Cowmpetency
Farmer John is hiring a new herd leader for his cows. To that end, he has interviewed N (2≤N≤105) cows for the position. After interviewing the ith candidate, he assigned the candidate an integer "cowmpetency" score ci ranging from 1 to C inclusive (1≤C≤109) that is correlated with their leadership abilities.
Because he has interviewed so many cows, Farmer John does not remember all of their cowmpetency scores. However, he does remembers Q (1≤Q<N) pairs of numbers (aj,hj) where cow hj was the first cow with a strictly greater cowmpetency score than cows 1 through aj (so 1≤aj<hj≤N).
Farmer John now tells you the sequence c1,…,cN (where ci=0 means that he has forgotten cow i's cowmpetency score) and the Q pairs of (aj,hj). Help him determine the lexicographically smallest sequence of cowmpetency scores consistent with this information, or that no such sequence exists! A sequence of scores is lexicographically smaller than another sequence of scores if it assigns a smaller score to the first cow at which the two sequences differ.
Each input contains T (1≤T≤20) independent test cases. The sum of N across all test cases is guaranteed to not exceed 3⋅105.
We can see that the given output satisfies all of Farmer John's remembered pairs.
- max(c1)=1, c2=2 and 1<2 so the first pair is satisfied
- max(c1,c2,c3)=2, c4=3 and 2<3 so the second pair is satisfied
- max(c1,c2,c3,c4)=3, c5=4 and 3<4 so the third pair is satisfied
There are several other sequences consistent with Farmer John's memory, such as
1 2 2 3 5 4 1 1 2 2 3 4 4 5
However, none of these are lexicographically smaller than the given output.
In test case 3, since C=1, the only potential sequence is
1 1
However, in this case, cow 2 does not have a greater score than cow 1, so we cannot satisfy the condition.
In test case 5, a1 and h1 tell us that cow 6 is the first cow to have a strictly greater score than cows 1 through 4. Therefore, the largest score for cows 1 through 6 is that of cow 6: 5. Since cow 7 has a score of 7, cow 7 is the first cow to have a greater score than cows 1 through 6. So, the second statement that cow 9 is the first cow to have a greater score than cows 1 through 6 cannot be true.
SCORING:
- Input 3 satisfies N≤10 and Q,C≤4.
- Inputs 4-8 satisfy N≤1000.
- Inputs 9-12 satisfy no additional constraints.
Problem credits: Suhas Nagar
Source: USA Computing Olympiad.
Input Format
The first line contains T, the number of independent test cases. Each test case is described as follows:
- First, a line containing N, Q, and C.
- Next, a line containing the sequence c1,…,cN (0≤ci≤C).
- Finally, Q lines each containing a pair (aj,hj). It is guaranteed that all aj within a test case are distinct.
Output Format
For each test case, output a single line containing the lexicographically smallest sequence of cowmpetency scores consistent with what Farmer John remembers, or −1 if such a sequence does not exist.