Problem A: Graph Coloring

Transcription

1 Problem A: Graph Coloring Source: Input: Output: coloring.{c cc cpp} stdin stdout You are to write a program that tries to find an optimal coloring for a given graph. Colors are applied to the nodes of the graph and the only available colors are black and white. The coloring of the graph is called optimal if a maximum of nodes is black. The coloring is restricted by the rule that no two connected nodes may be black. Input The graph is given as a set of nodes denoted by numbers 1... n, n 100, and a set of undirected edges denoted by pairs of node numbers (n 1, n 2 ), n 1 n 2. The input file contains m graphs. The number m is given on the first line. The first line of each graph contains n and k, the number of nodes and the number of edges, respectively. The following k lines contain the edges given by a pair of node numbers, which are separated by a space. Output The output should consists of 2m lines, two lines for each graph found in the input file. The first line of should contain the maximum number of nodes that can be colored black in the graph. The second line should contain one possible optimal coloring. It is given by the list of black nodes, separated by a blank. Sample Input Sample Output

2 Problem B: Risk Source: Input: Output: risk.{c cc cpp} stdin stdout Risk is a board game in which several opposing players attempt to conquer the world. The gameboard consists of a world map broken up into hypothetical countries. During a player s turn, armies stationed in one country are only allowed to attack only countries with which they share a common border. Upon conquest of that country, the armies may move into the newly conquered country. During the course of play, a player often engages in a sequence of conquests with the goal of transferring a large mass of armies from some starting country to a destination country. Typically, one chooses the intervening countries so as to minimize the total number of countries that need to be conquered. Given a description of the gameboard with 20 countries each with between 1 and 19 connections to other countries, your task is to write a function that takes a starting country and a destination country and computes the minimum number of countries that must be conquered to reach the destination. You do not need to output the sequence of countries, just the number of countries to be conquered including the destination. For example, if starting and destination countries are neighbors, then your program should return one. The following connection diagram illustrates the first sample input. Input Input to your program will consist of a series of country configuration test sets. Each test set will consist of a board description on lines 1 through 19. The representation avoids listing every national boundary twice by only listing the fact that country I borders country J when I < J. Thus, the Ith line, where I is less than 20, contains an integer X indicating how many higher-numbered countries share borders with country I, then X distinct integers J greater than I and not exceeding 20, each describing a boundary between countries I and J. Line 20 of the test set contains a single integer (1 N 100) indicating the number of country pairs that follow. The next N lines each contain exactly two integers (1 A, B 20; A B) indicating the starting and ending countries for a possible conquest. There can be multiple test sets in the input file; your program should continue reading and processing until reaching the end of file. There will be at least one path between any two given countries in every country configuration

3 Output For each input set, your program should print the following message "Test Set #T" where T is the number of the test set starting with 1 (left-justified starting in column 11). The next NT lines each will contain the result for the corresponding test in the test set that is, the minimum number of countries to conquer. The test result line should contain the start country code A right-justified in columns 1 and 2; the string " to " in columns 3 to 6; the destination country code B rightjustified in columns 7 and 8; the string ": " in columns 9 and 10; and a single integer indicating the minimum number of moves required to traverse from country A to country B in the test set left-justified starting in column 11. Following all result lines of each input set, your program should print a single blank line. Sample Input Sample Output Test Set #1 1 to 20: 7 2 to 9: 5 19 to 5: 6 18 to 19: 2

4 16 to 20: 2

5 Problem C: Identifying Concurrent Events Source: Input: Output: concurrent.{c cc cpp} stdin stdout It is important in distributed computer systems to identify those events (at identifiable points in time) that are concurrent, or not related to each other in time. A group of concurrent events may sometimes attempt to simultaneously use the same resource, and this could cause problems. Events that are not concurrent can be ordered in time. For example, if event e 1 can be shown to always precede event e 2 in time, then e 1 and e 2 are obviously not concurrent. Notationally we indicate that e 1 precedes e 2 by writing e 1 e 2. Note that the precedes relation is transitive, as expected. Thus if e 1 e 2 and e 2 e 3, then we can also note that e 1 e 3. Sequential events in a single computation are not concurrent. For example, if a particular computation performs the operations identified by events e 1, e 2 and e 3 in that order, then clearly and e 1 e 2 and e 2 e 3. Computations in a distributed system communicate by sending messages. If event esend corresponds to the sending of a message by one computation, and event erecv corresponds to the reception of that message by a different computation, then we can always note that esend erecv, since a message cannot be received before it is sent. In this problem you will be supplied with lists of sequential events for an arbitrary number of computations, and the identification of an arbitrary number of messages sent between these computations. Your task is to identify those pairs of events that are concurrent. Input A number of test cases will be supplied. For each test case the input will include first an integer, NC, specifying the number of computations in the test case. For each of these NC computations there will be a single line containing an integer NE i that specifies the number of sequential events in the computation followed by NE i event names. Event names will always contain one to five alphanumeric characters, and will be separated from each other by at least one blank. Following the specification of the events in the last computation there will be a line with a single integer, NM, that specifies the number of messages that are sent between computations. Finally, on each of the following NM lines there will be a pair of event names specifying the name of the event associated with the sending of a message, and the event associated with the reception of the message. These names will have previously appeared in the lists of events associated with computations, and will be separated by at least one blank. The last test case will be followed by the single integer 0 on a line by itself. Output For each test case, print the test case number (they are numbered sequentially starting with 1), the number of pairs of concurrent events for the test case, and any two pair of the concurrent event names. If there is only one concurrent pair of events, just print it. And if there are no concurrent events, then state that fact.

