Problem A Moogle. source: moogle.c or moogle.cpp or moogle.java

Transcription

1 Problem A Moogle source: moogle.c or moogle.cpp or moogle.java You got the original idea of making map software, called Moogle Maps, for the new cool Maple mphone. It will even be capable of indicating the location of a house address like Main Street 13. However, since the mphone has limited storage capacity, you need to reduce the data amount. You don't want to store the exact location of every single house number. Instead only a subset of the house numbers will be stored exactly, and the others will be linearly interpolated. So you want to select house numbers that will minimise the average interpolation error, given how many house locations you have capacity to store. We view the street as a straight line, and you will always store the rst and the last house location. Given that you've stored the locations x i and x j for the houses with numbers i and j respectively, but no other house in between, the interpolated value for a house with number k with i < k < j is x i + (x j x i ) k i j i. The rst line of input gives a single integer, 1 t 50, the number of test cases. For each test case, there are two lines. The rst contains h 00 and c h, where h is the number of houses in the street and c is the number of house locations that can be stored. The second contains h integers in increasing order giving the location of the h houses. Each location is in the interval [0, ]. For each test case, output the average interpolation error over all the h houses for the optimal selection of c house locations to store. The output should be given with four decimal places

2 Problem B Phone List source: phone.c or phone.cpp or phone.java Given a list of phone numbers, determine if it is consistent in the sense that no number is the prex of another. Let's say the phone catalogue listed these numbers: Emergency 911 Alice Bob In this case, it's not possible to call Bob, because the central would direct your call to the emergency line as soon as you had dialled the rst three digits of Bob's phone number. So this list would not be consistent. The rst line of input gives a single integer, 1 t 40, the number of test cases. Each test case starts with n, the number of phone numbers, on a separate line, 1 n Then follows n lines with one unique phone number on each line. A phone number is a sequence of at most ten digits. For each test case, output YES if the list is consistent, or NO otherwise NO YES

3 Problem C Nested Dolls source: dolls.c or dolls.cpp or dolls.java Dilworth is the world's most prominent collector of Russian nested dolls: he literally has thousands of them! You know, the wooden hollow dolls of dierent sizes of which the smallest doll is contained in the second smallest, and this doll is in turn contained in the next one and so forth. One day he wonders if there is another way of nesting them so he will end up with fewer nested dolls? After all, that would make his collection even more magnicent! He unpacks each nested doll and measures the width and height of each contained doll. A doll with width w 1 and height h 1 will t in another doll of width w and height h if and only if w 1 < w and h 1 < h. Can you help him calculate the smallest number of nested dolls possible to assemble from his massive list of measurements? On the rst line of input is a single positive integer 1 t 0 specifying the number of test cases to follow. Each test case begins with a positive integer 1 m 0000 on a line of itself telling the number of dolls in the test case. Next follow m positive integers w 1, h 1, w, h,..., w m, h m, where w i is the width and h i is the height of doll number i. 1 w i, h i for all i. For each test case there should be one line of output containing the minimum number of nested dolls possible

4 Problem D Fortune at El Dorado source: fortune.c or fortune.cpp or fortune.java In his fabulous trip to El Dorado, Kamran made his fortune. After helping the king solve a dicult math problem, the king granted a piece of royal garden to Kamran. The king wrote a letter to the gardener, so that a rectangular region of the royal garden with a specied area, be given to Kamran. You could guess that the trees in El Dorado are made from Gold! After taking the letter to the royal gardener, Kamran found out that, unfortunately, there is no specic pattern for the location of the trees in the garden. To get the most prot, Kamran talked to the gardener and convinced him that it would not matter much, if Kamran chose the location of his rectangular share, and if he would choose a share smaller than the letter suggests. However, the gardener insisted that Kamran's subgarden should have sides parallel to the garden sides (which is itself a rectangle), and that vertices should have integer coordinates. The area of the piece of land must be positive, so that the king would not suspect anything. Now, given the locations of the trees in the garden and the maximum allowed area of his share, Kamran should nd a rectangular sub-garden with maximum number of trees inside; if a tree is on the border of the sub-garden, he can safely claim it. The only number of the rst line, T, is the number of dierent instances to be solved. T blocks follow, which describe dierent independent problems. The rst line of a block, contains two integers 0 F 1000, the number of trees in the eld, and the specied area A. The following F lines, each describes the location of a tree, by two nonnegative integers x, y (1 x, y 1000). No two trees have the same position. For each block in the input, write a single integer, which is the maximum number of trees that Kamran can obtain

