Round 1: Online test
You have M identical ropes and N identical weights. You want to do an experiment to determine the strength S of the identical ropes, by testing how many weights can be held by one single rope. The strength S of the ropes is defined as:
1) If a rope breaks with the first weight, S=0;
2) If a rope can hold n weights and breaks at n+1 weights, S=n;
3) If a rope can hold all N weights, S=N.
In one test (“test” means to check if one rope can hold an amount of weights), if the rope breaks, you have to take another rope to continue the experiment; if the rope does not break, it can be used for the next test with no problem. If you used up all the ropes but still cannot determine the strength S of the ropes, the experiment fails.
Please write a C/C++ program that, given M and N (M and N are both integers, M>=1, N>=1), calculates the minimum number of tests T needed to guarantee you can determine S. Try to optimize the time complexity of your program, and explain:
1) What kind of “tricks” you have used to optimize the time complexity?
2) What is the time complexity without these “tricks” and what is the time complexity with these “tricks”?
Hint 1: If you have limited number of ropes, you don’t dare to take the risk. For example, if M=1, your only choice is to increase the weights one by one, from 1 to N, to make sure you can determine S in the worst case. In this case, T=N.
Hint 2: If you have lots of ropes, binary search is obviously helpful to calculate the minimum T efficiently.
Example 1: M=1, N=20
Command: calc_n_tests 1 20
Example 2: M=2, N=5
Command: calc_n_tests 2 5
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