Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!uunet!seismo!ut-sally!husc6!bloom-beacon!gatech!hubcap!Wen-King From: Wen-King@hubcap.UUCP Newsgroups: comp.hypercube Subject: Re: Hypercube routing (channel usage) (with PROOF) Message-ID: <312@hubcap.UUCP> Date: Fri, 17-Jul-87 04:47:49 EDT Article-I.D.: hubcap.312 Posted: Fri Jul 17 04:47:49 1987 Date-Received: Sat, 18-Jul-87 13:29:57 EDT Sender: fpst@hubcap.UUCP Lines: 54 Approved: hypercube@hubcap.clemson.edu [* If you have been getting duplicates of this posting, it is the fault of our system. Our system maintainer is looking into it. It is no reason, however, for interrupting normal and healthy exchange of information. **] In article<310@hubcap.UUCP> ogcvax!pase@seismo.css.gov (Douglas M. Pase) writes: > 110-------------10---------------111 < /| /| > / | / | < 1 | 2 | > / | / | < / 7 / 6 > 010---------------9--------------011 | <000 -(8)-> 001 | | | | >000 -(4)-> 010 | | | | <000 -(8)-> 001 -(5)-> 011 | | | | >000 -(0)-> 100 4 | 5 | <000 -(8)-> 001 -(3)-> 101 | 100-------------11----------|----101 >000 -(4)-> 010 -(9)-> 110 | / | / <000 -(8)-> 001 -(5)-> 011 -(2)-> 111 | / | / > | 0 | 3 < | / | / > |/ |/ < 000---------------8--------------001place the most messages on channels 8, 9, 10 and 11. The message traffic will 1, 2, and 3. I can see why you would say that edge #8 is be used four times as often as edge #0. (I use edge instead of channel to refer to the above pictuer because channel is used to refer to the output ports seen by each node). There are indeed four instance of edge #8 and one instance of edge #0 in the list of routing paths. However, there are four nodes feeding into edge #0 (in the direction of 000->100) as opposed to just one feeding into edge #8 (in the direction of 000->001). Let me try to give it another shot. Assuming we have random but uniform message flow from all nodes to all other node. Lets call the xor of the node numbers of the source and the destination node of each message the routing tag of the message. If the n-th bit of the tag is set, the message has to cross the n-th channel of some node somewhere on the cube. Since every node communicates with every other node with equal probability, every bit in a routing tag of a messages has the same probability of being 1 (slightly larger than 50% chance, actually (2**(N-1))/((2**N)-1) ). That means, the total amounts of messages crossing the n-th channels, for any valid values of n, in the cube are the same. By symetry, one node's n-th channel is no different from any other node's n-th channel. Therefore, given the prescribed assumption on message flow, channels in the cube are used equally. QED. +--------------------------------------------------------------------------+ | Wen-King Su wen-king@vlsi.caltech.edu Caltech Corp of Cosmic Engineers | +--------------------------------------------------------------------------+