Color the Picture solution codeforces – A picture can be represented as an 𝑛×𝑚n×m grid (𝑛n rows and 𝑚m columns) so that each of the 𝑛⋅𝑚n⋅m cells is colored with one color. You have 𝑘k pigments of different colors. You have a limited amount of each pigment, more precisely you can color at most 𝑎𝑖ai cells with the 𝑖i-th pigment.
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[Solution] Color the Picture solution codeforces
A picture is considered beautiful if each cell has at least 33 toroidal neighbors with the same color as itself.
Two cells are considered toroidal neighbors if they toroidally share an edge. In other words, for some integers 1≤𝑥1,𝑥2≤𝑛1≤x1,x2≤n and 1≤𝑦1,𝑦2≤𝑚1≤y1,y2≤m, the cell in the 𝑥1x1-th row and 𝑦1y1-th column is a toroidal neighbor of the cell in the 𝑥2x2-th row and 𝑦2y2-th column if one of following two conditions holds:
- 𝑥1−𝑥2≡±1(mod𝑛)x1−x2≡±1(modn) and 𝑦1=𝑦2y1=y2, or
- 𝑦1−𝑦2≡±1(mod𝑚)y1−y2≡±1(modm) and 𝑥1=𝑥2x1=x2.
Notice that each cell has exactly 44 toroidal neighbors. For example, if 𝑛=3n=3 and 𝑚=4m=4, the toroidal neighbors of the cell (1,2)(1,2) (the cell on the first row and second column) are: (3,2)(3,2), (2,2)(2,2), (1,3)(1,3), (1,1)(1,1). They are shown in gray on the image below:
The gray cells show toroidal neighbors of (1,2)(1,2).[Solution] Color the Picture solution codeforces
Each test contains multiple test cases. The first line contains the number of test cases 𝑡t (1≤𝑡≤1041≤t≤104). The description of the test cases follows.
The first line of each test case contains three integers 𝑛n, 𝑚m, and 𝑘k (3≤𝑛,𝑚≤1093≤n,m≤109, 1≤𝑘≤1051≤k≤105) — the number of rows and columns of the picture and the number of pigments.
The next line contains 𝑘k integers 𝑎1,𝑎2,…,𝑎𝑘a1,a2,…,ak (1≤𝑎𝑖≤1091≤ai≤109) — 𝑎𝑖ai is the maximum number of cells that can be colored with the 𝑖i-th pigment.
It is guaranteed that the sum of 𝑘k over all test cases does not exceed 105105.
For each test case, print “Yes” (without quotes) if it is possible to color a beautiful picture. Otherwise, print “No” (without quotes).
[Solution] Color the Picture solution codeforces
6 4 6 3 12 9 8 3 3 2 8 8 3 3 2 9 5 4 5 2 10 11 5 4 2 9 11 10 10 3 11 45 14
[Solution] Color the Picture solution codeforces
Yes No Yes Yes No No
In the first test case, one possible solution is as follows:
