So, you can see that this algorithm is slightly sub-linear, slightly better than the linear space that is required by the BFS and DFS and it is quite efficient, polynomial time is considered quite efficient in from the perspective of complexity theory. So, could this be improved that is the central question, can we have a polynomial time algorithm for reachability that requires space which is better than what is provided by the BBRS algorithm and in this regard this is the question that we try to answer. So, for general graph obviously this is the best known and there has been no developments but for grid graph that is a restricted class of graph there has been better algorithms are known. So, the first such algorithm was presented by a Sano endure in 2011 who showed that reachability in grid graphs can be solved in approximately root in space and polynomial time. Now, this was further improved by Ashida and Nakagawa in 2018 and they improve the space complexity to n to the power one-third and there comes our result we further improve the space complexity to roughly n to the power one-fourth. So, essentially this is our result that for every epsilon for every small epsilon you can you can select epsilon as small as you want there exists a polynomial time algorithm that decides reachability in grid graphs using n to the power one by four plus epsilon space. Now, okay one thing more that we are talking about the directed reachability problem because an indirect reachability problem has been solved in lock space by a researcher called Dean Gold. So, we are only interested in the directed reachability problem. Yeah, the running time depends on epsilon we treat epsilon as constant. So, n is the number of vertices in the grid. The number of vertices in the grid graph. Yeah, but since the reachability in undirected graph can be solved in lock space by Dean Gold we could just I mean yeah remove the isolated vertices they do not have any significance. So, yeah so let's see how how do we solve reachability. So, there is an important concept of a separator in a graph. The separator is basically a set of vertices which we remove from the graph. The graph gets divided into roughly half components or you can say it's divided into components whose size is a fraction of the original graph. So, the original the algorithm the first algorithm of root n size root n that solve reachability of in root n space used this idea that you divide the graph into the small components. And now what happens is now if you have such a separator then any path from vertex in one of the component to a vertex in other component would necessarily use one of these vertices of a separator. So, this is an important property. Now grid graph the size of the grid graph I mean in the in a grid graph it's known that a separator can have at like order root n size separator would exist in the grid graph. You would basically select the middle middle row or middle column and that would act as a separator of the grid graph. So, so as I know and do or you use this idea you have to solve reachability in grid graph using the separator. Basically they use this separator and to divide the graph into components solve each of the components recursively and then combine it and they do it in a space space bounded setting. So, basically you cannot store each of these components you have to kind of use solve it again and again whenever you require a solution for the smaller components. So, that was the initial algorithm by Asano Andrew. So, how to improve the solution? So, the the Ashida and Nakagawa gave an improvement to this algorithm. What they did was first there is some sort of preprocessing to make the graph smaller. So, you have a graph G of n vertices they preprocessed it. So, to make a graph G prime of n to the power 2 by 3 vertices. Now, this G prime was not a was not a grid graph it was it was it turned it was a planar graph, but planar graph also had a separator of root n size. So, they basically what happens is then the algorithm implicitly does this is that they find a separator of this n to the power 2 by 3 size graph and then solve it and use the basically the same divide and conquer technique that I previously explained. So, that is how they improve the space bound to n to the power 1 third. Let us see how we approach this problem. So, what we initially do is we partition the grid graph original grid graph into subgrids of size n to the power 1 minus 1 minus alpha. I am here taking alpha as some constant and what we do is we solve these subgrids though by solving the subgrids I mean that in each of the subgrid we only consider the vertices of which are at the boundary and if they have a path from one vertex from one vertex of the boundary to the other vertex at the boundary using the vertices of only that subgrid then we add an edge in this new graph. So, we call this graph an auxiliary graph. So, I have given an example I mean it is quite straight forward how we construct it. So, this auxiliary graph will have size root n to the power 1 by 2 plus alpha by 2. So, suppose now if this auxiliary graph had a separator of again root n size we would have got a n to the power 1 by 4 roughly n to the power 1 by 4 algorithm, but unfortunately this does not happen the subgrid might have a very large separator of root n not n to the power 1 4. Yes, the source and target nodes are pre-specified and we simply assume that they are present in one of the boundaries because if it is