A detachment of n soldiers must cross a wide and deep river with no bridge in sight. They notice two 12-year-old boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold two boys or one soldier. How can the soldiers get across the river and leave the boys in joint possession of the boat? How many times need the boat pass from shore to shore?
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- Four individuals wish to traverse an unstable bridge; they all start on the same side. You have 17 minutes to get everyone to the opposite side. Night has fallen, and they have only one torch. Two individuals may cross the bridge at once at most. Any crossing party, whether one or two individuals, must carry a torch. The torch must be carried back and forth; it cannot, for instance, be hurled. Person 1 crosses the bridge in one minute, person 2 in two minutes, person 3 in five minutes, and person 4 in ten minutes. A dyad must trek together at the slower individual's cadence. Create the algorithm specification that solves the problem.A detachment of n soldiers must cross a wide and deep river with no bridge in sight. They notice two 12-year-old boys playing in a rowboat by the shore. The boat is so tiny, however, that it can only hold two boys or one soldier. How can the soldiers get across the river and leave the boys in joint possession of the boat? How many times need the boat pass from shore to shore?To reduce traffic congestion in downtown Winburg, Indiana, the mayor convinces the city toset up a public transportation system. Bus-only lanes are to be established, and commuterswill be encouraged to “park and ride”; that is, to park their cars in suburban parking lots andthen take buses from there to work and back at cost of one dollar per ride. Each bus is tohave a fare machine that accepts only one dollar bills. Passengers insert a bill into the slotsas they enter the bus. Sensors inside the fare machine scan the bill, and the software in themachine uses an image recognition algorithm to decide whether the passenger has indeedinserted a valid dollar bill into the slot. It is important that the fare machine be accuratebecause, once the news gets out that any piece of paper will do the trick, fare income willplummet to effectively zero. Conversely, if the machine regularly rejects valid dollar bills,passengers will be reluctant to use the buses. In addition, the fare machine must be…
- Three prisoners have been sentenced to long terms in prison, but due to over crowed conditions, one prisoner must be released. The warden devises a scheme to determine which prisoner is to be released. He tells the prisoners that he will blindfold them and then paint a red dot or blue dot on each forehead. After he paints the dots, he will remove the blindfolds, and a prisoner should raise his hand if he sees at least one red dot on the other two prisoners. The first prisoner to identify the color of the dot on his own forehead will be release. Of course, the prisoners agree to this. (What do they have to lose?) The warden blindfolds the prisoners, as promised, and then paints a red dot on the foreheads of all three prisoners. He removes the blindfolds and, since each prisoner sees a red dot (in fact two red dots), each prisoner raises his hand. Some time passes when one of the prisoners exclaims, "I know what color my dot is! It's red!" This prisoner is then released. Your problem…On an island, a group of people are living until a visitor arrives with an odd directive: everyone with blue eyes must leave the island as quickly as possible. Every evening, a flight will depart at 8:00. Everyone can see the colour of everyone else's eyes, but they are blind to their own (nor is anyone allowed to tell them). They also do not know the number of persons who have blue eyes, but they do know that at least one person has them. How long will it take the folks with blue eyes to leave?Suppose a business person launches new cinema at Islamabad and ask his team to develop a ticket system for box office. He assigns some requirements about system that how should it work. The requirements are such a way that there are only '5' number of box office windows in the theatre. Each window can have at max '20' number of people waiting in line. To start with, only one window is opened. If the number of people waiting in line in that window exceeds 20, then the next window is opened and people can join the line in that window. Likewise, if both the first and second windows have n number of people waiting in each queue, then a third window is opened. This can go on until the maximum number of windows w is reached. Let us assume that once a window is opened it never closes. A new window is only opened if all open windows are full. Each person can buy only one ticket. So, the system should not allot more than one ticket per person. Let us assume that the system issues one ticket…
