Problem A: Optimal Array Multiplication Sequence

Transcription

1 Problem A: Optimal Array Multiplication Sequence Source: Input: Output: array.{c cc cpp} stdin stdout Given two arrays A and B, we can determine the array C = A B using the standard definition of matrix multiplication: C i,j = k A i,k B k,j The number of columns in the A array must be the same as the number of rows in the B array. Notationally, let s say that rows(a) and columns(a) are the number of rows and columns, respectively, in the A array. The number of individual multiplications required to compute the entire C array (which will have the same number of rows as A and the same number of columns as B) is then rows(a) columns(b) columns(a). For example, if A is a array, and B is a array, it will take , or 3000 multiplications to compute the C array. To perform multiplication of more than two arrays we have a choice of how to proceed. For example, if X, Y, and Z are arrays, then to compute X Y Z we could either compute (X Y ) Z or X (Y Z). Suppose X is a 5 10 array, Y is a array, and Z is a array. Let s look at the number of multiplications required to compute the product using the two different sequences: (X Y ) Z X (Y Z) = = 7000 multiplications to determine the product multiplications to determine the product (X Y ), a 5 20 array. (Y Z), a array. Then = 3500 Then = 1750 multiplications to determine the final result. multiplications to determine the final result. Total multiplications: 4500 Total multiplications: 8750 Given the size of each array in a sequence of arrays to be multiplied, you are to determine an optimal computational sequence. Optimality, for this problem, is relative to the number of individual multiplcations required. Input For each array in the multiple sequences of arrays to be multiplied you will be given only the dimensions of the array. Each sequence will consist of an integer N which indicates the number of arrays to be multiplied, and then N pairs of integers, each pair giving the number of rows and columns in an array; the order in which the dimensions are given is the same as the order in which the arrays are to be multiplied. A value of zero for N indicates the end of the input. N will be no larger than 10. Output Assume the arrays are named A 1, A 2,... A N. Your output for each input case is to be a line containing a parenthesized expression clearly indicating the order in which the arrays are to be multiplied. Prefix the output for each case with the case number (they are sequentially numbered, starting with 1). Your output should strongly resemble that shown in the samples shown below. If, by chance, there are multiple correct sequences, any of these will be accepted as a valid answer.

2 Sample Input Sample Output Case 1: (A1 x (A2 x A3)) Case 2: ((A1 x A2) x A3) Case 3: ((A1 x (A2 x A3)) x ((A4 x A5) x A6))

3 Problem B: Jury Compromise Source: Input: Output: jury.{c cc cpp} stdin stdout In Frobnia, a far-away country, the verdicts in court trials are determined by a jury consisting of members of the general public. Every time a trial is set to begin, a jury has to be selected, which is done as follows. First, several people are drawn randomly from the public. For each person in this pool, defence and prosecution assign a grade from 0 to 20 indicating their preference for this person. 0 means total dislike, 20 on the other hand means that this person is considered ideally suited for the jury. Based on the grades of the two parties, the judge selects the jury. In order to ensure a fair trial, the tendencies of the jury to favour either defence or prosecution should be as balanced as possible. The jury therefore has to be chosen in a way that is satisfactory to both parties. We will now make this more precise: given a pool of n potential jurors and two values d i (the defence s value) and p i (the prosecution s value) for each potential juror i, you are to select a jury of m persons. If J is a subset of {1,..., n} with m elements, then D(J ) = k J d k and P (J ) = k J p k are the total values of this jury for defence and prosecution. For an optimal jury J, the value D(J ) P (J ) must be minimal. If there are several jurys with minimal D(J ) P (J ), one which maximizes D(J ) + P (J ) should be selected since the jury should be as ideal as possible for both parties. You are to write a program that implements this jury selection process and chooses an optimal jury given a set of candidates. Note: If your solution is based on an inefficient algorithm, it may not execute in the allotted time. Input The input file contains several jury selection rounds. Each round starts with a line containing two integers n and m. n is the number of candidates and m the number of jury members. These values will satisfy 1 n 200, 1 m 20 and of course m n. The following n lines contain the two integers p i and d i for i = 1,..., n. A blank line separates each round from the next. The file ends with a round that has n = m = 0. Output For each round output a line containing the number of the jury selection round (Jury #1, Jury #2, etc.). On the next line print the values D(J ) and P (J ) of your jury as shown below and on another line print the numbers of the m chosen candidates in ascending order. Output a blank before each individual candidate number. Output an empty line after each test case.

4 Sample Input Sample Output Jury #1 Best jury has value 6 for prosecution and value 4 for defence: 2 3

5 Problem C: Supermarket Source: Input: Output: market.{c cc cpp} stdin stdout Mr. Jones is an exemplary husband. Every Saturday morning Mrs. Jones gives him a list of goods to be bought from the supermarket and he buys exactly what he has been asked for, always choosing the brands with lowest prices. But Mr. Jones hates going to the supermarket on a Saturday, since its aisles are packed with shoppers. He wants to change the way he does his shopping. Instead of going to and from to buy the products on his wife s list, he will try to get the goods on the list going through each aisle only once, picking up the products in the exact order given in the list. So he asked you to write a program to help him with his new style of shopping. Given the information about products available in the supermarket together with their prices in the order in which they appear in Mr. Jones way and the list of products given by his wife, your program must determine the least cost that he would pay. Mr. Jones buys the products in the order in which they appear in Mrs. Jones list and he never goes back as he walks down the aisles. Therefore, if he buys the i-th product on his way as the j-th item on the list, the next product to be bought is the (j + 1)-th item of the list and it must be bought from the products that come after i in his path. The figure below shows an example where products are identified by integers. Note that different brands of the same product may appear separately. In the example Mr. Jones must buy products 1, 1, 2, 20 (notice that product 1 appears twice in the list). For the example, the least cost for Mr. Jones following his constraints is Notice that with this new way of shopping it may be impossible for Mr. Jones to buy all the goods on Mrs. Jones list; in that case, your program should warn Mr. Jones (a) Mrs. Jones s list (b) List of products with respective prices, and order they appear on Mr. Jones way down the aisles. Input Your program should process data for several shopping sessions. The first line in the description of a shopping session contains two integers M and N; M indicates the number of items in Mrs. Jones list (1 M 100) and N represents the total number of products available in the supermarket (1 N 100, 000). The next line contains M integers X i representing the list of products in Mrs. Jones list (1 X i 100, 000, 1 i M). Then N lines follow, representing the supermarket products in the order in which they appear in Mr. Jones way. Each of those lines contains an integer K and a real P which represent respectively a product identifier and its price (1 K 100, 000). The end of input is indicated by M = N = 0.