not present on the boundary it could be taken care of I mean it is not it is not much of a big deal. Yeah, but this smaller grid graph this auxiliary graph might have a quite large separator. So, what do we do this? We first see we look at some of the properties that this auxiliary graph might have one is this that in this auxiliary graph if we pick a block of this auxiliary graph and see if these two edges that they cross each other at a certain point then it is guaranteed that there will be two more edges from the tail of the first edge to the head of the other edge from the tail of the second edge to the head of the first edge it is guaranteed because I mean if they are crossing it it would mean that they their paths would share a common vertices somewhere in the original graph and through that vertex those two new parts can be derived. The other observation which is basically an expansion of this if suppose a particular edge has been crossed by two edges or say multiple edges then the then consider the edge even which is closest to the tail of the of this edge f then there will exist an edge from the tail of even to the head of e2 and this is true if the edge f is being crossed by a multiple edges even to ek as well there will exist an edge from the tail of the for lowest edge to every other head of every other edges algorithm is basically this think of your favorite traversal algorithm if if you know that both these even and e2 are reachable from our original vertex s then you only need to traverse through one of the one of the vertices to the other side of f this is what our observation is and and this is how this is what we use to solve our reachability problem so we define a new component due new device called pseudo separator now what is the what is does is it's like a separator but quite not like it it divides the graph into small components but now the now in case of separator only one thing could be possible that if there is a path from one component to other component it would have it would pass through a vertex of the separator for the pseudo separator we require that the path could either either take a vertex of the separator or it could cross one of the edges of the pseudo separator now once you have such a pseudo separator you could think of that traversal algorithm we could simply find a pseudo separator divide it and so so let's look at any any suppose we have any vertex induced sub graph g alpha which has an H vertices so so because we require a pseudo separator of of every components as well we don't only require the pseudo separator of the of the graph g alpha we require once we divide the graph into multiple components these components are sub graph of g alpha and we require a we require such a separator for them also what we prove basically in our result is that that such a pseudo separator exists and it can be constructed in a space efficient manner and this is quite the this this this I will briefly comment on how we construct this pseudo separator what we do basically is we we kind of become maximal planar graph out of out of the graph that we have we pick maximal set of edges which makes the graph planar and then we find a separator of that maximal set of edges and we tweak it a little bit it turns out that it would act as a perfectly well pseudo separator so we find a pseudo separator and we construct our algorithm is simple we construct a pseudo separator of size of basically roughly root h size where h is the size of the original thing that we are looking at and then we divide the graph using see put pseudo separator now to combine this we first solve each of the components recursively and then since we talked about that traversal algorithm that we only require the edge that was closest to the tail of of an edge of the pseudo separator so we traverse using that that that particular that particular vertex so so I'll briefly restate our contribution so we basically develop a better algorithm for for reachability in grid graph that improves the space bound from n to the power 1 by 3 to n to power 1 by a 4 plus epsilon where epsilon can be as small as you like and we also introduce a new device this pseudo separator the workplace several open question is that can other classes of craft use this pseudo separator or something similar to pseudo separator something similar idea to to solve the reachability problem or why stick to reachability problem can be used so it can be used to separate or to solve other block problems in a space bounded setting or probably a different setting say perhaps can be solved matching I'm not sure I mean that is what we could look at and the thing that I am currently looking is can we improve the reachability for planar graph itself so currently it's known that planar graph the reachability can be solved in polynomial time that uses root and log n space so so and and see that our algorithm doesn't really use the fact that our graph was a grid graph it just used the fact that we could get this initial auxiliary graph fast so if you could somehow get this initial auxiliary graph of the planar graph as well then perhaps we could we could perform we could we could give a better space bounded algorithm for planar graph okay so these are the references no so obviously planar graphs are super set of grid graph but I mean so you require an embedding of the grid graph as well and it it can be proven that this specific reachability problem in planar graph can be reduced to grid