A 3/4-Approximation Algorithm for Multiple Subset Sum by Caprara A.

By Caprara A.

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Extra info for A 3/4-Approximation Algorithm for Multiple Subset Sum

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Kn ) is a consistent cut by way of contradiction. Suppose that the vector (k1 , . . , kn ) is an inconsistent cut. Then, by denition, there are processors pi and pj such that there is an instruction x of processor pi subsequent to instruction number ki , such that x happens before instruction number kj of processor pj . Recall that ki is the furthest entry of V Ti majorised by K . So entry ki + 1 is not majorised by K , and since all subsequent entries, including the one for instruction x, can have only larger coordinates, the entries are not majorised by K either.

Logical time, causality, and consistent state 615 Proof We rst show the forward implication. Suppose that x

Suppose that (k1 , . . , kn ) is not a consistent cut. Then there is a processor pj such that instruction number kj + 1 or later sends a message other than , and the message is received on or before a processor pi executes instruction number ki . So the message must have been sent after the message was sent from pj to pi . But messages are received in the order they are sent, so pi processes before it processes . But then message arrives after snapshot was taken at pi .

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