MIT Programming Contest Team Contest 1 Problems 2008

Transcription

1 MIT Programming Contest Team Contest 1 Problems 2008 October 5, Edit distance Given a string, an edit script is a set of instructions to turn it into another string. There are four kinds of instructions in an edit script: Add ( a ): Output one character. This instruction does not consume any characters from the source string. Delete ( d ): Delete one character. That is, consume one character from the source string and output nothing. Modify ( m ): Modify one character. That is, consume one character from the source string and output a different character. Copy ( d ): Copy one character. That is, consume one character from the source string and output the same character. A shortest edit script is an edit script that minimizes the total number of adds, deletes, and modifies. Given two strings, generate a shortest edit script that changes the first into the second. Input The input consists of two strings on separate lines. The strings contain only alphanumeric characters. Each string has length between 1 and 17000, inclusive. 1

2 Output The output is a shortest edit script. Each line is one instruction, given by the one-letter code of the instruction (a, d, m, or c), followed by a space, followed by the character written (or deleted if the instruction is a deletion). In case of a tie, you may generate any shortest edit script. Example Sample Input abcde xabzdey Sample Output a x c a c b m z c d c e a y 2

3 2 Fish Catch Fishermen of New England like to catch so called SLSS fish. SLSS stands for straight line same speed fish, because such fish is very lazy and rarely changes its speed and direction of travel. Fishermen bring a net in a shape of a circle, that has an adjustable radius. They locate the best position for the center of the net, based on their visual observation of high concentration of fish. There are N fish. They also write down the initial position and the direction of travel of each fish. Their plan is to catch K fish in order to make a nice dinner for their families. However, they don not want to catch many more fish, so that there is enough fish left in the Ocean. They need to set a proper radius length of the net, which leads them to a hard computational problem. As a fishermen s friend, please help them to find the smallest radius of the net such that there is a moment when they can dip the net into the Ocean and catch at least K fish. Input The first line of the input contains two integers C x and C y separated by a space character (1 C x, C y 10 4 ), that represent the coordinates of the center of the net. The second line of the input contains two integers N and K separated by a space character (5 N 1000, 1 K N). Each of the next N lines contain four integers A x, A y, B x, B y separated by a space character (0 A x, A y, B x, B y 10 4 ). A x and A y represent the initial coordinates of a fish (at time zero), while B x and B y represent the estimated coordinates of a fish after 1 second. Note that fishermen can use the net at any moment starting from time zero. Output In the first line of the output write one real number R using exactly 5 decimal digits of precision which is the desired radius of the net. It should be followed by a newline. Example Sample Input Sample Output

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5 3 The (Bayesian) Hound and the Hare Researchers have discovered that hounds are extremely intelligent so intelligent, in fact, that they use Bayesian statistics to hunt down their prey. As an experiement, researchers studying these hounds blindfold one and put it in a maze laid out on a square grid. They put a hare somewhere else in the maze. Once every second, the hare takes a step one unit in a random direction (no diagonals), chosen uniformly at random among the directions that aren t blocked by walls. Sometimes, when the hare takes a step, it makes a little bit of noise at its new position. The hound hears the noise with a certain amount of uncertainty σ that is, if the hare is at position (x, y), then the hound thinks that it heard the hare at position (x, y ) with probability N exp (((x x ) 2 + (y y ) 2 )/(2σ 2 )), where N is an appropriate normalization factor. After every step the hare takes, the hound moves one unit in the direction (North, East, South, or West, or not moving at all) that mininizes the expected shortest-path distance to the hare. In case of a tie, the hound favors not moving, then North, then East, then South, then West. (That is, the hound first considers whether to stay go West; then, if the expected distance if it goes South is better or within 10 5, the hound considers going West; then South; then North; then staying still.) Neither the hound nor the hare can walk into a wall or on a diagonal, and the hound knows this. The hound has a perfect memory and knows the exact layout of the maze. The hound assumes that the hare starts at a uniform random location within the maze. The hound assumes that it could occupy the same space as the hare without realizing it. Write a program that determines what path the hound takes given a sequence of observations. Input The maze has dimension X units West to East by Y units North to South. The first line of input is two numbers, X and Y. You may assume that 3 X, Y 100. This is followed by a map of the maze. The first line corresponds to the northernmost strip of the maze, and the first column is the western side. A # indicates a wall, a. indicates open space, and a h marks the starting position of the hound. The sides of the maze are always walls, and the interior of the maze is connected. The map of the maze is followed by σ, an integer, on one line. You may assume that σ 1. The remainder of the input is a series of observations, one line per second. A line that contains silence indicates that the hound hears nothing, and a line that contains two integers gives the coordinates column, row where northwest corner of the maze is (0,0) in which the hound thinks it heard the hare. Note that the coordinates may be outside the maze. 5

