2879: Hamming Triples
Time Limit: 2 Sec Memory Limit: 256 MBSubmit: 0 Solved: 0
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Description
Little Chris is having a nightmare. Even in dreams all he thinks about is math.
Chris dreams about m binary strings of length n, indexed with numbers from 1 to m. The most horrifying part is that the bits of each string are ordered in either ascending or descending order. For example, Chris could be dreaming about the following 4 strings of length 5:
The Hamming distance H(a,b) between two strings a and b of length n is the number of positions at which the corresponding symbols are different.
Chris thinks that each three strings with different indices constitute a single triple. Chris's delusion is that he will wake up only if he counts the number of such string triples a, b, c that the sum H(a,b)+H(b,c)+H(c,a) is maximal among all the string triples constructed from the dreamed strings.
Help Chris wake up from this nightmare!
The first line of input contains two space-separated integers n and m (1≤n≤109; 3≤m≤105), the length and the number of strings. The next m lines contain the description of the strings. The i-th line contains two space-separated integers si and fi (0≤si≤1; 1≤fi≤n), the description of the string with index i; that means that the first fi bits of the i-th string are equal to si, and the remaining n-fi bits are equal to 1-si. There can be multiple equal strings in Chris's dream.
Output a single integer, the number of such string triples among the given that the sum of the Hamming distances between the strings of the triple is maximal.
5 4
0 3
0 5
1 4
1 5
3
10 4
1 5
0 5
0 5
1 5
4