, 3) return? c. What will mystery(200, 300) return? d. What does mystery(a, b) return in general about a and b? Problem 4. Consider the following program: // Circle. java import edu. princeton.cs. algs4. StdOut ; public class Circle { public static void main ( String [] args ) { double r = Double. parsedouble ( args [0]); double c = 2 * Math. PI * r; double a = Math. PI * r * r; StdOut. printf (" radius = %.2f, circumference = %.2f, area = %.2 f\ n", r, c, a); 2 of 29

3 a. What does Circle compute and print in general? b. What will Circle write when run with the command-line argument 5? Problem 5. Write a program RandomInts.java that takes command-line arguments n, a, and b as integers and writes n random integers (one per line) from the range [a, b]. For example $ java RandomInts Problem 6. Write a program Stats.java that reads integers from standard input, and computes and writes their mean, variance, and standard deviation, each up to 3 decimal places. For example $ java Stats <ctrl -d> mean = 3.000, var = 2.000, std = Problem 7. a. What is the command for generating random integers from the interval [1, 24] using the RandomInts program from Problem 5? b. What is the command for generating random integers from the interval [1, 24] and saving the output in a file called numbers.txt? c. What is the command for using the Stats program from Problem 6 to calculate stats for the numbers in numbers.txt? d. What is the command to perform the last two tasks in one shot? a. The sum of squares of the integers in a. b. 55 Solution 2. a. An n-by-n matrix in which the elements below the main diagonal are zeros and the rest of the elements are k. b. {{5, 5, 5, 5, {0, 5, 5, 5, {0, 0, 5, 5, {0, 0, 0, 5 Solution 3. a. 0 b. 30 c d. a * b Solution 4. a. Reads a command-line argument r as a double, computes the circumference c and area a of a circle with radius r, and prints the values of r, c, and a up to 2 decimal places. 3 of 29

4 b. radius = 5.00, circumference = 31.42, area = Solution 5. # RandomInts. java import edu. princeton.cs. algs4. StdOut ; import edu. princeton.cs. algs4. StdRandom ; public class RandomInts { public static void main ( String [] args ) { int n = Integer. parseint ( args [0]); int a = Integer. parseint ( args [1]); int b = Integer. parseint ( args [2]); for ( int i = 0; i < n; i ++) { int r = StdRandom. uniform (a, b + 1); StdOut. println (r); Solution 6. # Stats. java import edu. princeton.cs. algs4. StdIn ; import edu. princeton.cs. algs4. StdOut ; public class Stats { public static void main ( String [] args ) { int a[] = StdIn. readallints (); double mean = 0.0, var = 0.0, std = 0.0; for ( int x : a) { mean += x; mean /= a. length ; for ( int x : a) { var += (x - mean ) * (x - mean ); var /= a. length ; std = Math. pow (var, 0.5); StdOut. printf (" mean = %.3f, var = %.3f, std = %.3 f\ n", mean, var, std ); Solution 7. a. java RandomInts b. java RandomInts > numbers.txt c. java Stats < numbers.txt d. java RandomInts java Stats 2 Data Abstraction (Algorithms 1.2) Problem 1. Implement an immutable and comparable data type Card that represents a playing card. Each card is represented using an integer rank (0 for 2, 1 for 3,..., 11 for King, and 12 for Ace) and an integer suit (0 for Clubs, 1 for Diamonds, 2 for Hearts, and 3 for Spades). Your data type must support the following API: 4 of 29

5 method/class Card(int suit, int rank) boolean equals(card that) String tostring() int compareto(card that) static class SuitOrder static class ReverseRankOrder description construct a card given its suit and rank is this card the same as that a string representation of the card; for example Ace of Hearts is this card less than, equal to, or greater than that card a comparator for comparing cards based on their suits a comparator for comparing cards based on the reverse order of their ranks A card c 1 is less than a card c 2 if either the suit of c 1 is less than the suit of c 2 or c 1 and c 2 are of the same suit and the denomination of c 1 is less than the denomination of c 2. We refer to this as the natural order. Problem 2. Suppose deck is an array of Card (the data type from Problem 1) objects. a. Write down a statement that uses Arrays.sort() to sort deck by the natural order. b. Write down a statement that uses Arrays.sort() to sort deck by suit order. c. Write down a statement that uses Arrays.sort() to sort deck by reverse rank order. Problem 3. Complete the implementation of the iterable data type Range that allows clients to iterate over a range of integers from the interval [lo, hi] in increments specified by step. For example, the following code for ( int i : new Range (1, 10, 3)) { StdOut. println (i); should output import java. util. Iterator ; public class Range implements Iterable < Integer > { private int lo; private int hi; private int step ; // Construct a range given the bounds and the step size. public Range ( int lo, int hi, int step ) {... // A range iterator. public Iterator < Integer > iterator () {... // Helper range iterator. private class RangeIterator implements Iterator < Integer > { private int i; public RangeIterator () {... public boolean hasnext () {... public Integer next () {... 5 of 29

