Score : 1200 points
Takahashi is a magician. He can cast a spell on an integer sequence (a_1,a_2,...,a_M) with M terms, to turn it into another sequence (s_1,s_2,...,s_M), where s_i is the sum of the first i terms in the original sequence.
One day, he received N integer sequences, each with M terms, and named those sequences A_1,A_2,...,A_N. He will try to cast the spell on those sequences so that A_1 < A_2 < ... < A_N will hold, where sequences are compared lexicographically. Let the action of casting the spell on a selected sequence be one cast of the spell. Find the minimum number of casts of the spell he needs to perform in order to achieve his objective.
Here, for two sequences a = (a_1,a_2,...,a_M), b = (b_1,b_2,...,b_M) with M terms each, a < b holds lexicographically if and only if there exists i (1 ≦ i ≦ M) such that a_j = b_j (1 ≦ j < i) and a_i < b_i.
The input is given from Standard Input in the following format:
N M A_{(1,1)} A_{(1,2)} … A_{(1,M)} A_{(2,1)} A_{(2,2)} … A_{(2,M)} : A_{(N,1)} A_{(N,2)} … A_{(N,M)}
Print the minimum number of casts of the spell Takahashi needs to perform. If he cannot achieve his objective, print -1
instead.
3 3 2 3 1 2 1 2 2 6 3
1
Takahashi can achieve his objective by casting the spell on A_2 once to turn it into (2 , 3 , 5).
3 3 3 2 10 10 5 4 9 1 9
-1
In this case, Takahashi cannot achieve his objective by casting the spell any number of times.
5 5 2 6 5 6 9 2 6 4 9 10 2 6 8 6 7 2 1 7 3 8 2 1 4 8 3
11