You decide to give up computer science, and instead go into environmental engineering. Luckily, your computer science skills will come in handy! Your rst job is to deal with modeling the water run-o { or drainage { in a basin area. Given a representation of the area to model, your task is to determine how far the water will ow.
The land will be represented by height map, which is a two-dimensional grid of heights. Each grid will have r rows and c columns. Each grid location { or grid cell { will have an integer height elevation. Because you will be dealing with multiple grids, each one will have a title as well.
Your task is to gure out the longest sequence of grid lo-cations that water can ow between. Water will ow from a higher elevation to a lower elevation. For the purposes of this problem, water will never ow from a given elevation to the same elevation, nor will it ow uphill. Futhermore, water can only ow from one grid cell to an adjacent cell (adjacent cells are above, below, left, and right; not diago-nal!).
As an example, consider the following 5x5 grid. Note that the input in this example is justi ed to help illustrate
the grid; there will only be one space between heights in the actual input.
667841 377
49041 868
12 11 29 24 53
05158 928
97 99 96 58 92
There are many such valid drainage paths in this grid. One starts in the second cell of the second row, and ows from 90-78-41-3. Note that 90-41-41-3 is not a valid drainage ow, as water is not always owing downhill (41-41 is now downhill). The longest drainage path in this example is of length 7, and ows from the 99 in the bottom row to the 3 in the top row; the full path is 99-96-58-29-24-8-3.
This homework MUST be a dynamic programming solution. There isn’t much choice, though { a brute-force solution (recursive or not) will time out long before it could complete.
Input
The rst line of the input will be the number of test cases. There will not be more than 100 test cases provided.
Each test case will start with three values space-separated on a single line: the title string of the test case (all letters and numbers; no punctuation or whitespace) followed by r and c, the number of rows and
columns, respectively. The following r lines will contain c numbers each, de ning the heights of the grid. Both r and c will be positive integers not greater than 100. Each number in the grid itself will be an integer height ranging from 0 to 100.
Output
For each test case, your output should contain a single line that contains the test case title, a colon, a space, and the length of the longest drainage run.
Sample Input
Sample Output
4
Charlottesville: 7
Charlottesville 5 5
Richmond: 1
66
78
41
3 77
WashingtonDC:
5
4
9041868
Wintergreen:
25
12
11
29
24
53
0
51
58
9
28
97
99
96
58
92
Richmond
3 3
1
1 1
1
1 1
1
1 1
WashingtonDC 5 5
10
81
28
2 49
64
59
61
85
82
77
14
81
6 76
37
86
99
11
92
85
95
78
13
57
Wintergreen
5 5
1
2 3
4
5
10
9
8 7
6
11
12
13
14
15
20
19
18
17
16
21
22
23
24
25
2
