CS 124 Final exam: Spring 2022
Your name: Instructions:
• Spend at most 180 minutes on the exam, from the time you first see a problem to the last time you write anything in your solutions. (Time spent uploading doesn’t count against your time.)
Time started: Time ended:
• Do not communicate with anyone but the course staff about the contents of the exam.
• Do not use a computer or other electronic aids during the exam for anything except for:
1. (optionally) typesetting your solutions,
2. (optionally) accessing offline notes prepared before the exam, without using a search function, 3. (optionally) asking clarification questions of the instructors on Ed.
• (You may use your notes and/or books prepared before the exam.)
• After you finish the exam, type or sign your name in affirmation of the following statement:
I, , pledge my honor that I have obeyed all instructions on this page in taking this exam.
Reminder for extension students:
After you finish, upload your exam to Gradescope as if it were a problem set.
• You have 180 minutes to earn a maximum of 180 points. Do not spend too much time on any single problem: we recommend roughly n minutes on an n point problem. Read all the problems first, and attack them in the order that allows you to make the most progress.
• You can ask clarification questions of the instructors on Ed/in person.
• Be neat and write legibly and clearly. You will be graded not only on the correctness of your answers,
but also on the clarity with which you express them.
• Do not waste time re-deriving results presented in class, psets, or sections. You may simply state and cite them.
1 True or False (18 points)
For each of the following statements, say whether the statement is TRUE or FALSE. Each correct answer
is worth 2 points. Incorrect answers are worth 0 points. No explanations are needed.
1. On a connected, undirected graph G, if a breadth-first search starting from a vertex s finds a path of
length 124 to a vertex t, then every path from s to t has length at most 124.
True False
2. If every problem in NP can be solved in polynomial time, then there is a polynomial-time algorithm to break RSA encryption.
True False
3. If we insert at least mk elements into an initially-empty Bloom filter with m ≥ 1 bits of memory and k ≥ 1 hash functions and then look up an element (not necessarily one of the ones we inserted), then it is possible for the lookup to be a false negative.
True False
4. If we insert 0 elements into an initially-empty Bloom filter with m ≥ 1 bits of memory and k ≥ 1 hash functions and then look up an element (necessarily not one we inserted) and then look up an element, then it is possible for the lookup to be a false negative.
True False
5. Let ODD be the function that takes an integer as input and returns True iff x is odd. Then, for every decision problem f solvable in polynomial time, there’s a polynomial-time reduction from f to ODD.
True False
6. Let G be a directed graph with positive even integer edge weights, and let s and t be two vertices of G. There is necessarily a maximum-valued flow from s to t in which the flow on every edge is even.
True False
7. Let G be a directed graph with positive odd integer edge weights, and let s and t be two vertices of G. There is necessarily a maximum-valued flow from s to t in which the flow on every edge is odd.
True False
8. In the simulated annealing algorithm for Number Partition, if we choose the temperature (probability of moving to a worse neighbor) to be 0, the resulting algorithm is equivalent to the hill-climbing algorithm—that is, they have the same asymptotic runtime and distribution of outputs.
True False
9. Assuming P ̸= NP, there is a polynomial-time algorithm that, given a directed graph, vertices s and t and positive integer edge capacities (as in a maximum flow problem), finds the maximum-valued flow from s to t that puts flow at least 1 on every edge.
True False
2 False (18 points)
Each of the following statements is FALSE. For each of them, prove it’s false by providing a counterexam-
ple. No explanation is necessary; only the example itself will be graded.
1. For every undirected graph G with |V (G)| > 1, if G has a vertex v whose degree is greater than every
other vertex’s degree, then no maximum independent set of G contains v.
2. For every undirected graph G with |V (G)| > 1, if G has a vertex v whose degree is less than every other vertex’s degree, then no minimum vertex cover of G contains v.
3. For every undirected graph G with positive integer edge weights, if G has an edge e whose weight is greater than every other edge’s weight, and G has a Hamiltonian cycle (i.e. a Traveling Salesperson Tour) that doesn’t include e, then the shortest Hamiltonian cycle (TSP tour) doesn’t include e.
