Problem 1. (10 points)

We have a set of n heroes H and a set of m dragons D (both n and m are at least 1). Each hero has a persuasion value pi  and each dragon has a determination dj . All pi and dj are positive integers. To avoid confusing arguments, each hero is limited to persuading only one dragon, but multiple heroes can work together to persuade the same dragon (if they do so, their combined persuasion is the sum of their indi- vidual persuasion values). A dragon is persuaded if the total persuasion value of the heroes persuading them is at least as large as the dragon’s determination. We need to find out if the heroes can persuade all dragons to not eat them and play a game of "Houses & Humans" instead.

Example:

When the heroes have persuasion H = {4, 7, 1, 2, 2} and the dragons have deter- mination D =  {4, 1, 10}, we can combine the heroes with persuasion 4 and 7 to overcome the dragon with determination 10 (as 4 + 7 ≥ 10), the two 2s can work together to persuade 4 (as 2 + 2 ≥ 4), and 1 can persuade 1. Hence, in this case we should return true.

When the heroes have persuasion {1, 1, 1} and the dragons have determination {1, 2, 3, 4}, there is no way to persuade the dragons and thus the heroes were never heard from again. In this case we should return false.

Your friend says that they’ve got two algorithms that can solve this problem easily:  wrongAlgorithm sorts the heroes by their persuasion in non-increasing order and the dragons by their determination also in non-increasing order.  It then re- peatedly assigns the unassigned hero with highest persuasion to the unpersuaded

dragon with highest determination. If at the end all dragons are persuaded (i.e., we reached the last dragon and persuaded it), it returns true, and false otherwise.

heapAlgorithm creates two max-heaps HH and HD and inserts the heroes into HH  and the dragons into HD .  While HH  is not empty, it removes the hero and dragon with the highest persuasion p and determination d and inserts d − p into HD (reflecting that the hero has partially convinced the dragon). Once it has processed all heroes, it checks whether the highest determination left in HD is at most 0 (i.e., whether the heroes together have persuaded all dragons).

a) Convince your friend that wrongAlgorithm doesn’t always return the cor- rect answer by giving a counterexample, i.e., an instance where the algorithm should return true, but returns false (or vice versa). Remember that describing the allocation of heroes to dragons that the algorithm computes and explain- ing why the output is incorrect are also part of your counterexample.

b) Argue whether heapAlgorithm always returns the correct answer by either arguing its correctness (if you think it’s correct) or by providing a counterex- ample (if you think it’s incorrect).

Problem 2. (25 points)

We’re attending a trading card game expo and we’re looking to complete the col- lection of our favourite game.  To make collecting the cards easier, the distributor decided to number all cards. Every seller sells cards in a specific range, say [k1, k2]. To make collecting more challenging, we’ve come up with the rule that when we visit a seller we want to get the card with the smallest integer number that we don’t already have. As we want to do this efficiently, we’re going to design a data struc- ture that supports the basic operations we need: insert(i) which inserts the card numbered i that we don’t own already, delete(i) which removes the card numbered i (if we own it) as we might want to trade cards, and smallestMissingInRange(k1, k2) which returns the smallest integer in the range [k1, k2] that we don’t already own.

Example execution:

insert(23)                                                        [23]

insert(4)                                                          [23, 4]

smallestMissingInRange(4, 15)   returns 5    [23, 4]

insert(5)                                                          [23, 4, 5]

smallestMissingInRange(4, 15)   returns 6    [23, 4, 5]

smallestMissingInRange(2, 15)   returns 2    [23, 4, 5]

remove(4)                                                        [23, 5]

smallestMissingInRange(4, 15)   returns 4    [23, 5]

Your task is to design a data structure that supports the above operations, more formally defined as follows:

• insert(i): Inserts integer i into the data structure. You can assume the integers we insert are all distinct, i.e., i is not yet present in the data structure upon insertion.

• delete(i): Deletes i from the data structure.

• smallestMissingInRange(k1, k2):  Returns the smallest integer i such that k1  ≤ i ≤ k2 and i isn’t present in the data structure. The function returns null if all integers in the specified range are present in the data structure.

For full marks, each operation should run in O(log n) time and your data struc- ture should use O(n) space, where n is the number of integers stored in the data structure.

a) Design a data structure that supports the above operations in the required time and space.

b) Briefly argue the correctness of your data structure and operations.

c) Analyse the running time of your operations and the total space of the data structure.

Problem 3. (25 points)

You and your best friend just got out of the movies and are very hungry; unfortu- nately, it is now very late at night and all restaurants are closed.  You manage to find, by chance, some vending machine still full of very...  nutritious foods (bags of chips, and the occasional cereal bar). Looking into your pockets, you look what change you have, in order to try and buy something (anything!) to eat.

You have n coins of various values.  So does your friend!  As for the vending machine... it contains n different "food items" each with its own price. Looks like you may eat tonight... except for two things:

• the vending machine is old and somewhat broken: it only accepts at most two coins, and does not return change: you must put exactly the right price to get the item you ask for.

• out of fairness, you and your friend refuse to pay for the whole thing alone. So each of you has to contribute (no matter how little): to buy the food, each of you has to contribute at least one coin.

Which means that, if you want to eat tonight, you must figure out if there is an item in the vending machine whose price is exactly equal to the sum of two coins, one from you and one from your friend. And you are very hungry, so you want to figure that out fast.

Your task:  given three arrays Y,  F,  and V  (You, Friend, Vending machine) each containing n positive integers, decide if there exist 0 ≤ i, j, k < n such that Y[i] + F[j] = V[k]. You can assume that all integers in all arrays are distinct (also

between different arrays).  Since you want to eat soon, you want an algorithm for this task which solves your problem fast: running in (expected) time O(n2 ).

Example:

Y = [3, 2, 1], F = [4, 5, 6], V = [50, 8, 13].

We need to return true, since Y[1] + F[2] = V[2] (i.e., 2 + 6 + 8).

For the same Y and F, but with V = [50, 10, 9823], we’d return false, as there are no i, j, and k such that Y[i] + F[j] = V[k].

a) As a warm-up, describe an O(n3 )-time algorithm in plain English.

b) Describe your efficient O(n2 ) (expected) time algorithm in plain English. Hint: use hashing and assume that you can compute the hash value of a key in constant time.

c) Prove the correctness of your O(n2 ) (expected) time algorithm.

d) Analyze the expected time complexity of your O(n2 ) (expected) time algorithm.

e) Analyze the worst-case time complexity of your O(n2 ) (expected) time algorithm.