6 // Card. java import java. util. Comparator ; public class Card implements Comparable <Card > { private static String [] RANKS = {"2", "3", "4", "5", "6", "7", "8", "9", "10", " Jack ", " Queen ", " King ", " Ace "; private static String [] SUITS = {" Clubs ", " Diamonds ", " Hearts ", " Spades "; private final int suit ; private final int rank ; public Card ( int suit, int rank ) { this. suit = suit ; this. rank = rank ; public boolean equals ( Card other ) { return compareto ( other ) == 0; public String tostring () { return RANKS [ rank ] + " of " + SUITS [ suit ]; public int compareto ( Card that ) { if ( suit < that. suit suit == that. suit && rank < that. rank ) { return -1; else if ( suit == that. suit && rank == that. rank ) { return 0; else { return 1; public static class SuitOrder implements Comparator <Card > { public int compare ( Card c1, Card c2) { return c1. suit - c2. suit ; public static class ReverseRankOrder implements Comparator <Card > { public int compare ( Card c1, Card c2) { return c2. rank - c1. rank ; Solution 2. a. Arrays. sort ( deck ); b. Arrays. sort (deck, new Card. SuitOrder ()); c. Arrays. sort ( deck, new Card. ReverseRankOrder ()); 6 of 29

7 Solution 3. // Range. java import java. util. Iterator ; public class Range implements Iterable < Integer > { private int lo; private int hi; private int step ; public Range ( int lo, int hi, int step ) { this. lo = lo; this. hi = hi; this. step = step ; public Iterator < Integer > iterator () { return new RangeIterator (); private class RangeIterator implements Iterator < Integer > { private int i; public RangeIterator () { i = lo; public boolean hasnext () { return i <= hi; public Integer next () { int r = i; i += step ; return r; 3 Analysis of Algorithms (Algorithms 1.4) Problem 1. Consider an array a[] with 10 4 integers. a. Roughly how many comparisons are involved if one performs 10 6 linear search operations on a? b. Roughly how many comparisons (sorting and searching included) are involved if one performs 10 6 binary search operations on a? Problem 2. Consider the following table, which gives the running time T (N) in seconds for a program for various values of the input size N: What is the order of growth of T (N)? N T (N) ,599 Problem 3. Provide the order-of-growth classification (constant, logarithmic, linear, linearithmic, quadratic, cubic, or exponential) for the following tasks: 7 of 29

8 a. Adding two N-by-N matrices. b. Enumerating the subsets of a set with N items. c. Finding the average of N numbers. d. Finding distinct triples (a, b, c) in a collection of N positive integers such that a 2 + b 2 = c 2. e. Finding the maximum key in a maximum-oriented heap-ordered binary tree with N items. f. Inserting a new key into a heap-ordered binary tree with N items. g. Multiplying two numbers. h. Sorting an array of N items using quick sort. i. Sorting an array of N items using insertion sort. j. Summing up the diagonal elements of an N-by-N matrix. Problem 4. Give the order of growth (as a function of N) of the running times of each of the following code fragments: a. int sum = 0; for ( int n = N; n > 0; n /= 2) { for ( int i = 0; i < n; i ++) { sum ++; b. int sum = 0; for ( int i = 1; i < N; i *= 2) { for ( int j = 0; j < i; j ++) { sum ++; c. int sum = 0; for ( int i = 1; i < N; i *= 2) { for ( int j = 0; j < N; j ++) { sum ++; Problem 5. Consider a data type Planet with the attributes String name and int moons. What is the memory footprint (in bytes) of the array planets of Planet objects, created and initialized in the following manner? Planet [] planets = new Planet [8]; planets [0] = new Planet (" Mercury ", 0); planets [1] = new Planet (" Venus ", 0); planets [2] = new Planet (" Earth ", 1); planets [3] = new Planet (" Mars ", 2); planets [4] = new Planet (" Jupiter ", 67); planets [5] = new Planet (" Saturn ", 62); planets [6] = new Planet (" Uranus ", 27); planets [7] = new Planet (" Neptune ", 14); a = b lg 10 4 = of 29

