 So, in the previous lecture, what we had done was canter network and its cross point complexity. And we ultimately came up with the bound that for any strictly non-blocking switch, you will have number of cross points which are required will always be less than or equal to this. And if the number of cross points are more than this, you can always build up a strictly non-blocking switch. That was the thing and this was through a canter network. And I think one of the things which you I think should appreciate that the way this was made a strictly non-blocking is you took something which is a rearrangeable non-blocking and you repeated in multiple times vertical. So, that you create more alternate path still it becomes strictly non-blocking. And only basically the contribution was that figure out to figure out how many number of how many number of times it has to be repeated in vertical direction. So, now we move on to something which is different and we call it wide sense non-blocking. I have not defined it so far. So, let me first of all explain what is it wide sense non-blocking. So, technically the meaning of this is you will have a switch and important thing is the algorithm to set up the path. Usually what happens whenever switch is in one particular state and a new path has to be set up. All paths are not set up simultaneously. That is also possible if all paths have to be set up simultaneously, rearrangeable non-blocking switch is always better actually. You can anyway find out the all non-constructing arrangements and use that and rearrangeable non-blocking switch can be used if all paths are set up together. But if they are not it has been done one by one or suddenly a request comes you do not know what you because what happens a connection set up request will come you set up the connection you set a few more and then some connections will go away. So, coming and going main you that sequence cannot be guaranteed that sequence is random because users are setting up the connection they are making the request and they are also relinquishing the connection. So, the question is sometimes now each one of this configuration where is a path is set up is known as a state of the switch. So, with one single with no connection there is only one state of the switch that there is no connection with one possible connection between an input and output port. For one particular I O pair there may be in many possible ways you can set up even one single path all those form an state actually and then all pairs can actually have one single path. So, now the idea is that when we want to set up the connection out of you all these possible paths which one which particular option should I choose whenever a new connection has to be set up when connection is going away it is being relinquished I cannot do much actually. So, I am just moving on to a state with lower number of connection but whenever a new connection comes I need to have a definite algorithm to ensure that the switch always remains in non-blocking state that non-blocking state means in that state whatever free input ports and whatever free output ports are there if I want to set up the connection between them I can always set up the connection. So, that should be possible. So, that is the idea. So, we are going to now look at an example there is no mathematics I am actually now going through what we call state space. So, I am going to build up the states of the switches going to iterate them that how you move from one state to another state and based on that we will figure out that the switch is wide sense non-blocking if I avoid certain states of the switch there is going to be exactly one state. Now, before going to this I have to understand that many switch states are equivalent. So, I will give few examples of them and then what we will do is all equivalent switch states will be represented by only one single state. So, if by some certain transformation I can actually converge from those set of switch state to only one state. So, it is like forming sets of equivalent states from any state you can move to any other state by just doing transformation they are equivalent in the sense that functionally or mathematically they are same actually, but when you draw them they looks different they only look different, but they are not different. So, let us start with that now the wide sense non-blocking switch we already know this switch 4 by 4 switch is a Bain's network actually and this does satisfy slap and do get theorem this is a rearrangeable non-blocking switch. I can always set up the paths I can give an example for this one. So, I set up the path say from 1 to 1 prime in this fashion and 4 to 4 prime in this fashion this path is already set up in this case. Now, can I set up a path from 2 to 3 prime and 3 to 2 prime it cannot be done you can observe that I cannot set up a path from 2 to 3 prime this is not possible in this case. So, this is a blocking switch, but I can always do the rearrangements. So, once I do the rearrangement it is possible to set up the path. So, rearrangement will be a pretty simple you can actually use let me do even that pulse matrix. So, that becomes an example how this is done. So, pulse matrix for this will contain only. So, this switch number 1 switch number 2 these are not port number. So, this is 1 1 prime and 2 prime 1 prime 1 and 2 middle stress switches are also written as same as 1 and 2. So, in this case 1 is already connected to 1 using 1 and 2 is already connected to the switch 2 is connected to 2 again by 2 that is what has happened when I want to set up a connection between 2 to 3 prime. So, what happens is I am not able to set up because 2 to 3 prime means from here to here I need to put up an element here which is not which is free, but you look into this column and this row you will find out all elements are