6 Sample Input 2 2 e1 e2 2 e3 e4 1 e3 e1 0 Sample Output Case 1, 2 concurrent events: (e1,e4) (e2,e4)

7 Problem D: A Node Too Far Source: Input: Output: node.{c cc cpp} stdin stdout To avoid the potential problem of network messages (packets) looping around forever inside a network, each message includes a Time To Live (TTL) field. This field contains the number of nodes (stations, computers, etc.) that can retransmit the message, forwarding it along toward its destination, before the message is unceremoniously dropped. Each time a station receives a message it decrements the TTL field by 1. If the destination of the message is the current station, then the TTL field s value is ignored. However, if the message must be forwarded, and the decremented TTL field contains zero, then the message is not forwarded. In this problem you are given the description of a number of networks, and for each network you are asked to determine the number of nodes that are not reachable given an initial node and TTL field value. Consider the following example network: Input There will be multiple network configurations provided in the input. Each network description starts with an integer N C specifying the number of connections between network nodes. An N C value of zero marks the end of the input data. Following NC there will be NC pairs of positive integers. These pairs identify the nodes that are connected by a communication line. There will be no more than one (direct) communication line between any pair of nodes, and no network will contain more than 30 nodes. Following each network configuration there will be multiple queries as to how many nodes are not reachable given an initial node and TTL field setting. These queries are given as a pair of integers, the first identifying the starting node and the second giving the initial TTL field setting. The queries are terminated by a pair of zeroes. Output For each query display a single line showing the test case number (numbered sequentially from one), the number of nodes not reachable, the starting node number, and the initial TTL field setting.

8 Sample Input Sample Output Case 1: 5 nodes not reachable from node 35 with TTL = 2. Case 2: 1 nodes not reachable from node 35 with TTL = 3. Case 3: 8 nodes not reachable from node 1 with TTL = 1.

9 Problem E: Network Wars Source: Input: Output: wars.{c cc cpp} stdin stdout It is the year 2126 and comet Swift-Tuttle has struck the earth as predicted. The resultant explosion emits a large cloud of high energy neutrons that eliminates all human life. The accompanying electro-magnetic storm causes two unusual events: many of the links between various parts of the electronic network are severed, and some postgraduate AI projects begin to merge and mutate, in much the same way as animal life did several million years ago. In a very short time two programs emerge, Paskill and Lisper, which move through the network marking each node they visit: Paskill activates a modified Prolog interpreter and Lisper activates the Hello World program. However Hello World has mutated into an endless loop that so ties up the node that no other program, not even Lisper, can re-enter that node and the Prolog interpreter immediately reverse compiles (and destroys) any program that enters. However, Paskill knows which nodes it has visited and never tries to re-enter them. Thus if Lisper attempts to enter a node already visited by Paskill it will be annihilated; neither can enter a node already visited by Lisper, if either (or both) cannot move both will halt and if they ever arrive at a node simultaneously they annihilate each other. Both programs move through the network at the same speed. Write a program to simulate these events. All nodes in the the network are labelled with a single uppercase letter as shown below. When moving to the next node, Paskill searches alphabetically forwards from the current node, whereas Lisper searches alphabetically backwards from the current node, both wrapping round if necessary. Thus, (in the absence of the other) if Paskill enters the network below at A, it would visit the nodes in the order A, B, C, D, G, H, E, F ; if Lisper enters the network at H it would visit them in the order H, G, E, F. Simulation stops when one or more of the above events occurs. If more than one event occurs, mention Paskill first. C D G F B A E H Input Input will consist of a series of lines. Each line will describe a network and indicate the starting nodes for the two programs. A network is described as a series of nodes separated by ; and terminated by a period.. Each node is described by its identifier, a : and one or more of the nodes connected to it. Each link will be mentioned at least once, as will each node, although not all nodes will be described. After the period will appear the labels of the starting nodes first

10 Paskill and then Lisper. No line will contain more than 255 characters. The file will be terminated by a line consisting of a single #. Output Output will consist of one line for each network. Each line will specify the terminating event and the node where it occurs. The terminating event is one or two of the following: Lisper destroyed in node? {Paskill/Lisper} trapped in node? Both annihilated in node? Sample Input A:BD;C:BD;F:E;G:DEH;H:EG. A H # Sample Output Paskill trapped in node D Lisper trapped in node F