# Thus t i h9275 w t j t i h9275 u t j and t j h9275 w

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Thus, T i H9275 w T j T i H9275 u T j (and T j H9275 w T i T j H9275 u T i ). Now by induction: T i H9275 F ( T i , T j ) T j T i H9275 w T j and the result follows. Now, we prove thatthe given protocolensures serializabilityanddeadlock freedom by induction on the length of minimal cycle. a. m = 2 : The protocol ensures no minimal cycles as shown in the above Lemma 2 b. m > 2 : Assume by contradiction that T 0 H9275 T 1 H9275 T 2 ...... H9275 T m 1 H9275 T 0 is a minimal cycle of length m . We will con- sider two cases: i. F ( T i , T i + 1 )’s are not all distinct. It follows that, L ( T i ) intersectiontext L ( T j ) intersectiontext L ( T k ) negationslash= H9278 for 0 i j k m 1 A. Assume < i , j , k > = < i , i + 1 , i + 2 > . Then it easily follows that T i H9275 T i + 2 and (*) is not a minimal cycle. B. Assume < i , j , k > negationslash= < i , i + 1 , i + 2 > . If say | j i | > 1 then as either T i H9275 T j or T j H9275 T i and (*) is not a minimal cycle. If | j i | = 1 and | k j | > 1 then the proof is analogous.. ii. All F ( T i , T i + 1 )’s are distinct. Then for some i H9257 ( F ( T i , T i + 1 )) < H9257 ( F ( T i + 1 , T i + 2 )) (**) and H9257 ( F ( T i + 1 , T i + 2 )) > H9257 ( F ( T i + 2 , T i + 3 )) (***) By(**), E ( T i + 1 ) negationslash= F ( T i + 1 , T i + 2 ) andby(***), E ( T i + 2 ) negationslash= F ( T i + 1 , T i + 2 ). Thus, F ( T i + 1 , T i + 2 ) / ∈ { E ( T i + 1 ) , E ( T i + 2 ) } , a contradiction to Lemma 1. We have thus shown that the given protocol ensures serializability and deadlock freedom. 15.8 Answer: The proof is Silberschatz and Kedem, A Family of Locking Protocols for Database Systems that Are Modeled by Directed Graphs , IEEE Trans. on Software Engg. Vol. SE-8, No. 6, Nov 1982. The proof is rather complex; we omit details, which may be found in the above paper.

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• Spring '13
• Dr.Khansari
• TI, Two-phase locking

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