4 research outputs found
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
Efficient parallel algorithms for dead sensor diagnosis and multiple access channels
We study parallel algorithms for identifying the dead sensors in a mobile ad hoc wireless network and for resolving broadcast conflicts on a multiple access channel (MAC). Our approach involves the development and application of new group-testing algorithms, where we are asked to identify all the defective items in a set of items when we can test arbitrary subsets of items. In the standard group-testing problem, the result of a test is binary—the tested subset either contains defective items or not. In the versions we study in this paper, the result of each test is non-binary. For example, it may indicate whether the number of defective items contained in the tested subset is zero, one, or at least two (i.e., the results are 0, 1, or 2+). We give adaptive algorithms that are provably more efficient than previous group testing algorithms (even for generalized response models). We also show how our algorithms can be implemented in parallel, because they possess a property we call conciseness, which allows them to be used to solve dead sensor diagnosis and conflict resolution on a MAC. Dead sensor diagnosis poses an interesting challenge compared to MAC resolution, because dead sensors are not locally detectable, nor are they themselves active participants. Even so, we present algorithms that can be applied in both contexts that are more efficient than previous methods. We also give lower bounds for generalized group testing