Here is code :
#include <string>
#include <regex>
#include <iostream>
int main()
{
    std::string pattern("[^c]ei");
    pattern = "[[:alpha:]]*" + pattern + "[[:alpha:]]*";
    std::regex r(pattern); 
    std::smatch results;   
    std::string test_str = "cei";
    if (std::regex_search(test_str, results, r)) 
        std::cout << results.str() << std::endl;      
    return 0;
}
Output :
cei
The compiler used is gcc 4.9.1.
I'm a newbie learning regular expression.I expected nothing should be output,since "cei" doesn't match the pattern here. Am I doing it right? What's the problem?  
Update:
This one has been reported and confirmed as a bug, for detail please visit here : https://gcc.gnu.org/bugzilla/show_bug.cgi?id=63497
It's a bug in the implementation. Not only do a couple other tools I tried agree that your pattern does not match your input, but I tried this:
#include <string>
#include <regex>
#include <iostream>
int main()
{
  std::string pattern("([a-z]*)([a-z])(e)(i)([a-z]*)");
  std::regex r(pattern);
  std::smatch results;
  std::string test_str = "cei";
  if (std::regex_search(test_str, results, r))
  {
    std::cout << results.str() << std::endl;
    for (size_t i = 0; i < results.size(); ++i) {
      std::ssub_match sub_match = results[i];
      std::string sub_match_str = sub_match.str();
      std::cout << i << ": " << sub_match_str << '\n';
    }
  }
}
This is basically similar to what you had, but I replaced [:alpha:] with [a-z] for simplicity, and I also temporarily replaced [^c] with [a-z] because that seems to make it work correctly.  Here's what it prints (GCC 4.9.0 on Linux x86-64):
cei
0: cei
1:
2: c
3: e
4: i
5:
If I replace [a-z] where you had [^c] and just put f there instead, it correctly says the pattern doesn't match.  But if I use [^c] like you did:
std::string pattern("([a-z]*)([^c])(e)(i)([a-z]*)");
Then I get this output:
cei
0: cei
1: cei
terminate called after throwing an instance of 'std::length_error'
  what():  basic_string::_S_create
Aborted (core dumped)
So it claims to match successfully, and results[0] is "cei" which is expected.  Then, results[1] is "cei" also, which I guess might be OK.  But then results[2] crashes, because it tries to construct a std::string of length 18446744073709551614 with begin=nullptr.  And that giant number is exactly 2^64 - 2, aka std::string::npos - 1 (on my system).
So I think there is an off-by-one error somewhere, and the impact can be much more than just a spurious regex match--it can crash at runtime.
The regex is correct and should not match the string "cei".
The regex can be tested and explained best in Perl:
 my $regex = qr{                 # start regular expression
                 [[:alpha:]]*    # 0 or any number of alpha chars
                 [^c]            # followed by NOT-c character
                 ei              # followed by e and i characters
                 [[:alpha:]]*    # followed by 0 or any number of alpha chars    
               }x;               # end + declare 'x' mode (ignore whitespace)
 print "xei" =~ /$regex/ ? "match\n" : "no match\n";
 print "cei" =~ /$regex/ ? "match\n" : "no match\n";
The regex will first consume all chars to the end of the string ([[:alpha:]]*), then backtrack to find the NON-c char [^c] and proceed with the e and i matches (by backtracking another time).
Result:
 "xei"  -->  match
 "cei"  -->  no match
for obvious reasons. Any discrepancies to this in various C++ libraries and testing tools are the problem of the implementation there, imho.
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