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You are kayaking through a system of underground caves and suddenly realize that the tide is coming in and you are trapped! Luckily, you have a map of the cave system. You are stuck until the tide starts going out, so you will be here for a while. In the meantime, you want to determine the fastest way to the exit once the tide starts going out.
The cave system is an N by M grid. Your map consists of two N by M grids of numbers: one that specifies the height of the ceiling in each grid square, and one that specifies the height of the floor in each grid square. The floor of the cave system is porous, which means that as the water level falls, no water will remain above the water level.
You are trapped at the north-west corner of the map. The current water level is Hcentimeters, and once it starts going down, it will drop at a constant rate of 10 centimeters per second, down to zero. The exit is at the south-east corner of the map. It is now covered by water, but it will become passable as soon as the water starts going down.
At any time, you can move north, south, east or west to an adjacent square with the following constraints:
Note that you can go up or down as much as you want with your kayak. (You're very athletic from all this kayaking!) For example, you can go from a square with floor at height 10 centimeters to an adjacent square with floor at height 9000 centimeters (assuming the constraints given above are met).
These constraints are illustrated below:
When moving from one square to another, if there are at least 20 centimeters of water remaining on the current square when you start moving from it, it takes 1 second to complete the move (you can use your kayak). Otherwise, it takes 10 seconds (you have to drag your kayak). Note that the time depends only on the water level in the square you are leaving, not in the square you are entering.
It will be a while before the tide starts going out, and so you can spend as much time moving as you want before the water starts going down. What matters is how much time you will need from the moment the water starts going down until the moment you reach the exit. Can you calculate this time?
For each test case, output one line containing "Case #x: t", where x is the case number (starting from 1), and t is the time, in seconds, starting from when the tide begins going out, that it takes you to make your way out of the cave system. Answers within an absolute or relative error of 10-6 of the correct answer will be accepted.
4 200 1 2 250 233 180 100 100 3 3 500 500 500 500 500 600 500 140 1000 10 10 10 10 10 490 10 10 10 100 3 3 500 100 500 100 100 500 500 500 500 10 10 10 10 10 10 10 10 10 100 2 2 1000 1000 1000 1000 100 900 900 100
Case #1: 11.7 Case #2: 3.0 Case #3: 18.0 Case #4: 0.0
It is possible that you can go through the whole cave system before the tide starts dropping. In this case you will be able to wait at the exit for the tide to start dropping, so the answer in this case should be zero (this is the case in the fourth of the sample test cases).
In the first sample test case, there are initially only 33 centimeters between the water level and the ceiling of the eastern square, so after the tide starts going down, you have to wait for at 1.7 seconds to enter it. Once it is accessible, you can start going in - but the water level in the western square is now so low (only 3 centimeters above the floor) that you have to drag your kayak for the next 10 seconds to get to the exit point.
The initial situation in the second case is better - you have a lot of headroom in adjacent squares, so you can move, for example, to (1, 1) before the tide starts dropping. Once there, you have to wait for the tide to start going down, and the water level to go down to 90cm (that takes one second). Then you can kayak south and then east and get out (in a total of three seconds). Note that you cannot go through the cave at (2, 1), even though the ceiling there is high enough, because there is too little space between the floor of this cave and the ceiling of any caves you could try to enter from ((1, 1) and (2, 0)) - only 10 centimeters in each case.
The third case is somewhat similar to the first - you have to wait at the starting position until the tide goes down to 50cm. After that you can kayak for the exit - but after three moves (taking three seconds) the water is at 20cm, which is only 10cm above the floor, which means the fourth move will be dragging instead of kayaking.
In the fourth case you are really lucky! You can immediately go the exit, even before the tide starts leaving, and wait there.