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When I reading a document:

The number of routing entries in a routing table may be just a few in a typical host, and in a router, this number can reach hundreds of thousands.

How to understand the "in a router, this number can reach hundreds of thousands"?

why is the gap number between them so big?

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A router may just have a few hosts that are connected to it in its routing table plus a default gateway to a router which providers connectivity to the rest of the internet.

However, a router connected to the DFZ (Default Free Zone) does not have a default route, it has routes to all routable IP prefixes. Currently, that's well over 700 000 (see this twitter feed for a current number).

Basically, it all comes down to the fact that there are very different types of routers used in very different parts of networks.

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If you are buliding a network with a stable heirachical structure, centralised up-front design and a plentiful supply of addresses you can keep routing tables pretty small. Routers towards the bottom of the heiracty only need to know about their local environment and which ports led upstream. Routers towards the top of the heiracty only need to know about the broad structure of the network.

The Internet is not that network. It's a loose jumble of interconnected networks run by different entities with constantly shifting relationships. While smaller customers do get addresses allocated by their providers most larger networks get their addresses allocated in a way that has nothing to do with network structure. Worse on the IPv4 internet the long history and constrained address supply mean that many networks have acquired their addresses in multiple seperate allocations.

The result is that the Internet routing table is large and growing. Some routers can avoid having the full table relying on default routes to their upstreams to do the heavy lifting for them but ultimately default routes can only take you so far, sooner or later you need an explicit route to tell you where the destination is.

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