6 Output For each observation, output N, S, E, W, or. if the hound goes North, South, East, West, or stays still, respectively, after each observation. Example Sample Input #################### #################### ####...h...##### ####.#########.##### ####.#########.##### ####.#########.##### ####.#########.##### ####.#########.##### ####...##### ####.#########.##### ####.#########.##### ####.#########.##### ####...##### #########.########## #########.########## #########.########## #########.########## #################### #################### #################### 5 silence silence silence Sample Output E E E E E S 6

7 4 Infinite Game Krzysztof had a little too much to drink tonight and is now walking around aimlessly. He has two sets of moves A, B. Each of A, B is a finite subset of the positive integers. Krzysztof starts off standing at 0, and at each step he either takes a step to the right by an integer in A, or takes a step to the left by an integer in B. In odd-numbered steps his move must be from A, and in even-numbered steps his move must be from B (the first step he makes is considered step number 1). Is it true that for every integer x (both the positives and the negatives), Krzysztof can eventually reach x? Input The first line contains a positive integer t, the number of test cases (1 t 500). t sets of lines then follow. Each set of lines start with a line with two space-separated integers n, m (1 n, m 10000), denoting the sizes of A and B, respectively. n lines then follow listing the elements of A, then m lines follow listing the elements of B. Each element of A, B will be an integer between 1 and 10 9, inclusive. Output For each test case output YES if Krzysztof can eventually reach all integers; else output NO. The word you output should be followed by a newline. Example Sample Input Sample Output NO YES 7

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9 5 Minesweeper In the game of minesweeper, you are shown a grid. Some cells in the grid contain mines. If you think that a cell does not contain a mine, you can reveal that cell. If the cell contained a mine, you lose, but if not, then you learn how many neighboring cells, including diagonals, also contain mines. Given a minesweeper grid, determine which cells must contain mines and which cells cannot contain mines. Input The first line of input is two integers, the width and height of the grid. The remainder of the input is an image of the grid. A cell containing a digit is revealed, which means that it contains no mine and that the number of adjacent mines (including diagonals) is that digit. A cell containing a period is not revealed. The grid is at most 30 columns by 16 rows. This problem is, in general, hard, so we will not give input in which the number of possible arrangements of mines adjacent to revealed cells is more than a few tens of thousands. We also guarantee that there is at least one valid arrangement of mines. Output Output an image of the grid. Each revealed cell should have its digit printed, cells that must contain mines are marked by *, and cells that cannot contain mines but are not revealed are marked by a space. Example Sample Input Sample Output * * * **

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11 6 Space Boomerang Far outside of our universum, kids of a hyperspace creatures play the following game. They have a boomerang toy that they can program and throw. Naturally, they want it to come back to the same location. To program it, they have a bunch of modules that they can install into it. Each module gives it power along a specific direction. For each module they install, they enter a non-zero distance that the toy will move along that direction (obviously, they do not have to install all of them in one game). Distance can be both positive and negative, where negative distance means that it travels in the opposite direction. After programming, they throw the boomerang. It travels thorugh the space in a funny way (it moves using modules simultaneously, until they are all out of power). Of course, they repeat the game many times. Modules can be reused. They would like to maximize their fun by using different modules. However, some of them might be useless, because it not possible to use them and make boomerang coming back to the same location. Please, help the kids to find such modules! Input The first line of the input contains two integers M and N separated by a space character (1 M 3000, 1 N 300). M is the number of modules, and N is the number of dimensions of the hyperspace. Each of the next M lines contain N real numbers separated by a space character that represent coordinates of a direction in N-dimenional space for the coresponding module. Modules are indexed by numbers 1 through M, in the order given in the input. Each coordinate is given with at most two decimal digits, and its absolute value is not grater than Output In the first line of the output write one integer K, which is the number of useless modules. In the next K lines write indices of those modules, one per line, in the same order as they appear in the input. Terminate each line with a newline character. Example Sample Input Sample Output

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13 7 Rectangles There are N axis-parallel rectangles in the plane. Calculate the total area they cover. Input The first line contains a positive integer N (1 N 200, 000). N lines then follow, each with four space-separated values x 1 x 2 y 1 y 2 denoting the rectangle [x 1, x 2 ] [y 1, y 2 ]. Each of the values x 1, x 2, y 1, y 2 are integers between 0 and 10 9, inclusive, and for a given rectangle x 1 < x 2 and y 1 < y 2 always. Output Output the total area covered by the given N rectangles. Any region covered by multiple rectangles should only be counted once. Example Sample Input Sample Output 5 13

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15 8 Spanning Trees Count the number of minimum-weight spanning trees of a connected weighted multigraph (there can be self-loops, as well as multiple edges between any two vertices). In the input graph, no edge weight appears more than 4 times. Input The first line of the input contains two integers N and M separated by a space character (1 N , 1 M 10 5 ). N is the number of vertices, and M is the number of edges of a graph. Vertices are indexed with numbers 1 through N. Each of the next M lines contain three integers a, b, w, separated by a space character (1 a, b N, 1 w 2 30 ), meaning that there is an edge of weight w between vertices a and b. Output Write one integer K followed by a newline, which is the number of minimum spanning trees modulo Example Sample Input Sample Output 5 15

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