Car Parking. Task Description IOI 2000 Beijing PROBLEM

Transcription

1 CAR Car Parking PROBLEM A parking center by the Great Wall has a long row of parking places. One end of the row is considered left and the other end is considered right. The row is full of cars. Each car has a type and several cars may be of the same type. The types are identified by integer values. A number of workers decide to order the cars parked in the row in ascending order from left to right by the car types using the following method. In what is called a round, each of the workers can simultaneously drive one car out of its place and then park it in a place from where a car has been moved out in the same round. It may be that some workers are not moving a car in a round. For efficiency, a small number of rounds is preferable. Suppose that N is the number of cars and W is the number of workers. You are to write a program which, given the types of the parked cars and the number of workers, finds such a way to sort the cars that the number of rounds needed is at most N /( W 1), that is N/(W-1) rounded up. The minimal number of rounds is never greater than N /( W 1). Consider the following example. There are 10 parked cars of types 1,2,, and with workers. The initial placement of the cars from left to right identified by their types is The minimal number of rounds is three, and the rounds can be implemented so that the placement after each round is as follows: after round 1, after round 2, and 2 2 after round. INPUT The input file name is CAR.IN. The first line in the input file contains three integers. The first integer is the number of cars N, 2 N The second integer is the number of types M, 2 M 50. The car types are identified by the integers from 1 to M. There is at least one car of each type. The third integer is the number of workers W, 2 W M. The second line contains N integers, where the ith integer is the type of the ith car in the row, starting from the left end of the row.

2 CAR OUTPUT The output file name is CAR.OUT. The first line of the output file contains one integer R, which is the number of rounds in the solution. The next R lines describe the rounds ordered from 1 to R. In each line, the first integer is the number of cars C, which are moved in that round. After that follow 2C integers, identifying car positions. The car positions are identified by the integers from 1 to N starting at the left end. The first two are a pair describing how one of the cars is moved: the first integer is the position from the left end before the round and the second is the position from the left after the round. The next two integers are a pair describing how another car is moved, and so on. There may be several different solutions for these R lines, and your program only needs to output one of them. EXAMPLE INPUT AND OUTPUT CAR.IN CAR.OUT PARTIAL CREDIT Suppose that your program s output for an evaluation run is R and N /( W 1) is Q. If in your program s output the R rounds are not described correctly or they do not produce the desired order for the cars, then your score is 0. Otherwise, your score will be calculated from the maximum score as follows.!"r Q!"R=Q+1!"R=Q+2!"R>=Q+ 100% Score 50% Score 20% Score 0% Score

3 MEDIAN PROBLEM Median Strength A new space experiment involves N objects, labeled from 1 to N. It is known that N is odd. Each object has a distinct but unknown strength expressed by a natural number. For each strength Y, it holds that 1 Y N. The object with median strength is the object X such that there are equally many objects having smaller strength than X as there are objects having greater strength than X. You are to write a program that determines the object with median strength. Unfortunately, the only way to compare the strengths is by a device that, given three distinct objects, determines the object with median strength among the three objects. LIBRARY You are given a library named device with three operations: GetN, to be called once at the beginning without arguments; it returns the value of N. Med, called with three distinct object labels as arguments; it returns the label of the object with median (middle) strength. Answer, to be called once at the end, with one object label as argument; it reports the label of object X with median strength and properly ends the execution of your program. The library device produces two text files: MEDIAN.OUT and MEDIAN.LOG. The first line of file MEDIAN.OUT contains one integer: the label of the object passed to the library in the call to Answer. The second line will contain one integer: the number of calls to Med that have been performed by your program. The dialogue between your program and the library is recorded in the file MEDIAN.LOG. Instruction for Pascal programmers: Include the import statement uses device; in the source code. Instructions for C/C++ programmers: Use the instruction #include device.h in the source code, create a project MEDIAN.PRJ and add the files MEDIAN.C (MEDIAN.CPP) and DEVICE.OBJ into this project.

