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CSpace division switching  TEach sample takes a different path through the switch, depending on its destination T    DCrossbar   LSimplest possible space-division switch Crosspoints can be turned on or off $M( *(       G!Crossbar - example  H Crossbar   Advantages: simple to implement simple control strict sense non-blocking Drawbacks number of crosspoints, N2 large VLSI space vulnerable to single faults = G =  -*]     2  J$Time-space switching  Precede each input trunk in a crossbar with a TSI Delay samples so that they arrive at the right time for the space division switch s schedule  K%Time-Space: Example    M&Finding the schedule  Build a graph nodes - input links session connects an input and output nodes. Feasible schedule Computing a schedule compute perfect matching.L@'@'  L#Time-space-time (TST) switching   rAllowed to flip samples both on input and output trunk Gives more flexibility => lowers call blocking probability r    N'"Circuit switching - Space division#  graph representation transmitter nodes receiver nodes internal nodes Feasible schedule edge disjoint paths. cost function number of crosspoints internal nodest0%0% *       O(Example    Z4 Clos Network   Q,(Clos Network - strict sense non-blocking)  Holds for k >= 2n-1 Proof: Consider and idle input and output Input box connected to at most n-1 middle layer switches output box connected to at most n-1 middle layer switches There exists a "free" middle switch. Tfffff  R-Proof    [5Example  S*Clos Network - rearrangable*       Holds for k >= n Proof: Consider all input and output find a perfect matching. route the perfect matching remaining network is Clos(N-n,n-1,k-1) summary: smaller circuit weaker guarantee Mulicast ?y ! fy f!f *      ^7ZRearrangable Clos Network  routing algorithm*       Start at some arbitrary 2x2 input switch, and route it to its destination through the upper switch. Route the other output port of the 2x2 switch you reached to its input port through the lower switch. If the other port in the input switched you reached has not been routed yet, route it through the upper middle switch. Otherwise, select an arbitrary input port switch that was not yet used, and repeat the procedure.   a;Recursive constructions  _8Algorithm Complexity eN routing steps for each level Log n levels to do ===> N log n Given parallel hardware: O(N) timedf" e `9Alternative View of the problem A graph coloring problem: Input/output switches = nodes A match = link Middle stage switches = colors This is the well known  Coloring of bi-partite graph problem. Heuristics fail miserably!!! b:@Another algorithm  matrix based  Specification matrix: Rows = input switches (n) Columns = middle switches (N/n = r) Content = output switches (1..n) We need an algorithms that will fill up the matrix with a feasible routing: Same number cannot appear in the same column Works also for CLOS with redundancy_L-$ L-$ rDAlgorithm Principle \For j=0,1,2,& balance the destination j among the columns Challenges: efficiency termination We describe an Algorithm by Lee, Hwang, and Carpinelli, T.COM. 44 (11), Nov 1996 t, Q, Q $ uGThe Matrix Meaning pCData Structures (j,e), j=0,1,& n-1, e=0,1,& ,r-1 :- the set of rows {i} such that sij=e 0(e), e=0,1,r-1 :- the set of columns {j} such that Si does not contain e 2(e), e=0,1,r-1 :- the set of columns {j} such that Si contains e at least twice 3 * '  H@7FsE Algorithm  fInit: e=0 If 2(e) empty ee+1 if e=r stop; else goto 1 if 2(e) j 1st element of 2(e) k 1st element of 0(e) (simple swap) i 1st element of (j,e) if e< sik swap sij with sik e sik remove i from (j,e) and (k,e ) add i to (j,e ) and (k,e) if |(j,e)|=1 remove j from 2(e) if |(j,e )|=1 remove j from 0(e ) if |(j,e )|=2 add j to 2(e ) if |(k,e )|=0 add k to 0(e ) if |(k,e )|=1 remove k from 2(e ) remove k from 0(e) goto step 1  ZnZnZ nZ/nZnZ5Z     ;0h  X(Next Simple Swap) If e> sik i 2nd element of sik repeat step 3 on i if e> si k goto step 5 (Successive Swap) u e; remove k from 0(u) if if |(j,u)|=2 remove j from 2(u) v sik swap sij with sik remove i from (j,u) and (k,v) add i to (j,v) and (k,u) if e<v if |(k,v)|=0 add k to 0(v) if |(k,v)|=1 remove k from 2(v) if |(j,v)|=1 remove j from 0(v) if |(j,v)|=2 add j to 2(v) goto step 1 if e>v uv goto step 5B nZ Z.ZZnZ nZ6nZ nZInZnZnZnZnZ     7 $$((,,,,,, ` tFAlgorithm Complexity adding an element to a set, choosing/removing the 1st/2nd element from a set take O(1) steps 5A 5B and 5D each take O(1) time removing a generally positioned element from an r-set takes O(r) time step 5C time complexity is in O(r) [ 2(v)-{k}, 0(v)-{j} ] the looping in step 5 does not contain step 5C, only 5B and 5D the time complexity for step 5 is O(r) Algorithm time complexity O(nr2)A'#3m T.'Recursive constructions - Benes Network(  c<Clos network size Number of switching elements is given by for k=n (rearrangeable non-blocking) Optimal value for n is n=sqrt{2}N, which yields6e  (l i=3a lower bound for the number of switching elements?3 dAssume we have 2x2 switching units. 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