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If for one task not all its precedent tasks are finished, the task would still get notified via checkpoint barrier from the running precedent tasks. However, some of the precedent tasks might be already finished and the CheckpointBarrierHandler must consider this case. As a whole, EndOfPartition from one input channel would mark this channel as aligned for all the pending and following checkpoints. 

Failover and Recovery

Logically we could not restart the finished tasks on failover. We could only restart those tasks who have operators not fully finished. Specially, if all the source operators have been finished, the remaining operator would directly start the process of finishing. 

For the first version, we would still restart all the tasks, and the finished tasks would start finishing normally. This simplifies the implementation of the task side and scheduler. 

The basic algorithm to compute the tasks to trigger would be iterating over the ExecutionGraph to find the new root running tasks. However, the algorithm could be optimized by iterating over the JobGraph instead. The detailed algorithm is shown in Appendix. 

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Currently when a task finish, all the pending checkpoints would be canceled. Since now we would support checkpoints after tasks finished, tasks finish should not fail the checkpoints and we would change the behavior to waiting for all the pending checkpoints to finish before task get finished.

Failover and Recovery

Logically we could not restart the finished tasks on failover. We could only restart those tasks who have operators not fully finished. Specially, if all the source operators have been finished, the remaining operator would directly start the process of finishing. 

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