If N can take maximum value of 10^5 and T can take maximum value of 10 and I come up with an algo having O(T*N*log(N)) . Will It be able to solve the question in 1 second ?

Depends, but mostly Yes

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The value of TNlog(N) here is 0.166*10^8 , assume that constant here is 1 , it should get pass because generally 10^8 iterations are processed in 1s on Codechef. However this time(December Challenge), it looks that they have decreased the number of iterations allowed per sec thus making problems more challenging(like TLE with vector but AC with arrays)

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how much difference can using vectors and array cause ?