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Optional. Mark along the line to show your feelings Before exam: [L____________________J]. on the spectrum between L and J. After exam: [L____________________J]. UC Berkeley – Computer Science CS61B: Data Structures Midterm # 2 , Spring 2017 This test has 8 questions worth a total of 120 points, and is to be completed in 110 minutes. The exam is closed book, except that you are allowed to use two double sided written cheat sheets (front and back). No calculators or other electronic devices are permitted. Give your answers and show your work in the space provided. Write the statement out below in the blank provided and sign. You may do this before the exam begins. “I have neither given nor received any assistance in the taking of this exam.” Signature: ___________________________
Points # Points
0 0.5 5 5 5. 1 11 .5 6 12. 2 15.2 7 18 3 12 8 25 4 20 TOTAL 120 Name: __________________________ SID: ___________________________ Three-letter Login ID: _________ Login of Person to Left: _______ Login of Person to Right: ______ Exam Room: _____________________ Tips:
- There may be partial credit for incomplete answers. Write as much of the solution as you can, but bear in mind that we may deduct points if your answers are much more complicated than necessary.
- There are a lot of problems on this exam. Work through the ones with which you are comfortable first. Do not get overly captivated by interesting design issues or complex corner cases you’re not sure about.
- Not all information provided in a problem may be useful.
- Unless otherwise stated, all given code on this exam should compile. All code has been compiled and executed before printing, but in the unlikely event that we do happen to catch any bugs in the exam, we’ll announce a fix. Unless we specifically give you the option, the correct answer is not ‘does not compile.’
- ○ indicates that only one circle should be filled in.
- □ indicates that more than one box may be filled in.
- For answers which involve filling in a ○ or □, please fill in the shape completely.
- If the exam says “write only one statement per line”, a for loop counts as one statement.
UC BERKELEY Login: _________
0. So it begins (0. 55 points). Write your name and ID on the front page. Write the exam room. Write the IDs of your neighbors. Write the given statement. Sign when you’re done with the exam. Write your login in the corner of every page. Enjoy your free 0.5 5 points J. 1. Traversin (11.5 points). a) (6 pts) For the graph below, give the DFS preorder, postorder, and BFS order traversals starting from vertex 9. The BFS order is the order in which vertices are enqueued. Assume ties are broken in numerical order (i.e. the edge 15 à 16 would be considered before 15 à 17). DFS Preorder: __________________ DFS Postorder: ___________________ BFS Order: ___________________ b) (4 pts) Suppose we have a min heap of 7 unique items and we want to print the values of the heap in increasing order. For which of our standard tree traversals will we get the values of the heap in increasing order if we print when we visit a node? Fill in the bubbles completely. Preorder (^) ○ Never ○ Sometimes ○ Always Inorder (^) ○ Never ○ Sometimes ○ Always Postorder (^) ○ Never ○ Sometimes ○ Always Level order (^) ○ Never ○ Sometimes ○ Always c) (1.5 pts) Draw a tree with 5 nodes for which the preorder traversal is the reverse of the inorder traversal, and for which all values are unique. Or if this is not possible, simply write ”Impossible”.
UC BERKELEY Login: _________ 3. Weighted Quick Union (12 points) a. (2 pts) Suppose we have a weighted quick union object. What calls to connect(a, b) produce the following trees? Assume that each WQU starts with all items disconnected. Or fill in the “Impossible” option if the given tree is impossible. Assume that in case of a tie, the root of the left argument is placed below the root of the right argument. connect(_______, ________) connect(_______, ________) ○ Impossible connect( 2 , ________) connect( 4 , ________) connect(_______, ________) connect(_______, ________) connect(_______, ________) ○ Impossible b. ( 4 pts) Suppose we add a new operation undo(a, b) that undoes an earlier Disjoint Sets connect operation. If connect(a, b) has never been called, then this method has no effect. For each of the implementations of Disjoint Sets, mark the corresponding box if it is impossible to add the undo operation without adding additional data structures (i.e. instance variables) to that implementation. □Quick Union □Quick Find □Weighted Quick Union(WQU) □WQU with Path Compression c. (6 pts) Suppose we use a graph instead of a quick union tree to solve the disjoint sets problem. Assume we implement the connect(a, b) and isConnected(a, b) operations using a graph represented as an adjacency list with no duplicates allowed. Assume that we implement connect using addEdge, and isConnected using DFS with a marked array, where DFS terminates early if a connection is detected. Let N be the number of nodes in the graph, and let M be the total number of calls to either the connect or isConnected methods. For example, if we call connect 37 times, then isConnected 13 times, then connect 20 times, then M=70. What will be the worst case runtime of any single call to connect(a, b)? ○Θ(1) ○Θ(log N) ○Θ(log M) ○Θ(N) ○Θ(M) ○Θ(N + M) ○Θ(NM) ○Θ(N + M log* M) What will be the worst case runtime of any single call to isConnected(a, b)? ○Θ(1) ○Θ(log N) ○Θ(log M) ○Θ(N) ○Θ(M) ○Θ(N + M) ○Θ(NM) ○Θ(N + M log* M)
CS61B MIDTERM, SPRING 2017
Login: _______ 4. Asymptotics (20 points). a) ( 7 pts) For each code block below, fill in the blank(s) so that the function has the desired runtime. Do not use any commas. If the answer is impossible, just write “impossible” in the blank. public static void f1(int N) { // desired runtime: Θ(N) for (int i = 1; i < N; _________) { System.out.println(“hi”); } } public static void f2(int N) { // desired runtime Θ(log N) for (int i = 1; i < N; _________) { System.out.println(“hi”); } } public static void f3(int N) { // desired runtime Θ(1) for (int i = 1; i < N; _________) { System.out.println(“hi”); } } b) (8 pts) Give the runtime of the following functions in Θ or O notation as requested. Your answer should be as simple as possible with no unnecessary leading constants or lower order terms. For f5, your bound should be as tight as possible (so don’t just put O(N NM! ) or similar for the second answer). Don’t spend too much time on these! _Θ_____ public static void f4(int N) { if (N == 0) { return; } f4(N / 2); f4(N / 2); f4(N / 2); f4(N / 2); g(N); // runs in Θ(N 2 ) time } _ O_____ public static void f5(int N, int M) { if (N < 10) { return; } for (int i = 0; i <= N % 10; i++) { f5(N / 10, M / 10); System.out.println(M); } } c) (0 pts) This mostly subterranean building, designed by I.M. Pei, cost more than $250,000,000 to construct, and was built by a small religious group that believes that by “building architectural masterpieces in remote locations, they are restoring the Earth's balance”.
