Benes Networks
Permutation Networks and Rearrangeability
Kosta Tachtevrenidis
Introduction What is this paper about? This paper came out of a time when telephone companies were expanding like mad and were seeking ways to connect all nodes of a telephone network without using n 2 connections. Do I need to understand A this math to fully grasp the material? I certainly hope not because I don!t understand it either. "eah# I know it is my paper but if somebody can do a better $ob please stand up.
%reliminary Definitions stage &of switching' a series of switches that define a routing step link &pattern' the pattern of connections that links two switching stages s(uare switch a switch with the same amount of inputs as outputs non)blocking networks A set of one)to)one connection re(uests on an * + * network can be defined by a permutation. A network is non) blocking# if any permutation can be reali,ed by edge)dis$oint paths in the network. Depending on whether the permutation is reali,ed statically or dynamically we ha-e the following types of non)blocking networks.
Definitions continued... Rearrangeable non-blocking (benes network) A network is rearrangeable non)blocking if any permutation can be reali,ed by edge)dis$oint paths when the entire permutation is known. In other words# any permutation can be statically reali,ed. The word rearrangeable refers to that if the connection re(uests in a permutation arri-e dynamically# the permutation can be reali,ed with possible rearranging acti-e connections. This is e(ui-alent to reali,ing a permutation statically.
Wide)sense non)blocking the permutation can be reali,ed by edge)dis$oint paths wo rearranging acti-e connections sub$ect to the condition that a selected path is used for each new connection re(uest. In other words# any permutation can be dynamically reali,ed with the help of a wise algorithm.
/trict)sense non)blocking &n x n crossbar network' the permutation can be reali,ed by edge)dis$oint paths without rearranging acti-e connections# any idle path can be used for each new connection re(uest. In other words# any permutation can be dynamically reali,ed.
%ermutation *etworks Imagine a network with n inlets labeled 01...n. A permutation network is one in which the set of links used to route a call from a gi-en inlet# leads to an outlet which is a way is a permutation of those inlets. 3xample4 0 1 2 3
This is a butterfly network. The result of routing to the outlets is a permutation of the inlets.
5earrangeable *etworks Rearrangeable non-blocking (benes network) A network is rearrangeable non)blocking if any permutation can be reali,ed by edge)dis$oint paths when the entire permutation is known. In other words# any permutation can be statically reali,ed. The word rearrangeable refers to that if the connection re(uests in a permutation arri-e dynamically# the permutation can be reali,ed with possible rearranging acti-e connections. This is e(ui-alent to reali,ing a permutation statically. The abo-e is from a pre-ious slide. What that says is that gi-en a network with n inputs 01#2#...#n# if I know what the resulting permutation should be# I can !chart! a path for all inlets in a way that no path is blocked and e-erything is routed correctly.
Why /hould I 6are? 7enes networks pro-ide pro-ably congestion)free communication from inputs to a permutation of the outputs. Any n)permutation can be routed &off)line' on an n) input 7enes network with node)dis$oint paths. %roof? The proof relies on the fact that benes networks are composed of smaller benes networks. Induction is the key to the proof. An n)input 7enes network can simulate any n)node# degree)d network in 8&d lg n' time.
9ow do I construct one? /hort answer4 recursi-ely ong answer4 To construct one start with a small &2 input' benes networks and build on from there. To build a : input benes network# cascade two 2 input networks and add a stage to the left and one to the right. ;ultiplex the links as shown in the next slide. 6onfused yet? 5eally long answer4 read the paper.
2 inut benes network 1 2
6onstructing the network... ! inut benes network 0 1 2 3
Two -ersions of the same 7enes network are depicted here. The abo-e is the shuffle -ersion and the one below is the butterfly -ersion. 6an you tell they route in the exact same way and therefore they are e(ui-alent?
0 1 2 3
6onstructing the network... " inut benes network #butter$ly% 0 1 2 3 ! & ' (
The benes network is composed of the same sub) network mirrored. What kind of network is this?
6onstructing the network... " inut benes network #shu$$le%
6onstructing the network... recursion
1' inut benes network #butter$ly%
What are the ad-antages? The ad-antages of using 7enes networks are profound.
%ure non)blocking and rearrangeable network switching power. 7ut let!s see how it ranks on %feiffer!s *etwork 5anking Test4
ast = bandwidth = of simultaneous transmissions? 7enes networks are non)blocking so all nodes can be transmitting at the same time. O(n) 6heap = how many switches needed to implement? The number of switches re(uired to implement a benes network is logarithmically proportional to the number of nodes. O(n*log(n))
3xample %lease@
0 1
2 3
What if and B are already routed but I still want to reali,e the permutation &)C1# B)C2# 2)C# 1)CB'? 8b-iously it can!t be done with the existing state of things. 7ut this is a rearrangeable network so I can reroute 1 or : or both to create a routing scheme that does not block@
3xample 6ontinued@
0 1
2 3
Tada@
5eal World Applications The switching network &of this
AT;
D3/I<* 8> A ingerhut# ;argaret >lucke# Gonathan /. Turner Washington Hni-ersity# /t. ouis
What!s next? Is there something better? "ou bet there is... 6on-entional 7enes networks ha-e control complexity n&2log&n' )1'. This probably has to do with the cost of combining the two butterfly networks times the amount of nodes. 7ut can we do better than that? What if we could create a network that could create all the same permutations in a non)blocking way with less switches? 3nter 5 7enes *etworks@ This paper presents a network topology built on benes networks which has control complexity of 8&log&n''. 9ow do they do it? They exploit the locality property of permutations.
5)7enes *etwork with *J1
Alternati-es to 7enes *etworks Butterfly Networks &blocking'4 7enes networks ha-e the ad-antage that any permutation of inputs to outputs can be routed without conflicts. 9owe-er# computing the paths for a gi-en permutation takes a long time# and re(uires global information about all of the messages in the network# so typically can!t be done at run)time. 7utterflies ha-e some conflicts# but can be routed at run)time. They also ha-e lower latency because they ha-e only half as many stages as a 7enes network for the same number of nodes. 0 1 2 3
Alternati-es to 7enes *etworks Cantor Networks &strictly non)blocking'4
m copies of 7enes network.
*etwork si,e 4 * log2 *
m=3
5eferences annan# 5a$gopal. The 5)7enes *etwork4 A 6ontrol) optimal 5earrangeable %ermutation *etwork. Department of 6omputer /cience# ouisiana /tate Hni-ersity. 7oppana 5a$endra# 5agha-endra 6./. Designing 3fficient 7enes and 7anyan 7ased 8nput)7uffered AT; /witches. The Aero/pace 6orporation = Hni-ersity of Texas# /an Antonio. 6haney# >ingerhut# >lucke# Turner. Design of a