4 MEDIAN EXPERIMENTATION You can experiment with the library by creating a text file DEVICE.IN. The file must contain two lines. The first line must contain one integer: the number of objects N. The second line must contain the integers from 1 to N in some order: the ith integer is the strength of the object with label i. EXAMPLE DEVICE.IN The file DEVICE.IN above describes an input with 5 objects and strengths as below: Label Strength Here is a correct sequence of 5 library calls: 1. GetN (in Pascal) or GetN() (in C/C++) returns Med(1,2,) returns.. Med(,,1) returns.. Med(,2,5) returns. 5. Answer() CONSTRAINTS For the number of objects N we have 5 N 199 and N is odd. For the object labels i, we have 1 i N. For the object strengths Y, we have 1 Y N and all strengths are distinct. Pascal library file name: device.tpu Pascal function and procedure declarations: function GetN: integer; function Med(x,y,z:integer):integer; procedure Answer(m:integer); C/C++ library file names: device.h, device.obj (use large memory model) C/C++ function headers: int GetN(void); int Med(int x, int y, int z); void Answer(int m); No more than 7777 calls of function Med are allowed per run. Your program must not read or write any files.

5 PALIN Palindrome PROBLEM A palindrome is a symmetrical string, that is, a string read identically from left to right as well as from right to left. You are to write a program which, given a string, determines the minimal number of characters to be inserted into the string in order to obtain a palindrome. As an example, by inserting 2 characters, the string "Abbd" can be transformed into a palindrome ("dabbad" or "AdbbdA"). However, inserting fewer than 2 characters does not produce a palindrome. INPUT The input file name is PALIN.IN. The first line contains one integer: the length of the input string N, N The second line contains one string with length N. The string is formed from uppercase letters from A to Z, lowercase letters from a to z and digits from 0 to 9. Uppercase and lowercase letters are to be considered distinct. OUTPUT The output file name is PALIN.OUT. The first line contains one integer, which is the desired minimal number. EXAMPLE INPUT AND OUTPUT PALIN.IN 5 Abbd PALIN.OUT 2

6 POST Post Office PROBLEM There is a straight highway with villages alongside the highway. The highway is represented as an integer axis, and the position of each village is identified with a single integer coordinate. There are no two villages in the same position. The distance between two positions is the absolute value of the difference of their integer coordinates. Post offices will be bui lt in some, but not necessarily all of the villages. A village and the post office in it have the same position. For building the post offices, their positions should be chosen so that the total sum of all distances between each village and its nearest post office is minimum. You are to write a program which, given the positions of the villages and the number of post offices, computes the least possible sum of all distances between each village and its nearest post office, and the respective desired positions of the post offices. INPUT The input file name is POST.IN. The first line contains two integers: the first is the number of villages V, 1 V 00, and the second is the number of post offices P, 1 P 0, P V. The second line contains V integers in increasing order. These V integers are the positions of the villages. For each position X it holds that 1 X OUTPUT The output file name is POST.OUT. The first line contains one integer S, which is the sum of all distances between each village and its nearest post office as reported in the second line. The second line contains P integers in increasing order. These integers are the locations of the distinct villages in which the post offices will be built. There may be several different solutions for the locations, and your program needs to output only one of them. EXAMPLE INPUT AND OUTPUT POST.IN POST.OUT

7 POST PARTIAL CREDIT If your program s output does not satisfy the output requirements, then your score is 0. Otherwise, your score will be computed based on Table 1 as follows. If your program outputs sum S and the actual least possible sum is Smin, then calculate q=s/smin and find your score c in the bottom row. q=s/smin q= <q <q <q <q <q 1. 1.<q c Table 1.

