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In this study, we consider the problem of synthesising custom networks-on-chip (NoC) architectures that are optimised for a given application. Both unicast and multicast traffic flows are considered in the input specification. We formulate the joint multicast routing and network design problem using a rip-up and reroute procedure, where each multicast routing step is formulated as a minimum directed spanning tree problem, and we propose a very efficient algorithm called Ripup-Reroute-and-Router-Merging (RRRM). Our new formulation adopts a rip-up and reroute concept that provides us with a heuristic iterative mechanism to identify increasingly improving solutions. The minimum directed spanning tree formulation efficiently captures the best routing solutions for multicast flows during the topology synthesis procedure. Our design flow integrates floorplanning, and our solutions consider deadlock-free routing. Experimental results compared with our previous proposed algorithms CLUSTER and DECOMPOSE on a variety of NoC benchmarks showed that our new synthesis results are largely improved. RRRM can on average achieve a 9% reduction in power consumption over CLUSTER and a 17% reduction in power consumption over DECOMPOSE with 1786% and 57% faster execution times than CLUSTER and DECOMPOSE, respectively. Improvements in performance were also achieved, with an average of 3% reduction in hop counts over CLUSTER and 7% in hop counts over DECOMPOSE on all benchmarks.