consumed 1 and 2, but I can as there is slip and do it theorem says I can always find out a pair 1 is available here, but it is not available here 2 is used here, but it is not used there is 1 and 2 pair. So, I can simply put the 2 here and 1 here that is why it because the chaining cannot be done it is a very small pulse matrix and then I can set up a connection. So, this technically means 2 to 2 prime this one should be done. Obviously, this was a choice action. So, I can do it this way then 2 to 3 can actually be very done very easily and then of course, 3 to 2 prime also can be done very easily. So, by rearrangements I can always set up the connection, but I can now make it wide sense non blocking by adding one more stage. So, this is what I am going to prove and I am going to leave one question unanswered actually after this I add one more stage I am creating one more alternate path that is the only thing which I am doing this which becomes wide sense non blocking the proof we will come across my question is for which even I do not know the answer. Can I still add few more stages and make it strictly non blocking is it possible and I do not think it is possible there is a reason for that I have already given you the hint when I solved earlier. See, so far if you can get a switch state in which it is not possible to connect to one pair of free input and free output port. If you can even find out one state which cannot go into blocking state if that you end up in that state and this is always going to happen if you set up a connection like this and this these two pairs will always connect to the specific pairs here alternate cannot be done because they have to they always have to do criss cross across this stage. So, this cannot be made converted into strictly non blocking by a extending stage the state is actually further you cannot keep on adding more stages and make it a non blocking, but it becomes wide sense non blocking with 4 stages that is one thing which is sure. So, let us start with that. So, I will make the state diagram and then we will set up the equivalent. So, I am just going because I have to draw a lot of these figures now. So, one possible state is no connection and there is only one state of this kind when there is no connection this is a state. The connection between these switches is the one which I have already drawn earlier, but now I will only draw if a connection has been set up to be more clear. So, if I set up only one connection I can set up connection in this fashion that is one possible state. I can also make another state actually if with only one connection even this is also the state, but as I told you this is a matter of visualization where I put my node it is like graph theory. So, node placement gives us the visual picture, but mathematically if I swap these two I will again get back the same thing. So, this and these are equivalent instead of this if I have something like this I can swap these two I will again get back this state. So, whatever connection pattern you take from whichever input port to whichever output port so far there is only one connection all those connections can be always converted to this and how that is this is done is by simply moving this node here and this node here links are not disturbed. When I move this node here and this node here and this one on this side what happens this node will come here and this node will come here these two remain as it is and this state line has to be now connected to this that is one kind of transformation. Second transformation is I am looking from this side I can start looking from this side it is all the same visually whatever is true from the input is also true from output. So, I can also swap I can rotate this figure like this. So, I rotate the board and you see from the backside of the board that transformation is also fine. So, I can do this thing I can do the translation of the whole switch I can do it this way the switch state will not change it remains same. So, I will be only representing one single state which is equivalent of all these possible states. So, this is I think a very commonly done stuff when we try to merge all equivalent states and represented by only one single state otherwise I will not be able to draw actually this switch does not have large number of states that is why we can draw and do it graphically. So, that is a one single connection state now let us set up two connections and then what happens. So, I want to set up the second connection how it can be done I have to systematically do kind of all possible combination that is the only way. So, let me do it actually. So, I will start because there is another port on this side, but I cannot go straight that is not possible there is only one link I have to go up. Now, here there are two possibilities going up and going down. So, I will search for both actually I will create all possible combinations and then try to do the mergers. I will go first of all up again I can go up and down I will go up actually and then find out the combination this is one possibility fine with two connections. Second possibility let me draw it here the connection starts from here no option goes to only up I have to use even these remaining two also later on and draw the equivalence I have gone up I will still go up, but I can come down that is the next possibility and remember this merge state is only the one which I am considering for one connection, but same combinations can exist even with other equivalent states of the single connection state, but that does not matter I can represent them by this this defines the complete state of the space actually for two connections. So, this has been up and down I have taken care of. So, now the next one I have to go for the down there is no other option for him it has to go up and I think for all two