Frankly saying I don't know an answer to your question, but I will leave this as a fun fact that if we will allow also empty subsequences of form [i, i) (so in fact we increase number of occurences of zero by n) then answer will be always 0 :). It follows from an inequality which shows up in comparing number of occurences of zeros and some other value. Without adding those empty intervals that inequlity changes to , which in fact is not true :(. Or maybe I should say ":)", because that means that this problem has any sense :D.