Both CORE and TASKS nodes run TaskTrackers, but only CORE nodes run DataNodes, so yes, you need at least one CORE node.
So can you run point kernels?
./elastic-mapreduce --create --stream \ ... --instance-group master --instance-type m1.small --instance-count 1 \ --instance-group core --instance-type m1.small --instance-count 19 --bid-price 0.028
ps you can also run one CORE and many TASK nodes, but depending on how much you read / write, you will have pain, since 18 nodes will read / write 1 node.
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