********************************************************************** GOLD PROBLEMS ********************************************************************** Three problems numbered 1 through 3 ********************************************************************** Problem 1: Dividing the Path [Hal Burch, 2004] Farmer John's cows have discovered that the clover growing along the ridge of the hill in his field is particularly good. To keep the clover watered, Farmer John is installing water sprinklers along the ridge of the hill. To make installation easier, each sprinkler head must be installed along the ridge of the hill (which we can think of as a one-dimensional number line of length L (1 <= L <= 1,000,000); L is even). Each sprinkler waters the ground along the ridge for some distance in both directions. Each spray radius is an integer in the range A..B (1 <= A <= B <= 1000). Farmer John needs to water the entire ridge in a manner that covers each location on the ridge by exactly one sprinkler head. Furthermore, FJ will not water past the end of the ridge in either direction. Each of Farmer John's N (1 <= N <= 1000) cows has a range of clover that she particularly likes (these ranges might overlap). The ranges are defined by a closed interval (S,E). Each of the cow's preferred ranges must be watered by a single sprinkler, which might or might not spray beyond the given range. Find the minimum number of sprinklers required to water the entire ridge without overlap. PROBLEM NAME: divide INPUT FORMAT: * Line 1: Two space-separated integers: N and L * Line 2: Two space-separated integers: A and B * Lines 3..N+2: Each line contains two integers, S and E (0 <= S < E <= L) specifying the start end location respectively of a range preferred by some cow. Locations are given as distance from the start of the ridge and so are in the range 0..L. SAMPLE INPUT (file divide.in): 2 8 1 2 6 7 3 6 INPUT DETAILS: Two cows along a ridge of length 8. Sprinkler heads are available in integer spray radii in the range 1..2 (i.e., 1 or 2). One cow likes the range 3-6, and the other likes the range 6-7. OUTPUT FORMAT: * Line 1: The minimum number of sprinklers required. If it is not possible to design a sprinkler head configuration for Farmer John, output -1. SAMPLE OUTPUT (file divide.out): 3 OUTPUT DETAILS: Three sprinklers are required: one at 1 with spray distance 1, and one at 4 with spray distance 2, and one at 7 with spray distance 1. The second sprinkler waters all the clover of the range like by the second cow (3-6). The last sprinkler waters all the clover of the range liked by the first cow (6-7). Here's a diagram: |-----c2----|-c1| cows' preferred ranges |---1---|-------2-------|---3---| sprinklers +---+---+---+---+---+---+---+---+ 0 1 2 3 4 5 6 7 8 The sprinklers are not considered to be overlapping at 2 and 6. ********************************************************************** Problem 2: Fence Obstacle Course [USACO coaches, 2004] Farmer John has constructed an obstacle course for the cows' enjoyment. The course consists of a sequence of N fences (1 <= N <= 50,000) of varying lengths, each parallel to the x axis. Fence i's y coordinate is i. The door to FJ's barn is at the origin (marked '*' below). The starting point of the course lies at coordinate (S,N). +-S-+-+-+ (fence #N) +-+-+-+ (fence #N-1) ... ... +-+-+-+ (fence #2) +-+-+-+ (fence #1) =|=|=|=*=|=|=| (barn) -3-2-1 0 1 2 3 FJ's original intention was for the cows to jump over the fences, but cows are much more comfortable keeping all four hooves on the ground. Thus, they will walk along the fence and, when the fence ends, they will turn towards the x axis and continue walking in a straight line until they hit another fence segment or the side of the barn. Then they decide to go left or right until they reach the end of the fence segment, and so on, until they finally reach the side of the barn and then, potentially after a short walk, the ending point. Naturally, the cows want to walk as little as possible. Find the minimum