We assume that a message may be delivered by packets through multiple hops and investigate the feasibility and efficiency of an implementation of the Omega Failure Detector under such assumption. We prove that the existence and sustainability of a leader is exponentially more probable in a multi-hop Omega implementation than in a single-hop one. An implementation is: message efficient if all but finitely many messages are sent by a single process; packet efficient if the number of packets used to transmit a message in all but finitely many messages is linear w.r.t the number of processes, packets of different messages may potentially use different channels, thus the number of used channels is not limited; super packet efficient if the number of channels used by packets to transmit all but finitely many messages is linear. With deterministic assumption, we prove the following. If reliability and timeliness of one message does not correlate with another, i.e., there are no channel reliability properties, then a packet efficient implementation of Omega is impossible. If eventually timely and eventually reliable channels are allowed, we establish necessary and sufficient conditions for the existence of a message and packet efficient implementation of Omega. We also prove that the eventuality of timeliness of reliability of channels makes a super packet efficient implementation of Omega impossible. On the constructive side, we present and prove correct a deterministic packet efficient algorithm in a weaker model that allows eventually timely and fair-lossy channels.