9 Solution 2. T (N) N 3 (cubic) Solution 3. a. Quadratic b. Exponential c. Linear d. Cubic e. Constant f. Logarithmic g. Constant h. Linearithmic i. Quadratic j. Linear Solution 4. a. Linear b. Linear c. Linearithmic Solution ( ) = 430 bytes 4 Basic Data Structures (Algorithms 1.3) Problem 1. Consider the following method: public static int mystery ( Node < Integer > first ) { int x = 0; for ( Node y = first, int i = 0; y!= null ; y = y. next, i ++) { x += ( i % 2 == 0)? y. item : 0; return x; a. What does mystery() compute and return in general? b. What will mystery() return if the argument (a Node object) represents a linked list containing integers 1, 2, 3,..., 10? Problem 2. Consider the following method: public static int mystery ( Bag < Integer > bag ) { int x = 0; Itereator < Integer > iter = bag. iterator (); while ( iter. hasnext ()) { x += iter. next (); return x; a. What does mystery() compute and return in general? 9 of 29

10 b. What will mystery() return if the argument (a Bag object) contains the integers 1, 2, 3,..., 10? Problem 3. Suppose that a minus sign in the input indicates pop the stack and write the return value to standard output, and any other string indicates push the string onto the stack. Further suppose that following input is processed: a. What is written to standard output? it was - the best - of times it was - the - - worst - of times - b. What are the contents (top to bottom) left on the stack? Problem 4. Suppose that an intermixed sequence of (stack) push and pop operations are performed. The pushes push the integers 0 through 9 in order; the pops print out the return value. Which of the following sequence(s) could not occur? A B C D E F G H Problem 5. Consider the following code fragment: Stack < Integer > s = new Stack < Integer >(); while (n > 0) { s. push (n % 2); n = n / 2; while (!s. isempty ()) { StdOut. print (s. pop ()); StdOut. println (); a. What does the following code fragment print when n is 50? b. Give a high-level description of what it does when presented with a positive integer n. Problem 6. Suppose that a minus sign in the input indicates dequeue the queue and write the return value to standard output, and any other string indicates enqueue the string onto the queue. Further suppose that following input is processed: a. What is written to standard output? it was - the best - of times it was - the - - worst - of times - b. What are the contents (head to tail) left on the queue? Problem 7. Suppose that a client performs an intermixed sequence of (queue) enqueue and dequeue operations. The enqueue operations put the integers 0 through 9 in order onto the queue; the dequeue operations print out the return value. Which of the following sequence(s) could not occur? A B C D of 29

11 Problem 8. What does the following code fragment do to the queue q? Stack < String > s = new Stack < String >(); while (!q. isempty ()) { s. push (q. dequeue ()); while (!s. isempty ()) { q. enqueue (s. pop ()); a. Computes and returns the sum of every other integer in node, starting at the first. b. 25 Solution 2. a. Computes and returns the sum of the integers in bag. b. 55 Solution 3. a. was best times of the was the it worst times b. of it Solution 4. B, F, and G Solution 5. a b. Prints the binary representation of n. Solution 6. a. it was the best of times it was the worst b. of times Solution 7. B, C, and D Solution 8. Reverses the items on the queue. 5 Union-find (Algorithms 1.5) Problem 1. Let s say we are using the union-find (UF) data structure to solve the dynamic connectivity problem with 10 sites and input pairs (1, 2), (7, 8), (1, 6), (0, 5), (3, 8), (2, 3), (6, 7), (2, 7), and (4, 9), arriving in that order. a. How many components does UF identify? b. What are those components? c. Will the number of components or their membership change if the input pairs arrive in a different order than above? d. Suppose we process the pairs using QuickFindUF. What are the values in the id array after all the pairs are processed? e. Suppose we process the pairs using QuickUnionUF. What are the values in the parent array after all the pairs are processed? 11 of 29

