FUN2014.pdf - Normal Abby Normal Prefix Normal P´ eter...

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Normal, Abby Normal, Prefix Normal eter Burcsi, Gabriele Fici, Zsuzsanna Lipt´ ak , Frank Ruskey, and Joe Sawada FUN 2014 Lipari, 1-3 July 2014
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Prefix normal words Fici, Lipt´ ak (DLT 2011) Definition A binary word w is prefix normal (w.r.t. 1) if no substring has more 1s than the prefix of the same length. Example w = 10110001001101110010 w 0 = 11101001011001010010 Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 2 / 17
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Prefix normal words Fici, Lipt´ ak (DLT 2011) Definition A binary word w is prefix normal (w.r.t. 1) if no substring has more 1s than the prefix of the same length. Example w = 101 1000100110 111 0010 NO w 0 = 11101001011001010010 YES Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 2 / 17
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Prefix normal words Fici, Lipt´ ak (DLT 2011) Definition A binary word w is prefix normal (w.r.t. 1) if no substring has more 1s than the prefix of the same length. Example w = 101 1000100110 111 0010 NO w 0 = 11101001011001010010 YES all p.n. words of length 4: 1111 , 1110 , 1101 , 1100 , 1010 , 1001 , 1000 , 0000. Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 2 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Example ( n = 5) 1. start Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Example ( n = 5) 1. start 2. Alice 1 Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Example ( n = 5) 1. start 2. Alice 1 3. Bob 1 0 Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Example ( n = 5) 1. start 2. Alice 1 3. Bob 1 0 4. Alice 1 0 0 Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Let’s play a game. Alice and Bob are constructing a binary string of length n together. Alice wins if the word is prefix normal. Bob wins if it is not. Example ( n = 5) 1. start 2. Alice 1 3. Bob 1 0 4. Alice 1 0 0 Alice has won: 10000 , 10100 , 10101 are all p.n. Burcsi, Fici, Lipt´ ak, Ruskey, Sawada Normal, Abby Normal, Prefix Normal FUN 2014 3 / 17
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Prefix normal games Exercise Find n 0 s.t.
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