5 Problem E Parking source: parking.c or parking.cpp or parking.java When shopping on Long Street, Michael usually parks his car at some random location, and then walks to the stores he needs. Can you help Michael choose a place to park which minimises the distance he needs to walk on his shopping round? Long Street is a straight line, where all positions are integer. You pay for parking in a specic slot, which is an integer position on Long Street. Michael does not want to pay for more than one parking though. He is very strong, and does not mind carrying all the bags around. The rst line of input gives the number of test cases, 1 t 100. There are two lines for each test case. The rst gives the number of stores Michael wants to visit, 1 n 0, and the second gives their n integer positions on Long Street, 0 x i 99. for each test case a line with the minimal distance Michael must walk given optimal parking

6 Problem F Wavelet Compression source: wavelet.c or wavelet.cpp or wavelet.java The discrete wavelet transform is a popular tool for signal compression. In this problem, your job is to write a program to decompress a one-dimensional signal (a list of integers) that has been compressed by a simple wavelet transform. To understand how this simple wavelet transform works, suppose that we have a list of an even number of integers. We compute the sum and dierence of each pair of consecutive samples, resulting in two lists of sums and dierences each having half the original length. Formally, if the original samples are a(1),..., a(n) the i-th sum s(i) and dierence d(i) are computed as: for i = 1,..., n/ : s(i) = a( i 1) + a( i) d(i) = a( i 1) a( i) This is then rearranged to give the transformed signal by rst listing the sums and then the dierences. For example, if the input signal is: 5,, 3,, 5, 7, 9, 6 Then the sum and dierence signals are: Thus, the transformed signal is: s(i) = 7, 5, 1, 15 d(i) = 3, 1,, 3 7, 5, 1, 15, 3, 1,, 3 The same process is applied recursively to the rst half of the transformed signal, treating s(i) as the input signal, until the length of the input signal is 1. In the example above, the nal transformed signal is: 39, 15,, 3, 3, 1,, 3 It is assumed that the length of the original input is a power of, and the input signal consists of integers between 0 and 55 (inclusive) only. The input consists of a number of cases. Each case is specied on a line, starting with an integer N (1 N 56) indicating the number of samples. The next N integers are the transformed samples. The end of input is indicated by a case in which N = 0. For each test case, output the original samples on a single line, separated by a single space. 6

7

8 Problem G Server Relocation source: server.c or server.cpp or server.java Michael has a powerful computer server that has hundreds of parallel processors and terabytes of main memory and disk space. Many important computations run continuously on this server, and power must be supplied to the server without interruption. Michael's server must be moved to accommodate new servers that have been purchased recently. Fortunately, Michael's server has two redundant power supplies as long as at least one of the two power supplies is connected to an electrical outlet, the server can continue to run. When the server is connected to an electrical outlet, it can be moved to any location which is not further away from the outlet than the length of the cord used to connect to the outlet. Given which outlet Michael's server is plugged into initially and nally, and the locations of outlets in the server room, you should determine the smallest number of times you need to plug a cord into an electrical outlet in order to move the server while keeping the server running at all times. Note that, in the initial and nal conguration, only one cord is connected to the power outlet. The rst line of input is an integer giving the number of cases to follow. For each case, the rst line is of the form where OUT LET S OUT LET_INIT IAL OUT LET_F INAL LENGT H1 LENGT H OUT LET S is the number of outlets in the server room ( OUT LET S 1000). OUT LET_INIT IAL is the index (starting from 1) of the outlet the server is initially connected to. OUT LET_F INAL is the index (starting from 1) of the outlet the server is nally connected to. LENGT H1 and LENGT H are the positive lengths of the two power cords, with at most three digits of precision after the decimal point (0 < LENGT H1, LENGT H 30000). These are followed by OUT LET S lines giving the integer coordinates of the wall outlets, one per line, with the k-th line giving the location of the k-th outlet. All coordinates are specied as two integers (x- and y-coordinates) separated by a space, with absolute values at most You may assume that all coordinates are distinct, and that the initial outlet and the nal outlet are dierent. For each case, print the minimum number of times you need to plug a cord into an electrical outlet in order to move the server to the nal location while keeping the server running at all times. If this is not possible, print Impossible. 8