- On a chess board of r rows and c columns there is a lone white rook surrounded by a group of opponent's black knights. Each knight attacks 8 squares as in a typical chess game, which are shown in the figure - the knight on the red square attacks the 8 squares with a red dot. The rook can move horizontally and vertically by any number of squares. The rook can safely pass through an empty square that is attacked by a knight, but it must move to a square that is not attacked by any knight. The rook cannot jump over a knight while moving. If the rook moves to a square that contains a knight, it may capture it and remove it from the board. The black knights. never move. Can the rook eventually safely move to the designated target square? The figure illustrates how the white rook can move to the blue target square at the top-right corner in the first sample case. The rook captures one black knight at the bottom-right of the board on its way. Rok nd kight lcoes by Chunen Input The first line…A group of individuals are living on an island when a visitor arrives with an unusual order: all blue-eyed people must leave the island immediately. Every evening at 8:00 p.m., a flight will depart. Everyone can see everyone else's eye colour, but no one knows their own (nor is anyone allowed to tell them). Furthermore, they have no idea how many people have blue eyes, but they do know that at least one person has. How long will it take for the blue-eyed individuals to leave?Mr Monkey is standing in front of a row of banana trees on Skull Island which actually belong to his rival King Kong. The banana trees are unusually high on that island. He wants to steal as many bananas as possible. Here is his plan. He will climb up one of the trees and then keep jumping from one tree to the immediately next one, while collecting all the bananas from each of them. But he will not switch the direction of his jumps. No longer being the agile young monkey he once used to be, he can only jump a total distance of L, after which he will climb down and run away before Kong crushes his head. Let the trees be labelled as t1, t2, ·.· tn. Let v;, Vi = 1, 2, · ..n be the number of bananas in tree t;. Further, let l;, i = 1,2, Can you write an algorithm to help Mr. Monkey steal as many bananas as possible ? Your algorithm should be a polynomial in n, L. Please show all the steps of DP as in the sample solution - subproblem definition, recurrences, pseudocode, runtime . (Statutory…
- Page There is a famous railway station in PopPush City. Country there is incredibly hilly. The station was built in last century. Unfortunately, funds wwere extremely limited that time. It was possible to establish only a surface track. Moreover, it turned out that the station could be only a dead-end one (see picture) and due to lack of available space it could have only one track. 5, 4, 3, 2, 1 1, 2, 3, 4, 5 Station The local tradition is that every train arriving from the direction A continues in the direction B with coaches reorganized in some way. • Assume that the train arriving from the direction A has NS 1000 coaches numbered in increasing order 1, 2, . N. • The chieffor train reorganizations must know whether it is possible to marshal coaches continuing in the direction B so that their order will be a, a2. . aN. Help him and write a program that decides whether it is possible to get the required order of coaches. • You can assume that single coaches can be disconnected from…The Night Owls Society is arranging a conference. To minimize costs, they want to schedule as many sessions as possible in the main ballroom. Since night owls go to bed late and wake up late, they want sessions to begin as late as possible and don't care how late they run. Each session has a fixed start time and a fixed end time, though sessions may overlap. For example, the sessions may be as follows. (In a realistic version of this problem, there may be many more sessions.) Session Name Basket weaving Floral arrangements Cake decoration Cookie baking Pasta making Start time End time 10:30 AM 11:45 AM 10:45 AM 12:30 PM 12:15 AM 2:00 PM 12:45 PM 3:00 PM 2:15 PM 4:30 PM Table decoration 3:30 PM 5:45 PM a. Define what it means for this problem to have optimal substructure. b. Show that this problem does have optimal substructure. c. What algorithmic strategy can one use to solve a problem with optimal substructure? (Do not write out the algorithm.) d. What does it mean for a problem to…The Harvard robotics club has organized a Robothon. n robots are placed alongthe edge of a circular area at the middle of the OAT(open air theatre). Each robot will move along arbitrary tracksinside the circle while leaving behind a heat signature along its trail. However, they have beenprogrammed not to cross their own trail or the trail of another robot, neither will they ever moveout of the circle. In case a pair of robots i and j meet at any point, they are removed from the sceneand the club will pay a reward sum of M[i, j] to the owners of these robots. Note that some robotscan keep moving infinitely without ever meeting another one. Given the reward matrix M whereM[i, j] = M[j, i], design a polynomial time algorithm that determines the maximum money theclub might potentially end up spending. For this particular problem, give a very brief justificationof the recurrence. Give pseudo-code for a dynamic program which solves the recurrence efficiently Youdo not need to prove…