distance the cows have to travel back and forth to get from the starting point to the door of the barn. PROBLEM NAME: obstacle INPUT FORMAT: * Line 1: Two space-separated integers: N and S (-100,000 <= S <= 100,000) * Lines 2..N+1: Each line contains two space-separated integers: A_i and B_i (-100,000 <= A_i < B_i <= 100,000), the starting and ending x-coordinates of fence segment i. Line 2 describes fence #1; line 3 describes fence #2; and so on. The starting position will satisfy A_N <= S <= B_N. Note that the fences will be traversed in reverse order of the input sequence. SAMPLE INPUT (file obstacle.in): 4 0 -2 1 -1 2 -3 0 -2 1 INPUT DETAILS: Four segments like this: +-+-S-+ Fence 4 +-+-+-+ Fence 3 +-+-+-+ Fence 2 +-+-+-+ Fence 1 |=|=|=*=|=|=| Barn -3-2-1 0 1 2 3 OUTPUT FORMAT: * Line 1: The minimum distance back and forth in the x direction required to get from the starting point to the ending point by walking around the fences. The distance in the y direction is not counted, since it is always precisely N. SAMPLE OUTPUT (file obstacle.out): 4 OUTPUT DETAILS: Walk positive one unit (to 1,4), then head toward the barn, trivially going around fence 3. Walk positive one more unit (to 2,2), then walk to the side of the barn. Walk two more units toward the origin for a total of 4 units of back-and-forth walking. ********************************************************************** Problem 3: Cow Ski Area [Adam Rosenfield, 2004] Farmer John's cousin, Farmer Ron, who lives in the mountains of Colorado, has recently taught his cows to ski. Unfortunately, his cows are somewhat timid and are afraid to ski among crowds of people at the local resorts, so FR has decided to construct his own private ski area behind his farm. FR's ski area is a rectangle of width W and length L of 'land squares' (1 <= W <= 500; 1 <= L <= 500). Each land square is an integral height H above sea level (0 <= H <= 9,999). Cows can ski horizontally and vertically between any two adjacent land squares, but never diagonally. Cows can ski from a higher square to a lower square but not the other way and they can ski either direction between two adjacent squares of the same height. FR wants to build his ski area so that his cows can travel between any two squares by a combination of skiing (as described above) and ski lifts. A ski lift can be built between any two squares of the ski area, regardless of height. Ski lifts are bidirectional. Ski lifts can cross over each other since they can be built at varying heights above the ground, and multiple ski lifts can begin or end at the same square. Since ski lifts are expensive to build, FR wants to minimize the number of ski lifts he has to build to allow his cows to travel between all squares of his ski area. Find the minimum number of ski lifts required to ensure the cows can travel from any square to any other square via a combination of skiing and lifts. TIME LIMIT: 0.4 seconds PROBLEM NAME: skiarea INPUT FORMAT: * Line 1: Two space-separated integers: W and L * Lines 2..L+1: L lines, each with W space-separated integers corresponding to the height of each square of land. SAMPLE INPUT (file skiarea.in): 9 3 1 1 1 2 2 2 1 1 1 1 2 1 2 3 2 1 2 1 1 1 1 2 2 2 1 1 1 OUTPUT FORMAT: * Line 1: A single integer equal to the minimal number of ski lifts FR needs to build to ensure that his cows can travel from any square to any other square via a combination of skiing and ski lifts SAMPLE OUTPUT (file skiarea.out): 3 OUTPUT DETAILS: FR builds the three lifts. Using (1, 1) as the lower-left corner, the lifts are (3, 1) <-> (8, 2), (7, 3) <-> (5, 2), and (1, 3) <-> (2, 2). All locations are now connected. For example, a cow wishing to travel from (9, 1) to (2, 2) would ski (9, 1) -> (8, 1) -> (7, 1) -> (7, 2) -> (7, 3), take the lift from (7, 3) -> (5, 2), ski (5, 2) -> (4, 2) -> (3, 2) -> (3, 3) -> (2, 3) -> (1, 3), and then take the lift from (1, 3) - > (2, 2). There is no solution using fewer than three lifts. **********************************************************************