12 f. Suppose we process the pairs using WeightedQuickUnionUF. What are the values in the parent and size arrays after all the pairs are processed? {0, 5, {4, 9, and {1, 2, 3, 6, 7, 8 3. No 4. id = {5, 8, 8, 8, 9, 5, 8, 8, 8, 9 5. parent = {5, 2, 6, 8, 9, 5, 8, 8, 8, 9 6. parent = {0, 1, 1, 7, 4, 0, 1, 1, 7, 4, size = {2, 6, 1, 1, 2, 1, 1, 3, 1, 1 6 Elementary Sorts (Algorithms 2.1) Problem 1. Consider sorting an array a[] containing the following strings: E A S Y Q U T I O N a. (Selection sort.) What is a[] when i = 3 and after the exchange on line 10? 1 public static void sort ( Comparable [] a) { 2 int N = a. length ; 3 for ( int i = 0; i < N; i ++) { 4 int min = i; 5 for ( int j = i + 1; j < N; j ++) { 6 if ( less (a[j], a[ min ])) { 7 min = j; exch (a, i, min ); b. (Insertion sort.) What is a[] when i = 4 and the inner loop (lines 4-7) is complete? 1 public static void sort ( Comparable [] a) { 2 int N = a. length ; 3 for ( int i = 0; i < N; i ++) { 4 for ( int j = i; j > 0 && less (a[j], a[j - 1]); j - -) { 5 exch (a, j, j - 1); c. (Shell sort.) What is a[] after a 4-sort, ie, after the first iteration of the second while loop (lines 5-14)? 1 public static void sort ( Comparable [] a) { 2 int N = a. length ; 3 int h = 1; 4 while ( h < N / 3) { h = 3 * h + 1; 5 while (h >= 1) { 6 for ( int i = h; i < N; i ++) { 7 for ( int j = i; j >= h && less (a[j], a[j - h ]); j -= h) { 8 exch (a, j, j - h); h /= 3; of 29

13 a. A E I N Q U T S O Y b. A E Q S Y U T I O N c. E A S I O N T Y Q U 7 Merge Sort (Algorithms 2.2) Problem 1. Consider sorting an array a[] containing the following strings: E A S Y Q U T I O N What is a[] when the function call merge(a, aux, 0, 2, 4) returns? 1 public static void sort ( Comparable [] a) { 2 Comparable [] aux = new Comparable [ a. length ]; 3 sort (a, aux, 0, a. length - 1); private static void sort ( Comparable [] a, Comparable [] aux, int lo, int hi) { 7 if (hi <= lo) { return ; 8 int mid = lo + ( hi - lo) / 2; 9 sort (a, aux, lo, mid ); 10 sort (a, aux, mid + 1, hi ); 11 merge (a, aux, lo, mid, hi ); private static void merge ( Comparable [] a, Comparable [] aux, 15 int lo, int mid, int hi) { 16 int i = lo, j = mid + 1; 17 for ( int k = lo; k <= hi; k ++) { 18 aux [k] = a[k]; for ( int k = lo; k <= hi; k ++) { 21 if (i > mid ) { a[k] = aux [j ++]; 22 else if (j > hi) { a[k] = aux [i ++]; 23 else if ( less ( aux [j], aux [i ])) { a[k] = aux [j ++]; 24 else { a[ k] = aux [ i ++]; A E Q S Y U T I O N 8 Quick Sort (Algorithms 2.3) Problem 1. Consider partitioning an array a[] containing the following keys, by calling the partition() method (shown below) from quick sort, as partition(a, 0, a.length - 1): P A R T I O N E X M L 13 of 29

14 private static int partition ( Comparable [] a, int lo, int hi) { int i = lo; int j = hi + 1; Comparable v = a[ lo ]; while ( true ) { while ( less (a [++ i], v)) { if (i == hi) { break ; while ( less (v, a[--j ])) { if (j == lo) { break ; if (i >= j) { break ; exch (a, i, j); exch (a, lo, j); return j; a. What is the value of the pivot element v? b. What is the value returned by the function call, ie, what is the destination index of the pivot? c. What is the state of the array after the call? a. P b. 7 c. E A L M I O N P X T R 9 Priority Queues (Algorithms 2.4) Problem 1. Insert the following keys in that order into a maximum-oriented heap-ordered binary tree: E A S Y Q U T I O N a. What is the state of the array pq representing the resulting tree? b. What is the height of the tree (the root is at height zero)? Problem 2. Suppose that a letter in the input means insert the letter into an initially empty priority queue and an asterisk (*) means remove the minimum from the priority queue. What is left in the priority queue after the following input is processed? a. - Y S U O Q E T A I N b. 3 Solution 2. U P R I O * R * * I * T * Y * * * Q U E * * * U E * 14 of 29