9 Impossible 8 9

10 Problem H Best SMS to Type source: bestsms.c or bestsms.cpp or bestsms.java Using SMS today is more than a pleasing hobby. As the number of messages one sends through this service grows, the need to type them fast is better felt. Sometimes, one wonders how fast a message can be typed. Changing some words to their synonyms, might help type the whole message faster, if we were able to quickly calculate the time needed for a specic message. In the following, we assume that each message is a string of capital English letters and space character. The letters `A' through `Z' are assigned to keys `' to `9', as in Figure 1. To type a letter, one should press its key 1,, 3, or 4 times, depending on the position of the letter from left to right. Figure 1: Letter assignments. If two consecutive letters of the message are mapped to one key, one should wait for the rst letter to be xed on the screen and then use the key again to type the second one. For instance, to type the letter `X', one should press `9' twice. If the next letter of the message is not on the same key, one can continue to type the rest of the message. Otherwise, one has to wait for some time, so that the typed `X' is xed, and then the next letter (`W', `X', `Y', or `Z') can be typed. To type whitespace, we use the key `1'. As there is no letter mapped to the key `1', the whitespace needs no time to be xed. You are given the time needed to press any key, and the time one should wait for a letter to be xed. Your program should nd the minimum time needed to type a nonempty string, given the above rules. The input le contains multiple test cases. The rst line of the input, contains t, the number of test cases that follow. Each of the following t blocks, describes a test case. The rst line of each block contains p and w (1 p, w 1000), which show the amount of time in milliseconds for pressing a letter and waiting for it to be xed, respectively. The second line contains a non-empty string of length at most 1000, consisting of spaces or capital English letters. There is no leading or trailing spaces in a line. 10

11 For each test case, output one line showing the time needed to type the message in milliseconds ABBAS SALAM 7 11

12 Problem I Vacation Rentals source: vacation.c or vacation.cpp or vacation.java The Fifth Season Resort consists of a number of condominiums which are frequently occupied by their owners. At other times, however, they are available as vacation rentals. Since the resort has no more than 6 condominiums, they are identied by upper case letters. One day the resort manager's telephone rings. She receives a reservation request for a vacation rental with an arrival date of December and a departure date of December 9. She looks at the table of reservations, but doesn't nd a condominium that would be available for the entire period. Most of the existing reservations were made by the owners of the respective condominiums (who want to stay in their own units), so it is not desirable to move an existing reservation from one unit to another. As she continues to scrutinize the table of reservations, however, she has an idea and says: I can put you up in unit B for the rst three nights, and transfer you to unit F for the rest of your stay. Will that work? The person agrees and the reservation is made. Notice that reservations are done by nights, so that a one night reservation implies that the guest departs one day after the arrival. The goal of this problem is to satisfy such reservation requests (without changing existing reservations), with a minimum number of transfers (from one unit to another) during the requested period. The input consists of a number of cases. The rst line of each case contains two positive integers M and N. M is the number of consecutive days for which the resort's manager has a reservation table, and N is the number of units (condominiums) in the resort. The units are labeled by upper case letters starting at `A'. There are at most 100 days and at least 3 rooms in the reservation table. The days are numbered 1,,..., M for simplicity. The reservation table is given in the next M lines. Each line (row) of the table refers to a particular day (in the order 1,, 3, etc.), and each column of the table to a particular unit of the resort (in the order `A', `B', `C', etc.). An entry of 'X' means that the corresponding unit is reserved for that day, while an 'O' means that the unit is available. The reservation table is followed by one line of input, the reservation request, consisting of two integers: the arrival date and the departure date. The arrival date is in the range 1..M. The departure date is greater than the arrival date and less than or equal to M + 1. The end of input is indicated by M = N = 0. For each test case, rst print the case number followed by a colon and a blank line. If the reservation request can be met, the output will show a reservation schedule with a minimum number of transfers (from one unit to another) during the stay at the resort. Each line of the schedule corresponds to a consecutive stay in the same unit, and should be printed in the following way: <unit>: <start date>-<end date> where <unit> is the unit, <start date> is the date on which the guests moves into the unit, and <end date> is the date on which the guests moves out of the unit. The lines in the schedule should be ordered in ascending order by the start date. 1

13 Tie breaking rule. There may be several schedules with a minimum number of transfers. In these cases, choose the schedule which uses the lowest unit label in the rst day (so unit `A' is given preference to unit `B'). If there is still a tie, choose the schedule with the lowest unit number in the second day, and so on. If the reservation request cannot be met, print the line: Not available instead of the schedule. Separate the output of consecutive cases by a blank line XXXXXXX XOXXXXO XOXXXXO XOXXXOX OXXOXOX XOXOXOX OXXOXOX OXXXXOX XXXXXXX XXXXXXX Case 1: B: -5 F: