10/17/2014 Welcome Party Problem A: Welcome Party
Source file: welcome.{c, cpp, java} Input file: welcome.in
For many summers, the Agile Crystal Mining company ran an internship program for students. They greatly valued interns' ability to self-organize into teams. So as a get-to-know-you activity during orientation, they asked the interns to form teams such that all members of a given team either have first names beginning with the same letter, or last names beginning with the same letter. To make it interesting, they asked the interns to do this while forming as few teams as possible.
As an example, one year there were six interns: Stephen Cook, Vinton Cerf, Edmund Clarke, Judea Pearl, Shafi Goldwasser, and Silvio Micali. They were able to self-organize into three teams:
Stephen Cook, Vinton Cerf, and Edmund Clarke (whose last names all begin with C) Shafi Goldwasser and Silvio Micali (whose first names begin with S) Judea Pearl (not an interesting group, but everyone's first name in this group starts with J)
As a historical note, the company was eventually shut down due to a rather strange (and illegal) hiring practice---they refused to hire any interns whose last names began with the letter S, T, U, V, W, X, Y, or Z. (First names were not subject to such a whim, which was fortunate for our friend Vinton Cerf.)
Input: Each year's group of interns is considered as a separate trial. A trial begins with a line containing a single integer N, such that 1 ≤ N ≤ 300, designating the number of interns that year. Following that are N lines---one for each intern---with a line having a first and last name separated by one space. Names will not have any punctuation, and both the first name and last name will begin with an uppercase letter. In the case of last names, that letter will have an additional constraint that it be in the range from 'A' to 'R' inclusive. The end of the input is designated by a line containing the value 0. There will be at most 20 trials.
Output: For each trial, output a single integer, k, designating the minimum number of teams that were necessary.
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/welcome/welcome.html 1/2 10/17/2014 Welcome Party
Example Input: Example Output:
6 3 Stephen Cook 6 Vinton Cerf Edmund Clarke Judea Pearl Shafi Goldwasser Silvio Micali 9 Richard Hamming Marvin Minskey John McCarthy Edsger Dijkstra Donald Knuth Michael Rabin John Backus Robert Floyd Tony Hoare 0
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/welcome/welcome.html 2/2 10/17/2014 Round Robin Problem B: Round Robin
Source file: round.{c, cpp, java} Input file: round.in
Suppose that N players sit in order and take turns in a game, with the first person following the last person, to continue in cyclic order. While doing so, each player keeps track of the number of turns he or she has taken. The game consists of rounds, and in each round T turns are taken. After a round, the player who just had a turn is eliminated from the game. If the remaining players have all had the same number of turns, the game ends. Otherwise, they continue with another round of T moves, starting with the player just after the one who was most recently eliminated.
As an example, assume we begin a game with N=5 and T=17, labeling the players in order as A, B, C, D, and E, with all counts initially zero.
Player A B C D E Count 0 0 0 0 0
Beginning with A, 17 turns are taken. B will have taken the last of those turn, leaving our counts as follows:
Player A B C D E Count 4 4 3 3 3
Suppose that after every 17 turns, the player who just had a turn is eliminated from the game. All remaining players then compare their counts. If all of those counts are equal, everyone has had a fair number of turns and the game is considered completed. Otherwise, they continue with another round of 17 moves starting with the player just after the one who was most recently eliminated.
Continuing with our example, B will leave the game, and the next turn will be for C.
Player A C D E Count 4 3 3 3
After 17 more turns starting with C, we find that A, D and E received 4 turns, while C received 5 turns, including the last:
Player A C D E Count 8 8 7 7
Then C leaves, and since the remaining counts are not all the same, a new round beings with D having the next turn. http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/round/round.html 1/2 10/17/2014 Round Robin
Player A D E Count 8 7 7
The next 17 turns start with D and end with E. A adds 5 turns, while D and E add 6:
Player A D E Count 13 13 13
Then E leaves.
Player A D Count 13 13
At this point, notice that the two remaining players have the same count of 13. Therefore, the game ends. (We note that E's count was irrelevant to the decision to end the game.)
Input: The input will contain one or more datasets. Each dataset will be described with a single line containing two integers, N and T, where N (2 ≤ N ≤ 100) is the initial number of players, and T (2 ≤ T ≤ 100) is the number of turns after which the player with the most recently completed turn leaves. Following the last dataset is a line containing only 0.
Output: There is one line of output for each dataset, containing two numbers, p and c. At the time the game ends p is the number of players that remain in the game and c is the common count they all have.
Example input: Example output:
5 17 2 13 7 10 5 3 90 2 45 1 0
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/round/round.html 2/2 10/17/2014 Missing Pages Problem C: Missing Pages
Source file: missing.{c, cpp, java} Input file: missing.in
Long ago, there were periodicals called newspapers, and these newspapers were printed on paper, and people used to read them, and perhaps even share them. One unfortunate thing about this form of media is that every so often, someone would like an article so much, they would take it with them, leaving the rest of the newspaper behind for others to enjoy. Unfortunately, because of the way that paper was folded, not only would the page with that article be gone, so would the page on the reverse side and also two other pages that were physically on the same sheet of folded paper.
For this problem we assume the classic approach is used for folding paper to make a booklet that has a number of pages that is a multiple of four. As an example, a newspaper with 12 pages would be made of three sheets of paper (see figure below). One sheet would have pages 1 and 12 printed on one side, and pages 2 and 11 printed on the other. Another piece of paper would have pages 3 and 10 printed on one side and 4 and 9 printed on the other. The third sheet would have pages 5, 6, 7, and 8.
When one numbered page is taken from the newspaper, the question is what other pages disappear.
Input: Each test case will be described with two integers N and P, on a line, where 4 ≤ N ≤ 1000 is a multiple of four that designates the length of the newspaper in terms of numbered pages, and 1 ≤ P ≤ N is a page that has been taken. The end of the input is designated by a line containing only the value 0.
Output: For each case, output, in increasing order, the page numbers for the other three pages that will be missing.
Example input: Example output:
12 2 1 11 12 12 9 3 4 10 8 3 4 5 6 0
ACM Mid-Central Programming Competition 2013 http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/missing/missing.html 1/1 10/17/2014 Probability Paradox Problem D: Probability Paradox
Source file: paradox.{c, cpp, java} Input file: paradox.in
This problem considers repeated tosses of a fair coin. Each outcome, H or T, has probability 1/2. Any specific sequence of tosses of the same length, like HHH or THH, has the same probability (for example, 1/8 for a sequence of length 3).
Now consider the following two-player game. Each player chooses a distinct pattern of possible coin flips having a common fixed length. For example, the first player might predict HHH while the second predicts THH. The game then begins with both players observing the same coin as it is repeatedly flipped, until one of them witnesses their pattern. For example, if the sequence of observed flips begins HHTHTTHH... the second player in our example wins the game, having just witnessed the pattern THH.
The question that interests us is, given the two players' patterns, how likely is it that the first player wins the game? Because we are flipping a fair coin, many would assume that the patterns are irrelevant and that each player has probability 1/2 of winning the game. However, this is not the case, leading to what is known as the Probability Paradox.
For some patterns, it will be that the first player wins with probability precisely 1/2. For example, by symmetry, the patterns TT and HH have equal chances of occurring first.
However, consider again our original example in which the first player chooses HHH and the second player chooses THH. For this particular match-up, the only way that the first player can win is if each of the first three tosses are H. For if the earliest HHH were to come somewhere other than at the beginning of the game, the pattern could be represented as
...?HHH
where "..." means a possibly empty earliest part of the sequence, and "?" refers to the toss immediately before the HHH. The "?" can not refer to an H, as in
...HHHH
because there would have been an earlier HHH that ended the game, the underlined part of ...HHHH. Yet if the preceding toss were a T, as in
...THHH.
then the second player would have already won, having observed pattern THH at the underlined ...THHH. Therefore, when considering pattern HHH vs. THH, the first player wins if and only if the first three flips are H, an event that happens with probability 1/8.
As one more example, if the first player chooses TTH and the second chooses THH, the first player will win with probability 2/3. Your job is to write a program that computes such a probability.
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/paradox/paradox.html 1/2 10/17/2014 Probability Paradox Input: The input will contain one or more datasets, each on a single line. Each dataset will consist of two equal-length yet distinct patterns using only characters H and T. The common pattern length will be in the range from 1 to 9, inclusive. There is a space between the two patterns. The input ends with a line containing only "$".
Output: For each data set, output a single line with the exact probability that the first sequence precedes the second in a random sequence of fair coin tosses. The probability must be stated as a rational number, reduced to lowest terms, with a "/" between the numerator and denominator. Because each player has a nonzero chance of winning, this probability will always be strictly between 0 and 1.
Warning: The numerator and denominators for all of the final probabilities can be expressed as 32-bit integers. However, depending on your approach, you may need 64-bit integers (type long in Java or long long in C++) for some intermediate calculations, and even then you must be careful.
Example input: Example output:
TT HH 1/2 HHH THH 1/8 TTH THH 2/3 THHTH HTHTT 15/28 HTTHHHTHT THHHTHTHH 259/452 $
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/paradox/paradox.html 2/2 10/17/2014 Letter Cubes Problem E: Letter Cubes
Source file: cubes.{c, cpp, java} Input file: cubes.in
This problem is based on a puzzle by Randall L. Whipkey.
In the game of Letter Cubes, there are a set of cubes, with each face of each cube having a letter of the alphabet, such that no letter appears more than once within the entire set. The maximum number of cubes is 4, allowing for up to 24 of the 26 letters of the alphabet to occur.
Words are formed by rearranging and turning the cubes so that the top letters of all the cubes together spell a word. The 13 words below have been made using a particular set of cubes.
CLIP CLOG CONE DISH FAZE FURL MARE MOCK QUIP STEW TONY VICE WARD
Only 23 distinct letters were used in the above words, so we will tell you the extra information that a B is included on one cube. Can you now determine the letters on each cube? For the above set of words, there is indeed a unique set of cubes. We will state this solution in canonical form as
ABCHTU DEKLQY FGIMNW OPRSVZ
Note that the letters on each individual cube are stated as a string of characters in alphabetical order, and the four 6-letter strings representing the four cubes are also listed in alphabetical order.
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cubes/cubes.html 1/3 10/17/2014 Letter Cubes A simpler example relies on two cubes, forming the following 11 two-character strings (although the puzzles are more fun when the strings are actual words, they do not need to be):
PI MU HO WE WO BE MA HI RE AB PY
The only solution for the two cubes forming these strings is
AEIOUY BHMPRW
The same two cubes could be determined without the last pair PY being listed, as long as you were told that there was a Y on one cube. Your job is to make similar deductions.
Input: The input will contain from 1 to 20 datasets. The first line of each dataset will include a positive integer n (6 ≤ n ≤ 30) and a character c, described below. The next n lines will each contain a string of uppercase letters. Each string will be the same length, call it k, with 2 ≤ k ≤ 4. Following the last dataset is a line containing only 0.
Returning to the issue of the special character, c, on the first input line for each dataset, there will be two cases to consider. Recall that the implicit set of k cubes must use 6*k distinct letters on their collective faces If all 6*k of those letters appear within the set of strings, then the character c on the first line of input is a hyphen, '-'. Otherwise, the strings have been chosen so that only one letter on the cubes does not appear. In this case, the character c on the first line of input will be that undisplayed letter. (For example, the B in our opening puzzle.)
Output: There is one line of output for each dataset, containing a 6-letter string for each cube, showing the letters on the faces of that cube. Each of those strings should have its letters in alphabetical order, and the set of strings should be given in alphabetical order with respect to each other, with one space between each pair. We have chosen datasets so that each has a unique solution.
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cubes/cubes.html 2/3 10/17/2014 Letter Cubes
Example input: Example output:
13 B ABCHTU DEKLQY FGIMNW OPRSVZ CLIP AEIOUY BHMPRW CLOG AEIOUY BHMPRW CONE DISH FAZE FURL MARE MOCK QUIP STEW TONY VICE WARD 11 - PI MU HO WE WO BE MA HI RE AB PY 10 Y PI MU HO WE WO BE MA HI RE AB 0
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cubes/cubes.html 3/3 10/17/2014 Digit Sum Problem F: Digit Sum
Source file: digitsum.{c, cpp, java} Input file: digitsum.in
When Grace was in third grade, her elementary school teacher assigned her the following problem:
What is the smallest possible sum of two numbers that together use the numerals 1, 2, 7, 8, and 9?
Grace figured out that the answer to this problem is 207 (for example, as 78 + 129), but when the teacher assigned four pages of similar problems as homework, Grace got bored. It turns out that Grace was a rather advanced third grader, so she decided that it would be more fun to write a computer program to solve such problems. Surely you can do the same!
Input: Each problem is described on a single line. The line begins with an integer N, such that 2 ≤ N ≤ 14, designating the number of numerals included in the problem. Following that are those N numerals. There will always be at least 2 numerals that are nonzero. The end of the input is designated by a line containing only the value 0.
Output: For each case, output a line with the minimum sum S that can be achieved. Please keep in mind that by standard convention, the numeral 0 cannot appear as the first digit of either summand.
Example input: Example output:
5 1 2 7 8 9 207 6 3 4 2 2 2 2 447 9 0 1 2 3 4 0 1 2 3 11257 0
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/digitsum/digitsum.html 1/1 10/17/2014 Cash Cow Problem G: Cash Cow
Source file: cashcow.{c, cpp, java} Input file: cashcow.in
Years before Candy Crush became the wildly popular game that may lead developer Saga to a multi- billion dollar IPO, there was an online game named Cash Cow, which remains part of the Webkinz platform.
This single-player game has a board with 12 rows and 10 columns, as shown in Figure 1. We label the rows 1 through 12, starting at the bottom, and the columns a through j, starting at the left. Each grid location can either have a colored circle or be empty. (We use uppercase characters to denote distinct colors, for example with B=blue, R=red, and Y=yellow.) On each turn, the player clicks on a circle. The computer determines the largest "cluster" to which that circle belongs, where a cluster is defined to include the initial circle, any of its immediate horizontal and vertical neighbors with matching color, those circles' neighbors with matching colors, and so forth. For example, if a user were to click on the blue circle at cell (h10) in Figure 1, its cluster consists of those cells shown with empty circles in Figure 2.
Figure 1 Figure 2 Figure 3 Processing a click on cell h10.
The player's turn is processed as follows. If the indicated grid cell belongs to a cluster of only one or two circles (or if there is no circle at that cell), the turn is wasted. Otherwise, with a cluster of 3 or more circles, all circles in the cluster are removed from the board. Remaining circles are then compacted as follows:
1. Circles fall vertically, to fill in any holes in their column. 2. If one or more columns have become empty, all remaining columns slide leftward (with each nonempty column remaining intact), such that they are packed against the left edge of the board.
For example, Figure 3 shows the board after the cluster of Figure 2 was removed after the click on (h10).
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cashcow/cashcow.html 1/3 10/17/2014 Cash Cow As another example, Figure 4 below, portrays the processing of a subsequent click on cell (j1). During that turn, column (e) becomes empty, and the resulting columns (f) through (j) slide to become columns (e) through (i), respectively. Figure 5 provides one further example in which several columns are compacted.
Figure 4: Processing a click on cell j1.
Figure 5: Processing a click on cell g2.
Input: The input will consist of multiple games, each played with a new board. For each game, the input begins with a number T that denotes the number of turns that the player will be making, with 1 ≤ T ≤ 20. Following that will be an initial board configuration, which always has 12 rows and 10 columns per row, with uppercase letters used to denote distinct colors. There will never be empty cells within the initial board. Following the presentation of the initial board will be T additional lines of input, each designating a cell of the grid; we rely on the coordinate system illustrated in the above figures, with a lowercase letter, from a to j, denoting a column and a number from 1 to 12 that denotes a row. We note that if a player clicks on a grid cell that does not currently have any circle, that turn is simply wasted.
The end of the entire input will be designated by a line with the number 0.
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cashcow/cashcow.html 2/3 10/17/2014 Cash Cow Output: For each game, output a single line designating the the number of circles that remain on the board after all of the player's turns are processed.
Example input: Example output:
3 33 RYBBRBYYRY 62 RRRBBBBBRR 2 YRRBRBBBBR RYYBRYYRYY BRBBRBRBRY YYBYRBBRRB RYBBBBRYYY YBRBRBRYRB RYBBBBBBBY YBBRRRRRBB RBBRRBRYRR BBBRRYYYRR h 10 j 1 g 2 3 YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYBYYBBBBB YYBYYBBBBB c 2 c 12 g 1 2 YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYYYYBBBBB YYBYYBBBBB YYBYYBBBBB g 1 c 12 0
ACM Mid-Central Programming Competition 2013 http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/cashcow/cashcow.html 3/3 10/17/2014 Sort Me Problem H: Sort Me
Source file: sortme.{c, cpp, java} Input file: sortme.in We know the normal alphabetical order of the English alphabet, and we can then sort words or other letter sequences. For instance these words are sorted:
ANTLER ANY COW HILL HOW HOWEVER WHATEVER ZONE
The standard rules for sorting letter sequences are used:
1. The first letters are in alphabetical order. 2. Among strings with the same prefix, like the prefix AN in ANTLER and ANY, they are ordered by the first character that is different, T or Y here. 3. One whole string may be a prefix of another string, like HOW and HOWEVER. In this case the longer sequence comes after the shorter one.
The Gorellians, at the far end of our galaxy, have discovered various samples of English text from our electronic transmissions, but they did not find the order of our alphabet. Being a very organized and orderly species, they want to have a way of ordering words, even in the strange symbols of English. Hence they must determine their own order. Unfortunately they cannot agree, and every Gorellian year, they argue and settle on a new order.
For instance, if they agree on the alphabetical order
UVWXYZNOPQRSTHIJKLMABCDEFG
then the words above would be sorted as
WHATEVER ZONE HOW HOWEVER HILL ANY ANTLER COW
The first letters of the words are in their alphabetical order. Where words have the same prefix, the first differing letter determines the order, so the order goes ANY, then ANTLER, since Y is before T in their choice of alphabet. Still HOWEVER comes after HOW, since HOW is a prefix of HOWEVER. http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/sortme/sortme.html 1/2 10/17/2014 Sort Me Dealing with the different alphabetical orders each year by hand (or tentacle) is tedious. Your job is to implement sorting with the English letters in a specified sequence.
Input: The input will contain one or more datasets. Each dataset will start with a line containing an integer n and a string s, where s is a permutation of the English uppercase alphabet, used as the Gorellians' alphabet in the coming year. The next n lines (1 ≤ n ≤ 20) will each contain one non-empty string of letters. The length of each string will be no more than 30. Following the last dataset is a line containing only 0.
Output: The first line of output of each dataset will contain "year " followed by the number of the dataset, starting from 1. The remaining n lines are the n input strings sorted assuming the alphabet has the order in s.
Example input: Example output:
8 UVWXYZNOPQRSTHIJKLMABCDEFG year 1 ANTLER WHATEVER ANY ZONE COW HOW HILL HOWEVER HOW HILL HOWEVER ANY WHATEVER ANTLER ZONE COW 5 ZYXWVUTSRQPONMLKJIHGFEDCBA year 2 GO TEAMS ALL GO ACM GO TEAMS ALL GO ACM 10 ZOTFISENWABCDGHJKLMPQRUVXY year 3 THREE ZERO ONE ONE NINE TWO FIVE THREE SEVEN FOUR ZERO FIVE TWO SIX FOUR SEVEN EIGHT EIGHT SIX NINE 0
ACM Mid-Central Programming Competition 2013
http://mcicpc.cs.atu.edu/archives/2013/mcpc2013/sortme/sortme.html 2/2 I - Udyr’s Counterjungling
Fizz is going about his own peaceful business in the jungle - however, his enemy, Udyr, won’t stand for that. He’d like to counterjungle Fizz - in other words, walk over and destroy him.
The jungle can be modelled as an infinite 2D plane, with Udyr initially standing at coordinates (XU , YU ), and Fizz standing at different coordinates (X , Y ). There are also N (0 N 3) vision wards belonging F F ≤ ≤ to Fizz in the jungle, the ith of which is at coordinates (Xi, Yi) and has a vision radius of Ri. Fizz can see everything going on within each ward’s circle of vision, which includes all points which are no further than Ri units away from (Xi, Yi). Due to the dense foliage, Fizz is unable to see anything else, even things right next to himself (unless he’s standing in range of one of his vision wards). All coordinates and radii have absolute values no greater than 104. Udyr would like to reach Fizz’s location undetected. If he spends any time in range of any of the vision wards, Fizz will notice his presence and get to safety. Determine whether or not Udyr may take some path to Fizz’s location without being seen. Note that Udyr and/or Fizz may be initially located in range of a ward, in which case Udyr’s task is surely impossible. Input
Line 1: 1 integer, T , the number of test cases (1 T 1000) ≤ ≤ For each test case:
Line 1: 5 integers, XU , YU , XF , YF , and N
Next N lines: 3 integers, Xi, Yi, and Ri, for i = 1..N Output
For each test case: Line 1: The string “Yes” if Udyr can reach Fizz, or the string “No” otherwise Sample Input
2 1 9 7 1 2 4 4 3 8 6 2 10 9 3 16 3 8 13 4 8 6 3 14 8 4 Sample Output
Yes No Explanation of Sample
The first case is illustrated below. As can be seen, Udyr can easily walk around both vision wards without entering their range and reach Fizz. y 6 10 U
...... t ...... 5 ...... q ...... q ...... F ...... - 0 t 0 5 10 x
The second case is illustrated below. As can be seen, Udyr cannot reach Fizz without passing through the range of at least one of the vision wards. y 6
...... F ...... 15 .. .. t ...... q ...... 10 ...... U ...... t...... q ...... 5 ...... q ......
0 - 0 5 10 15 x J - Lee Sin’s Tour
Lee Sin would like to take a tour of the battlefield (which can be represented as an infinite 2D plane), starting from coordinates (XL, YL) and ending by the safety of a friendly tower - in particular, he needs to reach any point no more than R units away from (XT , YT ). However, though he doesn’t care how long it’ll take, he also doesn’t like walking. As such, he will only move be repeatedly jumping from his current location to the location of a ward on the ground, no further than J units away from him. There are initially N (0 N 1000) wards on the battlefield, with the ith one ≤ ≤ at coordinates (Xi, Yi). All of the above values (except for N) are integers with absolute values no greater than 104, and R and J are positive. Any of the coordinates may be equal to one another. Before Lee Sin starts his tour, he can ask his friend Sona to place as many additional wards as he wants anywhere (even at non-integral coordinates). However, he doesn’t want to trouble her more than necessary. As such, determine the minimum number of wards which Sona must place such that Lee Sin will be able to reach the tower’s range from his initial location, by only jumping from ward to ward.
Input
Line 1: 1 integer, T , the number of test cases (1 T 50) ≤ ≤ For each test case:
Line 1: 3 integers, XL, YL, and J
Line 2: 3 integers, XT , YT , and R Line 3: 1 integer, N
Next N lines: 2 integers, Xi and Yi, for i = 1..N Output
For each test case: Line 1: 1 integer, the minimum number of additional wards which must be placed in order for Lee Sin to be able to reach the tower’s range
Sample Input
2 1 14 5 21 4 2 3 14 12 3 5 5 11 -5 -12 1 -6 -10 5 0
Sample Output
3 0
Explanation of Sample
The first case is illustrated below, with existing wards represented by filled circles, and a possible minimal set of locations at which Lee Sin could place new wards represented by empty circles. His corresponding path of travel is also shown. y 6
15 L Z Z s Z Z S vf 10 S v S S @ f @ ...... @...... 5 ...... T . . . .. f .. v ...... q......
0 - 0 5 10 15 20 x
In the second case, Lee Sin starts already in range of the tower, so he doesn’t need to place any additional wards. K - Morgana’s Hourglass
Morgana is under attack by her foes! They’re casting N (1 N 105) spells at her - the ith spell will ≤ ≤ 9 be cast Si seconds into the battle, and will remain active for Ti seconds (1 Si,Ti 10 ). During that 4 ≤ ≤ time, it will continuously deal Di (1 Di 10 ) damage to her per second. For example, the spell will deal Di ≤ ≤ 2 damage in half of a second, and DiTi total damage throughout its duration. Fortunately, Morgana is prepared for a fight - her friend Zhonya lent her a magical hourglass! Morgana can activate its spell exactly once to prevent all damage that would be done to her during a contiguous period of H (1 H 109) seconds. Determine the minimum total amount of damage that Morgana can sustain if she wisely≤ chooses≤ when to use the hourglass.
Input
Line 1: 1 integer, T , the number of test cases (1 T 50) ≤ ≤ For each test case: Line 1: 2 integers, N and H
Next N lines: 3 integers, Si, Ti, and Di, for i = 1..N Output
For each test case: Line 1: 1 integer, the minimum amount of damage that Morgana can sustain
Sample Input
2 1 5 10 10 10 3 12 22 18 4 5 15 3 15 15 3
Sample Output
50 88
Explanation of Sample
In the first case, if Morgana uses the hourglass entirely during the only spell, she can block half of its 100 total damage. For example, if she decides to use it 12 seconds into the battle, she’ll take 20 damage from time 10 to 12, no damage from time 12 to 17, and 30 more damage from time 17 to 20. In the second case, Morgana can choose to use the hourglass 18 seconds into the battle. She’ll then take 48 damage before this point (39 from the second spell, and 9 from the third), and after its effect runs out at time 30, she’ll take 40 more damage (all from the first spell). L - Mordekaiser’s Brazilian Vegetation
Mordekaiser, in addition to being a fearsome warrior, is also the number one gardener in Brazil. Over the years, he’s nurtured a beautiful tree - a connected, undirected graph with N nodes (numbered 1..N) and 5 N 1 edges (1 N 10 ). The ith edge connects nodes Ai and Bi (1 Ai,Bi N). Each node is labelled with− a letter - either≤ ≤ an “H”, “U”, or “E”. ≤ ≤ Due to the nature of trees, it’s guaranteed that, for any pair of different nodes i and j, there’s exactly one shortest path connecting them. If this path is traversed from the starting node i to the ending node j, and the labels of the nodes passed on the way are concatenated in order (including those of both nodes i and j), a string is produced. Mordekaiser is a funny guy, and so he only likes strings which consist of one or more concatenations of the string “HUE”. For extremely necessary clarification, below is a list of the 8 shortest strings which Mordekaiser considers valid:
HUE HUEHUE HUEHUEHUE HUEHUEHUEHUE HUEHUEHUEHUEHUE HUEHUEHUEHUEHUEHUE HUEHUEHUEHUEHUEHUEHUE HUEHUEHUEHUEHUEHUEHUEHUE
Mordekaiser would like to know the longest valid string which his tree can produce, using any pair of start/end nodes - he knows that at least one such string will exist. Additionally, he wants to determine how many different pairs of start/end nodes yield that longest valid string.
Input
Line 1: 1 integer, T , the number of test cases (1 T 100) ≤ ≤ For each test case: Line 1: 1 integer, N Line 2: 1 string with N characters, the labels of nodes 1..N Next N-1 lines: 2 integers, A and B , for i = 1..(N 1) i i − Input
For each test case: Line 1: 1 string, the longest valid string which can be found in the tree Line 2: 1 integer, the number of pairs of start/end nodes which yield the longest valid string
Sample Input
2 5 HUEUH 1 2 2 3 3 4 4 5 11 UEHHUEEEUHH 1 2 2 3 1 4 4 5 5 6 4 7 4 8 8 9 10 9 11 9
Sample Output
HUE 2 HUEHUE 4
Explanation of Sample
The first tree looks like this:
H(1) U(2) E(3) U(4) H(5)
The longest valid string is just “HUE”, and it can be produced either by travelling from node 1 to node 3, or from node 5 to node 3. The second tree looks like this:
U(1) H(4) E(8)
E(2) U(5) E(7) U(9)
H(3) E(6) H(10) H(11)
The longest valid string, “HUEHUE”, can be produced from the four pairs of start/end nodes (10, 2), (10, 6), (11, 2) and (11, 6).