Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allod pattern matching to develop into the standard tool that it is today. #R##N#Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. Hover, a systematic study of tractable cases was made possible only recently, with the first conditional lor bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lor bound, with one exception known as the Word Break problem. #R##N#  complete their work as follows: #R##N#1)  present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. #R##N#2)  classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth. #R##N#3) For the Word Break problem  give an improved $\tilde{O}(n m^{1/3} + m)$ algorithm. Surprisingly,  also prove a matching conditional lor bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem. #R##N#In total,  prove matching upper and lor bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression membership testing.
['Karl Bringmann', 'Allan Grønlund', 'Kasper Green Larsen']
I am writing a research paper on a idea A Dichotomy for Regular Expression Membership Testing
[{"from": "human", "value": "I am writing a research paper on a idea A Dichotomy for Regular Expression Membership Testing"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study regular expression membership testing: Given a regular expression of size $m$ and a string of size $n$, decide whether the string is in the language described by the regular expression. Its classic $O(nm)$ algorithm is one of the big success stories of the 70s, which allod pattern matching to develop into the standard tool that it is today. #R##N#Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. Hover, a systematic study of tractable cases was made possible only recently, with the first conditional lor bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any \"type\" of homogeneous regular expressions of depth 2 or 3, they either presented a near-linear time algorithm or a quadratic conditional lor bound, with one exception known as the Word Break problem. #R##N#  complete their work as follows: #R##N#1)  present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. #R##N#2)  classify all types, except for the Word Break problem, into almost-linear time or quadratic time assuming the Strong Exponential Time Hypothesis. This extends the classification from depth 2 and 3 to any constant depth. #R##N#3) For the Word Break problem  give an improved $\\tilde{O}(n m^{1/3} + m)$ algorithm. Surprisingly,  also prove a matching conditional lor bound for combinatorial algorithms. This establishes Word Break as the only intermediate problem. #R##N#In total,  prove matching upper and lor bounds for any type of bounded-depth homogeneous regular expressions, which yields a full dichotomy for regular expression membership testing."}]