3 A Sneak Preview of this Exam’s Questions (18 points)
In the year 2124, CS124 will pilot an exam-hints system: each student will be allowed to ask one TF (before the exam) for a hint about what will be on the final exam. Each TF knows some fraction of the contents of each problem: for instance, Ashley might know 100% of problem 1, 50% of problem 2, and 200% of problem 3 (that is, she knows two solutions to problem 3), and Lawrence might know 100% of problem 1, 20% of problem 2, and 0% of problem 3.
You can ask for fractional hints from multiple TFs, as long as you get a total of only one hint: for instance, you could ask for 60% of a hint from Ashley and 40% of a hint from Lawrence, and you’d learn about 100% of problem 1, 38% of problem 2, and 120% of problem 3. You only care about knowing up to 100% of a solution to each problem, so even though you know 2.58 solutions, you only know and care about 2.38.
1. (9points)You’dliketomaximizethenumberofsolutionsyouknowandcareabout(like2.38above), given the level of knowledge ci, j by each TF i of each problem j. Give a reduction from this problem to a linear program. Your reduction should run in polynomial time, but you don’t need to prove this. You don’t need to prove the correctness of your reduction.
2. (9points)EachTFisresponsibleforcommunicatingwiththeotherTFswheneverastudentasksthem for a hint, but they’re not reliable at doing so. Instead, each TF only informs their friends (friends go both ways) of the tips they’ve distributed. (Their friends don’t pass that information along recursively.) You’d like to take advantage of this to cheat and ask for more than 1 total hint, but if any TF realizes that you’ve asked for more than 1 total hint, you will be sent to the Ad Board!
For instance, suppose Tarun and Evan are friends, Evan and Lavanya are friends, and Lavanya and Phyllis are friends, but no other pairs of TFs are friends. If you ask Tarun and Lavanya for .6 hints each, then Evan will realize that you’ve asked for 1.2 hints and report you. If you instead ask both Tarun and Phyllis for 1 hint each, you’ll get away with it.
Again, you’d like to maximize the number of solutions you know and care about, given the level of knowledge ci, j by each TF i of each problem j and the list of what pairs of TFs are friends.
Give a reduction from this problem to a linear program. Your reduction should run in polynomial time, but you don’t need to prove this. You don’t need to prove the correctness of your reduction.
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Calculations (48 points)
1. (12 points) Find an inverse of 124 modulo 1249 (that is, an integer x such that 124x ≡ 1 (mod 1249)),
and show the steps of whatever algorithm you use to do so.
2. (10 points) Suppose I search for a solution to the 2SAT formula (x∨¬y)∧(y∨¬z)∧(¬x∨z)
by choosing an assignment of the variables (uniformly at random from the 8 possible assignments), then repeatedly choosing an unsatisfied clause (uniformly at random from all unsatisfied clauses) and a random variable in it (uniformly at random from its two variables) and flipping the value of that variable. (If it happens that I guess a satisfying assignment correctly to start, I do 0 variable flips.) Compute the expected number of variable flips I need to find a satisfying assignment.
3. (11 points) I attempted to implement a hash table with chaining and table doubling. The table started at size 22 and contained 0 elements. If, after adding an element x to the table, the number of elements in the table equaled the size of the table, I intended to rebuild a table of size 2k for the next biggest value of k, and rehash everything into it. Unfortunately, due to a typo, I actually rebuilt a table of size k2 for the next biggest value of k, and rehashed everything into it. Calculate the amortized time per insert operation for the first n insert operations. (Give an asymptotic answer, like O(log n).) (Assume that computing one hash value takes time O(1).)
4. (15 points) Suppose I run the Edmonds-Karp algorithm to find a maximum flow from s to t in the graph below. In the breadth-first search that Edmonds-Karp uses to select paths to push flow along, break ties to explore alphabetically-earlier vertices before alphabetically-later ones. What paths does Edmonds-Karp push flow along, in order, and how much flow does it push along each? (Your answer should be of a form like “First, 3 flow on S-A-B-T, then 3 flow on S-B-A-T, then . . . ”.)