15 10 Sorting Applications (Algorithms 2.5) Problem 1. (Sorting and Shuffling Algorithms) The column on the left is the original input of strings to be sorted or shuffled; the column on the right are the string in sorted order; the other columns are the contents at some intermediate step during one of the 7 algorithms listed below. lynx bass lion bass bass bass gnat wren bass bass bear frog bear bear bear bass worm bear bear crab mole clam clam clam bear oryx clam crab lion hawk crab crab crab crab swan crab lion goat wren frog frog crow lion wolf crow goat duck lynx gnat goat deer goat mule deer mole frog crab goat hawk dove duck mole dove frog dove swan hawk lion duck frog puma duck swan clam bear lion lynx frog dove seal frog clam hawk clam lynx mole gnat clam deer gnat hawk deer bass lynx swan goat hawk lion goat wren crow goat mole wren hawk deer goat hawk mule gnat mule mule gnat mule crow bear lion oryx lynx oryx oryx lynx oryx lynx lynx lynx gnat lynx gnat swan mule lynx lynx gnat lynx lynx puma lynx wren oryx lynx oryx lynx mole puma worm puma puma puma puma puma frog mule worm seal worm worm worm worm worm crab oryx seal oryx seal seal crow seal seal bass puma crow mule crow crow deer lion mule crow seal deer wolf deer deer dove wren wren clam swan wolf wren wolf wolf duck wolf wolf hawk wolf dove swan dove dove seal mole swan dove worm duck mole duck duck wolf swan mole duck wren Original input 2. Selection sort 3. Insertion sort 4. Merge sort (top down) 5. Quick sort (standard, no shuffle) 6. Quick sort (3-way, no shuffle) 7. Heap sort 8. Shuffle 9. Sorted What is the sequence of numbers you get after you match up each algorithm s number with the corresponding column? Use each number exactly once. Problem 2. Give an N log N algorithm for computing the mode (value that occurs most frequently) of a sequence of N integers Solution 2. Sort the sequence using quick sort. Iterate through the sorted sequence to find the length of the longest stretch of numbers. The first step takes linearithmic (N log N) time and the second step can be performed in linear (N) time. The overall running time is N log N + N N log N. 15 of 29

16 11 Symbol Tables (Algorithms 3.1) Problem 1. Consider inserting the following key-value pairs into a sequential search symbol table st, an object of type SequentialSearchST. key: S Y M B O L T A B L E E X A M P L E value: a. What is the state of the linked list first? b. What is the value returned by st.size()? c. What is the value returned by st.get("b")? d. What is the state of the linked list first after the call st.put("e", null)? Problem 2. Consider inserting the following key-value pairs into a binary search (ordered) symbol table st, an object of type BinarySearchST. key: S Y M B O L T A B L E E X A M P L E value: a. What is the state of the keys and values arrays? b. What is the value returned by st.floor("g")? c. What is the value returned by st.ceiling("g")? d. What is the value returned by st.rank("g")? e. What is the value returned by st.select(5)? f. What is the state of the keys and values arrays after the call st.put("e", null)? a. (P, 16) -> (X, 13) -> (E, 18) -> (A, 14) -> (T, 7) -> (L, 17) -> (O, 5) -> (B, 9) -> (M, 15) -> (Y, 2) -> (S, 1) -> null b. 11 c. 9 d. (P, 16) -> (X, 13) -> (A, 14) -> (T, 7) -> (L, 17) -> (O, 5) -> (B, 9) -> (M, 15) -> (Y, 2) -> (S, 1) -> null Solution 2. a keys : A B E L M O P S T X Y values : b. E c. L d. 3 e. O f keys : A B L M O P S T X Y values : of 29