6 Output For each shopping session in the input, your program should produce one line of output, containing the least cost that Mr. Jones would pay. If it is not possible to buy all the goods for the session, print the word Impossible. The cost must be printed as a real number with two-digit precision, and the last decimal digit must be rounded. The input will not contain test cases where differences in rounding are significant. Sample Input Sample Output Impossible

7 Problem D: Always on the run Source: Input: Output: ontherun.{c cc cpp} stdin stdout Screeching tires. Searching lights. Wailing sirens. Police cars everywhere. Trisha Quickfinger did it again! Stealing the Mona Lisa had been more difficult than planned, but being the world s best art thief means expecting the unexpected. So here she is, the wrapped frame tucked firmly under her arm, running to catch the northbound metro to Charles-de-Gaulle airport. But even more important than actually stealing the painting is to shake off the police that will soon be following her. Trisha s plan is simple: for several days she will be flying from one city to another, making one flight per day. When she is reasonably sure that the police has lost her trail, she will fly to Atlanta and meet her customer (known only as Mr. P.) to deliver the painting. Her plan is complicated by the fact that nowadays, even when you are stealing expensive art, you have to watch your spending budget. Trisha therefore wants to spend the least money possible on her escape flights. This is not easy, since airlines prices and flight availability vary from day to day. The price and availability of an airline connection depends on the two cities involved and the day of travel. Every pair of cities has a flight schedule which repeats every few days. The length of the period may be different for each pair of cities and for each direction. Although Trisha is a good at stealing paintings, she easily gets confused when booking airline flights. This is where you come in. Input The input file contains the descriptions of several scenarios in which Trisha tries to escape. Every description starts with a line containing two integers n and k. n is the number of cities through which Trisha s escape may take her, and k is the number of flights she will take. The cities are numbered 1, 2,..., n, where 1 is Paris, her starting point, and n is Atlanta, her final destination. The numbers will satisfy 2 n 10 and 1 k Next you are given n(n 1) flight schedules, one per line, describing the connection between every possible pair of cities. The first n 1 flight schedules correspond to the flights from city 1 to all other cities (2, 3,..., n), the next n 1 lines to those from city 2 to all others (1, 3, 4,..., n), and so on. The description of the flight schedule itself starts with an integer d, the length of the period in days, with 1 d 30. Following this are d non-negative integers, representing the cost of the flight between the two cities on days 1, 2,..., d. A cost of 0 means that there is no flight between the two cities on that day. So, for example, the flight schedule means that on the first day the flight costs 75, on the second day there is no flight, on the third day it costs 80, and then the cycle repeats: on the fourth day the flight costs 75, there is no flight on the fifth day, etc. The input is terminated by a scenario having n = k = 0.

8 Output For each scenario in the input, first output the number of the scenario, as shown in the sample output. If it is possible for Trisha to travel k days, starting in city 1, each day flying to a different city than the day before, and finally (after k days) arriving in city n, then print The best flight costs x., where x is the least amount that the k flights can cost. If it is not possible to travel in such a way, print No flight possible.. Print a blank line after each scenario. Sample Input Sample Output Scenario #1 The best flight costs 460. Scenario #2 No flight possible.

9 Problem E: Transportation Source: Input: Output: train.{c cc cpp} stdin stdout Ruratania is just entering capitalism and is establishing new enterprising activities in many fields including transport. The transportation company TransRuratania is starting a new express train from city A to city B with several stops in the stations on the way. The stations are successively numbered, city A station has number 0, city B station number m. The company runs an experiment in order to improve passenger transportation capacity and thus to increase its earnings. The train has a maximum capacity n passengers. The price of the train ticket is equal to the number of stops (stations) between the starting station and the destination station (including the destination station). Before the train starts its route from the city A, ticket orders are collected from all onroute stations. The ticket order from the station S means all reservations of tickets from S to a fixed destination station. In case the company cannot accept all orders because of the passenger capacity limitations, its rejection policy is that it either completely accept or completely reject single orders from single stations. Write a program which for the given list of orders from single stations on the way from A to B determines the biggest possible total earning of the TransRuratania company. The earning from one accepted order is the product of the number of passengers included in the order and the price of their train tickets. The total earning is the sum of the earnings from all accepted orders. Input The input file is divided into blocks. The first line in each block contains three integers: passenger capacity n of the train, the number of the city B station and the number of ticket orders from all stations. The next lines contain the ticket orders. Each ticket order consists of three integers: starting station, destination station, number of passengers. In one block there can be maximum 22 orders. The number of the city B station will be at most 7. The block where all three numbers in the first line are equal to zero denotes the end of the input file. Output The output file consists of lines corresponding to the blocks of the input file except the terminating block. Each such line contains the biggest possible total earning.

10 Sample Input Sample Output 19 34