Error Detection and Correction: Nim; Secure Communication; RAID Greg Plaxton Theory in Programming Practice, Fall 2005 Department of Computer Science University of Texas at Austin
The Game of Nim • Initially, there are a number of piles of chips • Two players move alternately • A legal move consists of removing a positive number of chips from some pile • The first player who cannot make a legal move loses • Thus, the player who removes the last chip wins Theory in Programming Practice, Plaxton, Fall 2005
A Winning Strategy for Nim? • A state is losing if the player to move next can be forced to lose: – Examples: (1) The state in which all piles are empty; (2) Any state with an even number of nonempty piles, each of which has exactly one chip; (3) Any state with two piles of equal size • A state is winning if the player to move next can force a win – Examples: (1) Any state with exactly one nonempty pile; (2) Any state with an odd number of nonempty piles, each of which has exactly one chip; (3) Any state with two piles of unequal size • Is every state either losing or winning or could there be “uncertain” states? Theory in Programming Practice, Plaxton, Fall 2005
Every State is Either Losing or Winning • Proof by strong induction on the total number of chips • Base case: If there are no chips, the state is losing • Induction hypothesis: Let k be a nonnegative integer and assume that every state with a total of at most k chips is either losing or winning • Induction step: – Let S be an arbitrary state with a total of k + 1 chips – By the IH, every state reachable via a legal move from S is either losing or winning – If every state reachable via a legal move from S is winning, then S is losing; otherwise, it is winning Theory in Programming Practice, Plaxton, Fall 2005
Towards a Characterization of Losing/Winning States • Earlier we saw that if every pile has 0 or 1 chips, then the parity of the number of nonempty piles determines whether a state is losing or winning • Conjecture: A state is losing iff, for all nonnegative integers i , an even number of piles have the property that the binary representation of the number of chips in the pile has a 1 in bit position i • What is a more succinct statement of this conjecture? Theory in Programming Practice, Plaxton, Fall 2005
Bad and Good States • For any state S , let f ( S ) denote the XOR of the pile sizes • A state S is bad if f ( S ) = 0 , and good otherwise • Theorem: A state is losing (resp., winning) iff it is bad (resp., good) • In order to prove this claim, we will establish two lemmas – Lemma 1: Any legal move executed in a bad state yields a good state – Lemma 2: For any good state, there exists a legal move to a bad state • Why does the theorem follow? Theory in Programming Practice, Plaxton, Fall 2005
Proof of the Theorem (using Lemmas 1 and 2) • Proof by strong induction on the total number of chips • Base case: If there are no chips, the state is losing and bad • Induction hypothesis: Let k be a nonnegative integer and assume that every bad (resp., good) state with at most k chips is losing (resp., winning) • Induction step: Let state S have a total of k + 1 chips – If S is bad, then by Lemma 1, every state reachable via a legal move from S is good, and by the IH, winning; hence, S is losing – If S is good, then by Lemma 2, there is a legal move from S to a state that is bad, and by the IH, losing; hence, S is winning Theory in Programming Practice, Plaxton, Fall 2005
Proof of Lemma 1 • Lemma 1: Any legal move executed in a bad state yields a good state • Suppose we are in a bad state S and we move to a state S ′ by reducing the number of chips in some pile from x to y • Let i be a bit position where x has a 1 and y has a 0 – Such an i exists since x > y • Note that bit position i of f ( S ′ ) is 1 , so S ′ is good Theory in Programming Practice, Plaxton, Fall 2005
Proof of Lemma 2 • Lemma 2: For any good state, there exists a legal move to a bad state • Let S be an arbitrary good state • Let i be the highest bit position of f ( S ) containing a 1 • Let A be an arbitrary pile for which the number of chips, denoted by x , has a 1 in bit position i of its binary representation – Such a pile A exists since bit position i of f ( S ) is 1 • Note that x ⊕ f ( S ) < x and ( x ⊕ f ( S )) ⊕ ( x ⊕ f ( S )) = 0 , so we can reach a bad state by reducing the number of chips in A from x to x ⊕ f ( S ) Theory in Programming Practice, Plaxton, Fall 2005
Secure Communication: A Simple Approach • Suppose a sender wishes to securely send a block of data x to a receiver over an insecure communication channel • Let us call a block of data random if each bit is independently set to 0 (resp., 1) with probability 1 2 • Suppose the sender and receiver share a secret random block (sometimes called a key) k • The sender sends y = x ⊕ k over the channel • Note that the block y is random! – Therefore an eavesdropper gains no information whatsoever about the message x from observing y Theory in Programming Practice, Plaxton, Fall 2005
Disadvantages of this Approach • Key exchange problem: The sender and receiver need to agree in advance on the random block k , which must be sent over a secure channel • The key should not be reused – For example, suppose the eavesdropper sees x 1 ⊕ k and x 2 ⊕ k – The eavesdropper can calculate x 1 ⊕ x 2 , which is (presumably) not random and might yield statistical clues towards partially deciphering the content of x 1 and x 2 – Furthermore, an eavesdropper who knows the content of a single block can determine k and thereby decipher all blocks • Later in the course we will see a much better solution to the problem of secure communication Theory in Programming Practice, Plaxton, Fall 2005
Fault-Tolerant Storage • Suppose the records of a large corporate database are to be stored on a collections of disks • Occasionally one of the disks fails • We would like to store the data in such a way that the entire content of the database can be recovered whenever any single disk fails • Naive solution: – Store two copies of each record (on different disks) – This doubles the storage requirement • A better solution? Theory in Programming Practice, Plaxton, Fall 2005
Redundant Array of Independent Disks (RAID) • Store the database nonredundantly on k disks • Use one additional disk to store the XOR of these k disks – Here we are viewing the content of each disk as a block of bits • If any disk fails, its content can be recovered since it is the XOR of the remaining k disks Theory in Programming Practice, Plaxton, Fall 2005
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