17 12 Binary Search Trees (Algorithms 3.2) Problem 1. Consider inserting the following keys (assume values to be non null and arbitrary) into a binary search tree (ordered) symbol table st, an object of type BST. S Y M B O L T A E X P a. What is the binary search tree (BST) that results? List the keys along with their indices (starting at 1) in level order. b. What is the height of the BST (assume root to be at height 0)? c. What is the order in which the keys are visited if we traverse the BST in pre-order? d. What is the order in which the keys are visited if we traverse the BST in in-order? e. What is the order in which the keys are visited if we traverse the BST in post-order? f. What is the order in which the keys are visited if we traverse the BST in level-order? g. What is the BST that results if we delete the smallest key from the initial tree? h. What is the BST that results if we delete the largest key from the initial tree? i. What is the BST that results if we delete the key B from the initial tree? a. 1: S, 2: M, 3: Y, 4: B, 5: O, 6: T, 8: A, 9: L, 11: P, 13: X, 18: E b. 4 c. S M B A L E O P Y T X d. A B E L M O P S T X Y e. A E L B P O M X T Y S f. S M Y B O T A L P X E g. 1: S, 2: M, 3: Y, 4: B, 5: O, 6: T, 9: L, 11: P, 13: X, 18: E h. 1: S, 2: M, 3: T, 4: B, 5: O, 7: X, 8: A, 9: L, 11: P, 18: E i. 1: S, 2: M, 3: Y, 4: E, 5: O, 6: T, 8: A, 9: L, 11: P, 13: X 13 Balanced Search Trees (Algorithms 3.3) Problem 1. Consider inserting the following keys into an initially empty 2-3 search tree. E A S Y Q U T I O N a. What is the height of the tree that results (assume root to be at height zero)? What is the minimum and maximum number of keys that a 2-3 search tree of this height can hold? b. What are the 2-nodes in the tree? c. What are the 3-nodes in the tree? 17 of 29

18 Problem 2. Suppose you insert the key 23 into the following left-leaning red-black BST: a. Which of the following color flips result? Select all that apply. A. Color flip 18 B. Color flip 22 C. Color flip 23 D. Color flip 24 E. Color flip 26 F. Color flip 28 b. Which of the following rotations result? Select all that apply. A. Rotate 18 left B. Rotate 18 right C. Rotate 22 left D. Rotate 22 right E. Rotate 23 left F. Rotate 23 right G. Rotate 24 left H. Rotate 24 right I. Rotate 26 left J. Rotate 26 right K. Rotate 28 left L. Rotate 28 right a. 2; 7 and 14 b. (S), (U), (A), (Q), (T), (Y) 18 of 29

19 c. (E, O), (I, N) Solution 2. a. E (Color flip 24) b. A (Rotate 18 left), C (Rotate 22 left), J (Rotate 26 right), L (Rotate 28 right) 14 Hash Tables (Algorithms 3.4) Problem 1. Consider inserting the following keys into an initially empty hash table of M = 5 lists, using separate chaining. Use the hash function h(k) = k mod M to transform the kth letter of the alphabet into a table index, where 1 k 26. a. What is the value of h(18)? b. What is the state of the array st? a. 3 b. 0: O -> T -> Y -> E -> null 1: U -> A -> null 2: Q -> null 3: null 4: N -> I -> S -> null E A S Y Q U T I O N 15 Searching Applications (Algorithms 3.5) Problem 1. How is the following 10-dimensional sparse vector represented economically as a symbol table? Problem 2. How is the following 5-by-10 sparse matrix represented economically as an array of sparse vectors (symbol tables)? {2: , 4: Solution 2. 0: { 1: {2: 6, 8: 9 2: {0: 7, 5: 2, 8: 2 3: {0: 9, 3: 1 4: {9: of 29

20 16 Undirected Graphs (Algorithms 4.1) Problem 1. Consider the following undirected graph: a. What is the adjacency matrix representation of the graph? b. What is the adjacency list representation of the graph? List the adjacent vertices in decreasing order of their IDs. c. What is the state of the edgeto array if you run depth-first search (DFS) on the graph, starting at vertex 0? Is there a path from 0 to 3? If so, what is it? d. What is the state of the distto and edgeto arrays if you run breadth-first search (BFS) on the graph, starting at vertex 0? Is there a path from 0 to 3? If so, what is it? e. How many connected components does the graph have? f. What is the component identified by a DFS on the graph, starting at vertex 3? Problem 2. Consider building a SymbolGraph object sg from the following representation of a symbol graph as a file: JFK MCO ORD DEN ORD HOU DFW PHX JFK ATL ORD DFW ORD PHX ATL HOU DEN PHX PHX LAX JFK ORD DEN LAS DFW HOU ORD ATL LAS LAX ATL MCO HOU MCO LAS PHX a. What is the state of the sg.st symbol table and sg.keys array? b. Who are the neighbors of HOU? 20 of 29

