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The complexity of resolving conflicts on MAC
We consider the fundamental problem of multiple stations competing to
transmit on a multiple access channel (MAC). We are given stations out of
which at most are active and intend to transmit a message to other stations
using MAC. All stations are assumed to be synchronized according to a time
clock. If stations node transmit in the same round, then the MAC provides
the feedback whether , (collision occurred) or . When ,
then a single station is indeed able to successfully transmit a message, which
is received by all other nodes. For the above problem the active stations have
to schedule their transmissions so that they can singly, transmit their
messages on MAC, based only on the feedback received from the MAC in previous
round.
For the above problem it was shown in [Greenberg, Winograd, {\em A Lower
bound on the Time Needed in the Worst Case to Resolve Conflicts
Deterministically in Multiple Access Channels}, Journal of ACM 1985] that every
deterministic adaptive algorithm should take rounds
in the worst case. The fastest known deterministic adaptive algorithm requires
rounds. The gap between the upper and lower bound is
round. It is substantial for most values of : When constant and (for any constant , the lower bound is
respectively and O(n), which is trivial in both cases. Nevertheless,
the above lower bound is interesting indeed when poly(). In this
work, we present a novel counting argument to prove a tight lower bound of
rounds for all deterministic, adaptive algorithms, closing
this long standing open question.}Comment: Xerox internal report 27th July; 7 page
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