Summary: An interval routing scheme is a general method of routing messages in a distributive network using compact routing tables. In this paper, concepts related to optimal interval routing schemes are introduced and explored. Several problems concerning the insertion of nodes and joining of separate networks by a new link to form larger ones are considered. Various applications to distributed computing are given. In particular, leader-finding and generation of spanning trees in arbitrary networks are shown to require at most \(O(N+E)\) messages when a suitable interval routing scheme is available.