21 a b. 0: : 0 2: 0 3: 5 4 4: : : 4 0 7: 8 8: 7 9: : 9 11: : 11 9 c. v edgeto [ v] Yes, 0 -> 6 -> 4 -> 5 -> 3 d. v distto [ v] edgeto [ v] e. 3 Yes, 0 -> 5 -> 3 21 of 29

22 f. {0, 1, 2, 3, 4, 5, 6 Solution 2. a. sg. st: ATL -> 7 DEN -> 3 DFW -> 5 HOU -> 4 JFK -> 0 LAS -> 9 LAX -> 8 MCO -> 1 ORD -> 2 PHX -> 6 sg. keys : 0: JFK 1: MCO 2: ORD 3: DEN 4: HOU 5: DFW 6: PHX 7: ATL 8: LAX 9: LAS b. MCO, DFW, ATL, and ORD 17 Directed Graphs (Algorithms 4.2) Problem 1. Consider a directed graph with the following adjacency list: 0: 5 1 1: 2: 0 3 3: 5 2 4: 3 2 5: 4 6: : 6 8 8: 7 9 9: : 12 11: : 9 a. Is the graph a DAG (ie, a directed acyclic graph)? b. What are the pre-, post-, reverse-post depth first orders for the graph if the source vertex is 0? c. What is the adjacency list of the reverse graph? Problem 2. Consider the following acyclic digraph. Assume the adjacency lists are in sorted order: for example, when iterating through the edges pointing from 0, consider the edge 0 1 before 0 6 or of 29

23 a. Run DFS on the digraph, starting from vertex 2. List the vertices in preorder. b. Run DFS on the digraph, starting from vertex 2. List the vertices in postorder. c. Run DFS on the digraph, starting from vertex 2. List the vertices in topological order. d. Run BFS on the digraph, starting from vertex 2. List the vertices in the order in which they are dequeued from the FIFO queue. a. No b. Preorder: ; Postorder: ; Reverse postorder: c. 0: 6 2 1: 0 2: 4 3 3: 4 2 4: : 3 0 6: 7 7: 8 8: 7 9: : 9 11: 9 12: Solution 2. a b c d of 29

24 18 Minimum Spanning Trees (Algorithms 4.3) Problem 1. Consider the following edge-weighted graph with 9 vertices and 19 edges. distinct integers between 1 and 19. Note that the edge weights are a. Complete the sequence of edges in the MST in the order that Kruskal s algorithm includes them (by specifying their edge weights). b. What is the total weight of the edges on the MST? a b Shortest Paths (Algorithms 4.4) Problem 1. Consider the following edge-weighted digraph: Now suppose we fix the source vertex to be vertex 0 and run Dijkstra s algorithm on the above graph. 24 of 29

25 a. What are the states of the distto and edgeto arrays? b. Is there a path from the source (vertex 0) to vertex 6? If so, what is it and what is its length? a. v distto [ v] edgeto [ v] null > > > > > > > b. Yes, the path is 0 -> > > > , and its length is String Sorts (Algorithms 5.1) Problem 1. Consider sorting the following input array a by section using the key-indexed counting method: name section Becquerel 4 Bethe 2 Bloch 3 Bohr 4 Chadwick 2 Curie 2 Dirac 1 Einstein 3 Faraday 1 Fermi 3 Feynman 3 Galilei 1 Heisenberg 1 Maxwell 2 Newton 1 Planck 1 Rutherford 4 Schrodinger 4 Thomson 3 Walton 1 a. What is the state of the count array after the first step (compute frequency counts) of key-indexed counting? b. What is the state of the count array after the second step (transform counts to indices) of key-indexed counting? c. What is the state of the auxiliary array aux after the third step (distribute the data) of key-indexed counting? d. What is the state of the input array a after the fourth step (copy back) of key-indexed counting? Problem 2. Consider sorting the following array a of fixed-width strings using least-significant-digit (LSD) first string sort. 25 of 29