8 WALLS Walls PROBLEM In a country, great walls have been built in such a way that every great wall connects exactly two towns. The great walls do not cross each other. Thus, the country is divided into such regions that to move from one region to another, it is necessary to go through a town or cross a great wall. For any two towns A and B, there is at most one great wall with one end in A and the other in B, and further, it is possible to go from A to B by always walking in a town or along a great wall. The input format implies additional restrictions. There is a club whose members live in the towns. In each town, there is only one member or there are no members at all. The members want to meet in one of the regions (outside of any town). The members travel riding their bicycles. They do not want to enter any towns, because of the traffic, and they want to cross as few great walls as possible, as it is a lot of trouble. To go to the meeting region, each member needs to cross a number (possibly 0) of great walls. They want to find such an optimal region that the sum of these numbers (crossing-sum, for short) is minimized Figure 1 Figure 2 The towns are labeled with integers from 1 to N, where N is the number of towns. In Figure 1, the labeled nodes represent the towns and the lines connecting the nodes represent the great walls. Suppose that there are three members, who live in towns, 6, and 9. Then, an optimal meeting region and respective routes for members are shown in Figure 2. The crossing-sum is 2: the member from town 9 has to cross the great wall between towns 2 and, and the member from town 6 has to cross the great wall between towns and 7. You are to write a program which, given the towns, the regions, and the club member home towns, computes an optimal region and the minimal crossing-sum. INPUT The input file name is WALLS.IN. The first line contains one integer: the number of regions M, 2 M 200. The second line contains one integer: the number of towns N, N 250. The third line contains one integer: the number of club members L, 1 L 0,

9 WALLS L N. The fourth line contains L distinct integers in increasing order: the labels of the towns where the members live. After that the file contains 2M lines so that there is a pair of lines for each region: the first two of the 2M lines describe the first region, the following two the second and so on. Of the pair, the first line shows the number of towns I on the border of that region. The second line of the pair contains I integers: the labels of these I towns in some order in which they can be passed when making a trip clockwise along the border of the region, with the following exception. The last region is the outside region surrounding all towns and other regions, and for it the order of the labels corresponds to a trip in counterclockwise direction. The order of the regions gives an integer labeling to the regions: the first region has label 1, the second has label 2, and so on. Note that the input includes all regions formed by the towns and great walls, including the outside region. OUTPUT The output file name is WALLS.OUT. The first line contains one integer: the minimal crossing-sum. The second line contains one integer: the label of an optimal region. There may be several different solutions for the region and your program needs to output only one of them. EXAMPLE INPUT AND OUTPUT The following input and output files correspond to the example given in the text. WALLS.IN WALLS.OUT 2

10 IOI20000 BLOCK Building with Blocks PROBLEM A unit cube is a 1x1x1 cube, whose corners have integer x, y, and z coordinates. Two unit cubes are connected when they share a face. A -dimensional solid object (solid, for short) is a non-empty connected set of unit cubes (see Figure 1). The volume of a solid is the number of unit cubes it contains. A block is a solid with volume at most. Two blocks have the same type when one can be obtained from the other by translations and rotations (not reflections). There are exactly 12 block types (see Figure 2). The colors in the figures only help to clarify the structure of the solids; they have no other meaning. A set D of blocks is a decomposition of a solid S when the union of all blocks in D equals S, and no two distinct blocks in D have a unit cube in common. Your task is to write a program that, given a description of the block types and a solid S, determines a smallest set of blocks into which S can be decomposed. It only needs to report the types of these blocks as often as they occur in the decomposition. INPUT In the input files, we identify a unit cube by a line with three integers x, y, and z, being the coordinate triple of its corner that minimizes x + y + z. The input file describing the block types is named TYPES.IN. The contents of this file are listed below and are the same for all evaluation runs. It contains the descriptions of the 12 block types in Figure 2, sorted on type number. Each block type is described by a group of consecutive lines. The first line contains the integer I identifying the block type (1 I 12). The second line contains the volume V of the block type (1 V ). The remaining V lines contain three integers x, y and z, each being one unit cube of the block type (1 x, y, z ). The input file describing the solid is named BLOCK.IN. The first line contains the volume V of the solid (1 V 50). The remaining V lines contain three integers x, y, z, each being one unit cube of the solid (1 x, y, z 7). OUTPUT The output file is named BLOCK.OUT. The first line must contain one integer M, being the smallest number of blocks into which the input solid can be decomposed. The second line lists M type identifiers of the block types into which the input solid can be decomposed. There may be several solutions for each input file, and your program needs to output only one of them.

11 IOI20000 BLOCK EXAMPLE INPUT AND OUTPUT TYPES.IN BLOCK.IN BLOCK.OUT Note: 1.This input file BLOCK.IN describes the solid of the horse in Figure 1. 2.Other solutions for the second line of the output file, which describes the types of the blocks used, are any of the following:

12 IOI20000 BLOCK Z Y X Figure 1. Horse Z X Y Figure 2. The 12 block types