connections I have set I have figured out this state none of them are equivalent all three are unique by any transformation I cannot convert any one of these to anything else that is not possible these are unique. So, the first one which was there with white I call this is state as one and I call this is state as this one as two one you can give numbering anyway you want I am doing it as per my notebook actually the way I have solved it. So, I have taken the second connection from same port now I let me take it from a different switch and find out if there are more connections which are feasible. So, let me take it from here I can go up and down both. So, always I will go for up first actually go this way I will again go up first and up first that is one state and we know this is going to be a blocking state, but because now remaining two ports can only be connected into only certain map all possible combinations between two free input ports and two free output ports cannot be made. So, this is a blocking state, but this is not true with the remaining other states remember here I can take these two and connect with any one of these. Now, this can be done because there is a cross connect which is happening here they are sharing one single switch this by changing the state here itself it can be done same is true here, but this is not possible in this case. So, this one is a blocking state. So, let me put a circle around it and ultimately this is what we want to avoid I should not be able to come to the sorry not this one I should not be able to come to this state either by removal of a connection or by creation of a new connection that is the only thing which we have to do and that is what will give the algorithm there is one more which is left now the fifth one let me draw it here. So, I start from the top sorry goes to bottom I have to first of all go with the upper one and then the last one will be this which one you can so, from here you have to set up this way that is the only option you have. So, this can only be connected to this, but this cannot be connected to this three ports. So, there is one port three port here I want to connect this and this there is only one link between every stage it is not possible these are already occupied by this red color correction. So, this can only be connected to this and this one can only be connected to this actually. So, it is a blocking state for other cases this is not true now coming to the equivalence because I need not number everything this one I call two five. So, fifth kind of state now this state if you carefully observe is nothing but two three rotate. So, in from in this side instead of that you start looking from the right side you will get the same switch actually same switch state. So, this need not be considered I have already taken care of. So, all possible switch states which have been contracted by equivalence into five possible combinations have now been present. So, this I can remove because this is already being covered by two three. So, next will be so, I have done one I have done the two connections now the question is of three connections how this will be done. Now, this slightly elaborate ultimately will end up only with very few states actually, but every time there will be equivalence you should be able to find out. So, let me take up first of all two one and find out all possible things which can happen. So, what I will do is the final states I will maintain here, here I will do all homework and whatever is the final state I will just copy it there. So, let me take two one first this one this is a switch two one. So, I call it three one and I have to now set up the path into this third level the first one. Now, remaining two ports I will first of all try to go up and then down while trying to set up the connection and find out this. So, first is this. So, go straight there is no possibility of going this thing this is the only option available this is one state I call it three one one next possibility. So, this is state I am going to keep it as a final one that is what this three one one next possibility is I can do it on this itself instead of going up I can go down actually. So, if I do that the only option is going this way and this way, but instead of looking from left you look from right it is again three one one only instead of looking from left look from right it is three one one there is no other option. So, this is already been covered. So, need not bother now is there any other option in this case. So, no other option I think all I have covered with this. So, if I am starting from this state I have to take the next one which is two two. So, that is the next one. So, I call it three two. So, I will take that first connection and let me try for the state one first always this will be the first connection. So, I call it three two this will be equivalent to something actually and that equivalent will be yeah this is three two this is one combination next one let me try full then I will give the numbering I have numbered it slightly differently. Then I can go for a bottom one which is this and what I am trying is I am trying to make sure that because one of the port will have two input and two output I am trying to convert it to a standard form by transformation because otherwise it becomes difficult to track. So, that both there has to be one state line connection. So, where it is as if a minute from here and both the connections two ports will be should come from the bottom side I have to transform every situation to that thing. So, now do transform this one. So, this two lines has to come to bottom. So, that is why I have to do this trap once I do this only this particular line is going to be the state line in the bottom. So, I have to even do the trap of these once I do that. So, that is this particular line I let me use a