5 Non-adjacent subset sum (15 points)
Give an algorithm that takes as input an array of n integers and returns the maximum sum of a subset of them that doesn’t contain any three consecutive elements of the array. For instance, on input [4, 4, 5, 2, 3, −1, −3, 5], your algorithm should return 18 (because 18 = 4 + 4 + 2 + 3 + 5). Your algorithm should run in time O(n). Prove its correctness and analyze its runtime.
6 Almost a Majority (15 points)
Give an algorithm that returns True if any element in an input list of n integers appears more than n/124 times, and False otherwise. For full credit, your algorithm should run in worst-case linear time. For up to 12 points of partial credit, your algorithm can run in expected linear time. You don’t need to prove your algorithm’s runtime. Prove the correctness of your algorithm.
7 Orders under permutations (12 points)
Recall that the resemblance R(A, B) of two sets A and B is defined as R(A, B) = |A∩B| . |A∪B|
SupposethatforsomepositiveintegerN,AandBaresubsetsof[N](i.e. {0,1,…,N−1})with|A|≥2 and |B| ≥ 2. Prove or disprove: for every such A and B, if we choose a uniformly random permutation π:[N]→[N],theprobabilitythatsecond-largest({π(a):a∈A})=second-largest({π(b):b∈B})isR(A,B). (Given a set X, second-largest(X) is the second-largest element of X.)
8 Martian utilities (36 points)
Elon is planning to bring power and water to his Martian colonies! A colony consists of nm buildings arranged in an n × m grid, and the building at position (i, j) has a positive integer height ci, j .
1. (16 points) To bring water to the buildings, Elon plans to build pumps at some of them. The cost to build a pump at height h is $h. A pump provides water to its own building and the buildings immediately north, south, east, and west of it (up to 4 buildings, but fewer at the boundaries of the colony). The goal is to bring water to all buildings with the minimal cost. Give a 5-approximation algorithm for the problem: that is, an algorithm that returns a set of generators whose total cost is within a factor of 5 of the smallest possible total cost. For full credit, your algorithm should run in O(nm) time, assuming all relevant integers are small enough that integer arithmetic operations take O(1) time, but slower (polynomial-time) correct algorithms will receive some credit. You don’t need to prove a bound on your algorithm’s runtime. Prove the correctness of your algorithm.
(Hint: if each pump could only supply water to its own building or one of its four neighbors, but you could build multiple pumps in the same place, what would the optimal solution be?)
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2. (16points)Tobringelectricitytothecolonies,Elonplanstobuildwindturbinesatthemand/orpower
lines between them. The cost to build a wind turbine at height h is $h2. The cost to build a power line
between two buildings is the distance (in three dimensions) between them. A wind turbine provides
power to its own building and all buildings connected to it by (chains of) power lines. For instance, if
n=1,m=3,andtheheightsofthebuildingsarec1,1 =1,c1,2 =2,andc1,3 =3,thenthecheapestway
to power the colony is to build a wind turbine at (1,1,1) (for $1) and power lines connecting (1,1,1) to
(1,2,2) and (1,2,2) to (1,3,3) (for $ 2 each).
Give an algorithm to find the set of locations to build wind turbines and power lines that minimizes the total cost. For full credit, your algorithm should run in O(n2m2) time, assuming all relevant integers are small enough that integer arithmetic operations take O(1) time, but slower (polynomial-time) correct algorithms will receive some credit. You don’t need to prove a bound on your algorithm’s runtime. Prove the correctness of your algorithm.
3. (4 points) In the electricity-provision problem above, modern computer scientists do not know any polynomial-time algorithm to determine, given an integer $k, whether it’s possible to provide power to all the colonies for a cost of at most $k. Explain how this is consistent with the existence of an algorithm as above.
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