CS61B MIDTERM, SPRING 2017
Login: _______ 6. Bipartite Graphs (12.5 points). a) (3 pts) Suppose we want to color every vertex of a graph either blue or green such that no vertex touches anther vertex of the same color. This is possible for some graphs but not others. A graph where a valid coloring exists is called “bipartite”. Which of the graphs below are bipartite? ○ Bipartite ○ Not Bipartite ○ Bipartite ○ Not Bipartite b) ( 9 .5 pts) Suppose we are using the undirected Graph API from the lecture / optional textbook, shown below. public class Graph { public Graph(int V): Create empty graph with v vertices public void addEdge(int v, int w): add an edge v-w Iterable adj(int v): vertices adjacent to v int V(): number of vertices int E(): number of edges ... } Fill in the method twocolor below such that a correct assignment to the blue vertices is printed out when the code runs, or if no such assignment is possible, an exception is thrown. Write only one statement per line (note that the for loop counts as one statement by the rules of the exam on page 1). HashSet blue = new HashSet(); HashSet green = new HashSet(); twocolor(G, 0, blue, green); System.out.println(“Blue vertices are: “ + blue.toString()); public static void twocolor(Graph G, int v, Set a, Set b){
for (__________________________________________) { if (________________________________________) { throw new IllegalArgumentException("graph is not bipartite"); } if (_______________________________________) {
} } }
UC BERKELEY Login: _________ 7. Trees and Hashing (18 points). a. (4.5 pts) Suppose we implement a LLRBBucketHashSet where the hash table buckets are stored as left leaning red black binary search trees. Assume we resize by doubling the number of buckets whenever the load factor L exceeds 2, and that we never decrease the number of buckets. Assume that hashCode computation is constant time. Do not assume that the hash code nicely spreads out items! If there are currently N items and M buckets in the hash table, fill in the runtimes for each operation for a single call in the table below (i.e. the first box is the best case for one put call ). Give your answer in Θ notation in terms of N and M. You may not need both N and M. Not all of these facts may be relevant. put containsKey remove Best case Worst case Best case Worst case Best case Worst case b. (4.5 pts) Fill in the runtimes below for a single operation for an LLRBBucketNoResizeHashSet, which is the same as in part a, except that the number of buckets is never increased. Give your answer in terms of N and M. You may not need to use both of these parameters. Not all facts may be relevant. put containsKey remove Best case Worst case Best case Worst case Best case Worst case c. (2 pts) If we have an LLRBBucketHashSet that is initially empty and we perform Q insertions, what is the amortized (i.e. average) runtime for a single call to put assuming that our hash code spreads items nicely across the buckets? Give your answer in the blank below in terms of Q and M. You may not need to use both of these parameters. Not all facts may be relevant. Amortized time per put call after Q calls: ________________________ d. (2 pts) Same question as part c, but for an LLRBBucketNoResizeHashSet. Amortized time per put call after Q calls: ________________________ e. (3 pts) Is an LLRBBucketHashSet significantly worse, about the same, or significantly better than a standard HashSet that uses a linked list for buckets? Explain your answer. LLRBBucketHashSets are: (^) ○ Significantly Worse ○ About the same ○ Significantly Better Explanation: ______________________________________________________________ f. (2 pts) Suppose we wanted to implement LLRBBucketHashSet by using the RedBlackBST.java file provided in our optional textbook. Would it be more appropriate to use an extension based approach (i.e. extending RedBlackBST), a delegation approach (having instance variables that include a RedBlackBST), or either one? It’d be better to use: (^) ○ Extension ○ Delegation ○ Either is appropriate
UC BERKELEY Login: _________ d. (15 pts) Describe an algorithm in English for building a XelhaTree (i.e. createXelhaTree). Your answer will be graded on correctness, efficiency, and clarity. To keep things organized, you might consider using a numbered list of steps as all or part of your answer. If you didn’t figure out c on the previous page, there’s probably no point in working on this one. For full credit , your algorithm must take less than Θ(N 2 ) time in the worst case, though partial credit will be given for algorithms that complete in Θ(N^2 ) time in the worst case. Ungraded: To double check your answer, give the results of running your algorithm on the input [ 9 , 3 , 6 , 2 , 10 , 14 ] below. End of Exam Decompression Space