different color actually so that you can appreciate. So, this is this when I do this transformation all three nodes have been swept actually. So, the white color thing will go what way it will go up this node also goes up this node also goes up this will become this and the red one this has gone down. So, it will come here this has gone up. So, it has to go up this has gone down now these two are exactly same. So, these two combinations are equivalent again. So, I call this as 3 2 2 this is 3 2 1 and I take this as a standard because that is what I am referring that bottom line has to be state connection and two ports both the ports in the bottom switch has to be occupied at input and output. But that was my thing you can take any reference and try to do the transformation as per that. So, the standard connection will be now this I think all possible combinations which can emanate from quantity two has been taken care of by this and that is being represented by 3 2 2 because all the states are equivalent to that. So, next is coming to 2 1 2 2 2 3 let me check what happens with that. This one if you do swap this will become this and these two are same this is also 3 2 2. So, these two are equivalent actually only thing you I actually have done it very simply I said always the bottom line has to be state and both ports have to be occupied on the bottom side. So, once I try to make a comparison then I can immediately make comparison and then you can always look from input side and output side and try to make comparison for the equivalent. So, transformation becomes simple I can decide very easily which nodes to swap because this two this port has to come down and there is to be a state line which can only happen is if this connection comes down. So, this means all three have to be swapped or this has to be swapped and then you look from the top actually instead of the bottom you look from the top then also turns out to be same. So, now coming to the next one I think you all have to do this thing once unless you do it with your sit down and do it on your own you would not be able to appreciate how this actually happens because while doing here you may be grasping something, but not everything appreciating everything is not possible actually still I have been trying actually. So, next one is let me take two three this particular switch and let us put all combinations see if the equivalence is can be set up or not. So, I start the connection I start from the as usual always try to take the upper part, upper part and upper part there is one possibility and because I want these two points when they are occupied this node has to be in the bottom. So, I will transform this thing to swap between these two and this is the only state line going up. So, this has to remain here only this has to be swapped, but in the bottom I can get a straight line. So, the transformation will look let me change the color so that so with the swaps now you will have this thing as a straight the white colored one will because of the swap this remains down. So, it remains as it is here in the bottom this goes up and what happens to the red one. So, this is again one of the standard states and I call this as three three three is basically being derived from twenty three that is why I am calling it twenty three basically being used. So, there is a one state after transformation which you will get and this is also one of the standard actually. Yeah, because I want two ports to be occupied air two ports to be occupied air in one horizontal line for a reference well that was the purpose. You could have taken alternately not an issue that is possible, but be consistent you will end up in some transformed version of these states only. So, this one is and I call it three three three. So, next possibility is I have gone up actually first of all let me try downwards from here. Now, is it equivalent to something already existing this is nothing, but can be transformed to three twenty two. So, instead of look from input side look from the output side instead of looking from this side look from this side it is nothing, but three twenty two. So, it is already being covered. So, need not bother about it actually. So, next possibility is let us look this can only go horizontally does not matter. So, where the next possibility is when this goes down when this goes down it has to go up. Now, it can either go this way there is one option I have to do a transformation again because I want both the occupied ports to come down. So, this also has to be swept this need not be swept for the transformation and the transformation will give me all three have to be swept sorry all three have to be swept otherwise it cannot be done. So, this will go up this will also go up. The change will be in the last stage because there is only one output busy rather than the. So, if you see second stage and third stage the bottom line bottom nodes both are busy busy links. No, no I have to I will just figure it out I have just made some strapping problem actually. Third mode. So, this need not be strapped like this this strapping will not be done. Then it makes ok that is the mistake. So, that need not be strapped then to become a straight line. So, this one will come here this has not been strapped will come here this is nothing, but this just turn around make input to output and output to input same as 322. So, all the possibilities have been covered already sorry one only possibility which has been left is this. Now, this will give you nothing, but 333 this will give you 333. So, this is also covered. So, all possibilities have been covered for from 2 3 and from 2 5. Now, only thing which I have not kept the record here is from which state you are able to move to which state with 3 connections. So, but that I think you can again you actually must be remembering by this