26 3 LLT919 7 VTF645 5 HPH324 6 LYJ620 7 BVZ383 3 TYQ515 1 JNS516 3 RRI185 1 QEV852 8 FJC018 6 KOX096 7 YLW378 6 ZIG805 What is the state of the input array a at the end of the following iterations of the outer for loop in the sort() method? a. d = 6 b. d = 5 c. d = 0 Problem 3. Consider sorting the following array a of fixed-width strings using most-significant-digit (MSD) first string sort. Bohr Einsten Chadwick Curie Schrodinger Galilei Dirac Walton Rutherford Fermi Heisenberg Maxwell Planck Feynman Thomson Faraday Newton Becquerel Bethe Bloch What is the state of the input array a after the copy back step in the following calls? a. sort(a, 0, N - 1, 0, aux)? b. sort(a, 0, 3, 1, aux)? What is the state of the input array a when the call sort(a, 0, N - 1, 0, aux) returns? a. i: count [i]: b. i: count [ i]: of 29

27 c. i aux [i] 0 Dirac 1 1 Faraday 1 2 Galilei 1 3 Heisenberg 1 4 Newton 1 5 Planck 1 6 Walton 1 7 Bethe 2 8 Chadwick 2 9 Curie 2 10 Maxwell 2 11 Bloch 3 12 Einstein 3 13 Fermi 3 14 Feynman 3 15 Thomson 3 16 Becquerel 4 17 Bohr 4 18 Rutherford 4 19 Schrodinger 4 d. i a[ i] 0 Dirac 1 1 Faraday 1 2 Galilei 1 3 Heisenberg 1 4 Newton 1 5 Planck 1 6 Walton 1 7 Bethe 2 8 Chadwick 2 9 Curie 2 10 Maxwell 2 11 Bloch 3 12 Einstein 3 13 Fermi 3 14 Feynman 3 15 Thomson 3 16 Becquerel 4 17 Bohr 4 18 Rutherford 4 19 Schrodinger 4 Solution 2. a. 6 LYJ620 1 QEV852 7 BVZ383 5 HPH324 7 VTF645 3 TYQ515 3 RRI185 6 ZIG805 1 JNS516 6 KOX096 8 FJC018 7 YLW378 3 LLT919 b. 6 ZIG805 3 TYQ515 1 JNS516 8 FJC of 29

28 3 LLT919 6 LYJ620 5 HPH324 7 VTF645 1 QEV852 7 YLW378 7 BVZ383 3 RRI185 6 KOX096 c. 1 JNS516 1 QEV852 3 LLT919 3 RRI185 3 TYQ515 5 HPH324 6 KOX096 6 LYJ620 6 ZIG805 7 BVZ383 7 VTF645 7 YLW378 8 FJC018 Solution 3. a. i a[ i] 0 Bohr 1 Becquerel 2 Bethe 3 Bloch 4 Chadwick 5 Curie 6 Dirac 7 Einsten 8 Fermi 9 Feynman 10 Faraday 11 Galilei 12 Heisenberg 13 Maxwell 14 Newton 15 Planck 16 Rutherford 17 Schrodinger 18 Thomson 19 Walton b. i a[ i] 0 Becquerel 1 Bethe 2 Bloch 3 Bohr 4 Chadwick 5 Curie 6 Dirac 7 Einsten 8 Fermi 9 Feynman 10 Faraday 11 Galilei 12 Heisenberg 13 Maxwell 14 Newton 28 of 29

29 15 Planck 16 Rutherford 17 Schrodinger 18 Thomson 19 Walton i a[i] 0 Becquerel 1 Bethe 2 Bloch 3 Bohr 4 Chadwick 5 Curie 6 Dirac 7 Einsten 8 Faraday 9 Fermi 10 Feynman 11 Galilei 12 Heisenberg 13 Maxwell 14 Newton 15 Planck 16 Rutherford 17 Schrodinger 18 Thomson 19 Walton 21 Tries (Algorithms 5.2) 22 Substring Search (Algorithms 5.3) 23 Regular Expressions (Algorithms 5.4) 24 Data Compression (Algorithms 5.5) 29 of 29