I may not have noted it down. So, last state is now 2 5 in this state only if you set up a connection. So, it is to start from the bottom or top does not matter actually they start from the bottom you will yeah I will come to this what it will look like you can also come from here one of this we will not this one, but the upper one will come. This is 2 5 from 2 5 I am starting 2 4 I have left I will do it in the end. So, in this case you can start from any side I can start from the bottom first this is the only option I have I can go straight come down this is I think looks to be very similar to 311 that combination is taken care of 311. This is nothing but 311 actually go this way you have to do a swapping to find out the equivalence you swap this and swap this and of course last one this is also nothing but 322. So, that is also taken care of. So, this through top and bottom both options have been taken care of there is no other option in this case. So, already this can be mapped to one of those 3 states next you can try from the top side from here only possibilities go up you do this there is nothing but already covered the state 322. So, input output you have to just swap next possibility is this ok you have to do a do a swap of this and then I can just look from the top side let me go here this is one straight line one straight line look from the top this will be also nothing but 311. So, instead of from bottom you look from the up this is the top output is heading towards you swap now whole thing when you rotate it you rotate it actually you look from the top instead of from bottom currently I am looking with this reference. So, you stand on your head and then see this will be same as 311. So, it is already taken care of yeah there is no other possibility I have consumed all the possibilities now I have consumed all the possibilities. So, bottom one I had already taken care of with all possible combinations the top one also I have taken care of all possible roots which are there. So, I end up in one of these 3 in fact there are only 3 possibilities states with 3 connections they are not more. So, the last one we have to take 2 4. So, 2 4. So, you set up a connection this way and I want a straight line in the bottom as well as both connections. So, I will do the swap of this I will do the swap of this and this is nothing but 333 this one always get to 333. So, now I have to just draw the arrows the transition diagram. So, from here I can go to this state I can go to this state I can go to this state and I can go to this state as well as I can go to this state total 5 states in which I can go. So, I think now I can draw it once you understand this. So, from here you can come either this you can just keep on verifying from there actually. So, these 5 states you should be able to come from these you should be able to go to this one from here also you can come to this one actually. So, from this you can come here from this you can come here from this also you can come here all 3. So, from here you can come here from here you can come here from here again there is only one state which is possible that does not matter all these 3 states will be unique 4 states does not matter only important thing is that this which is a blocking state. Now, if your switch has to operate in this state and a connection goes away you might end up fall back into this particular state which is a blocking state you want to avoid to come into this state in any case. So, when you want to set up a path from here never come to this state. So, this is not permitted for you and since because this falling back is not in your hand you should not also enter into this state. So, this is also not permitted. So, if in your algorithm you avoid these transitions you will always end up in these 4 and these 2 and remaining 2 which will be on top of only in these states you will be operating and you will always be able to set up the connection without rearranging the existing one. Henceforth you got the algorithm and so it switches why it is not blocking. So, this is the simple argument which proves it. So, important thing which you have learnt is the state how to contract them into very few states all possible combinations and then how to build up a state transition diagram and basically from there to identify the algorithm of operation for a wide sense non-blocking state. So, different configurations again you will have different algorithms. So, this is one of those examples this one because if the connection goes off you might end up in this state which is a blocking one. See not only you are moving in the direction of arrow you are also going back when the connections are released. See there is no guarantee that connection will be released and you will always moving from here to here. You can also go from here to here then come back here and you might come back here and from there there is only one state in the ground when there is no connection. So, you are moving from one state to another state whenever connections are set up or they are released. So, avoid coming to this always ensure that you cannot come to this state and that will ensure that switch is non-blocking, but there is a algorithmic constant here. It is not strictly a strict sense non-blocking you cannot set up anything arbitrarily you have to follow certain rules. So, the rule is if you are in these states do not bother if you are in this state and you want to set up a connection never try setting up a connection so that you end up in this state always go to this configuration. There are two possible configurations where you can go. So, never go to this one always go to this one you can always set up the path. So, next class we will see how to implement a time switch using random access memory. So, that we can understand the control structures of the switches. Thank you.