graph I mean in log space however such a reduction would quite blow up the number of vertices by I guess n square or n to the power 4 but so the R algorithm wouldn't actually improve so one one way to improve the time with the space complexity of planar graph would be to try and come up a reduction from reachability in planar graph to a reachability in grid graph such that the number of vertices does not blow up substantially so we need to kind of do this recursive step right so we are dividing the graph into small components and then we are recursing over each small components so we need to make the components small enough so that the so that the depth of regression is constant and for that we need to slightly increase the size of the separator that's what it does in the time complex in the space complexity you know in time complexity I guess it was something 1 over epsilon or something like that but we treat it as constant whether it is reachable but I think I mean this algorithm be modified to solve for finding out the same space constant yes so no I'm assuming that the vertices are on placed on a grid and the edges are like you could have an edge or you could not have an edge but the edges can occur yeah you could see that I mean I defined the diff grid graph in a no no we don't require that all the edges are there but the edges can occur only between the edges and edges vertices that's why we are calling it yeah okay okay so how well in the earlier works such a definition was concerned I mean considered so perhaps in the other settings such a different definition is used in this we would have to look into it I mean I think it would work but I think it should work yeah perhaps you could you could embed this grid on on a surface of any genus I think but I mean I have to follow through I have to see if there is any problem I'm not sure okay I'm not sure if that would help so we tried solving it for three dimensional grid graph but it didn't have this property that if two edges are crossing each other then there would be an edge from tail to head of other so three dimensional grid no no no we were using this auxiliary graph right that we


Here we are in chapter 4 in the Gospel of John. We'll begin reading together at verse 46, and I'll read to verse 54, and we'll get into our study. What we're looking at is the healing of the nobleman's son found in these verses. So beginning at verse 46. So Jesus came again to Cane of Galilee where he had made the water wine.  When he heard that Jesus had come out of Judah into Galilee, he went to him and implored him to come down and heal his son for he was at the point of death. Then Jesus said to him, unless you people see signs and wonders, you will by no means believe. The nobleman said to him, sir, come down before my child dies. And Jesus said to him, go your way. Your son lives. So the man believed the word that Jesus spoke to him and he went his way. And as he was now going down, his servants met him and told him saying, your son lives. He inquired of them the hour when he got better. And they said to him yesterday at the seventh hour, the fever left him. So the father knew that it was at the same hour in which Jesus said to him, your son lives and he himself believed and his whole household. This again is the second sign Jesus did when he had come out of Judea into Galilee. So it's interesting as we begin this portion of scripture how John alludes to the miracle that Jesus had performed earlier in Cana. And that's to remind us that his works were intended to bring people to faith in him. When you read Matthew in chapter 11 verses 2 through 5, it reads when John had heard in prison about the works of Christ, he sent two of his disciples and said to him, are you the coming one or do we look for another? And Jesus answered and said to them, go tell John the things which you hear and see, the blind at sea and the lame walk, the lepers are cleansed, the deaf hear, the dead are raised up, and the poor have the gospel preached to them. Are you the coming one? All you need to do is tell him what they're seeing and what they're hearing. And so miracles were part of what Jesus would do to convince people that he was sent from the Father. Now we know that Jesus never performed miracles simply to satisfy people's curiosity. You see, there's always people who want to see a miracle and be honest with you, I don't blame them. You know, there are many who want to see miracles and when people were hearing while he was here, when they were hearing that he was performing miracles, they wanted to see him perform one. We even see that going into the court with Herod in Luke 23 verse 8 when it says when Herod saw Jesus, he was exceedingly glad for he had desired for a long time to see him because he had heard many things about him and he hoped to see some miracle done by him. And so people wanted to see miracles, but Jesus never performed miracles to satisfy curiosity. Many came to him because they had heard of his miraculous powers and it became quite common for people when they heard that Jesus was around, it became quite common for people to seek him out. Matthew tells us in chapter 8 verse 16, when evening had come, they brought to him many who were demon-possessed and he cast out the spirits with a word and he healed all who were sick. So people were beginning to follow him. They were hearing of the miracles that he performed and they wanted to see him perform miracles and they knew he could. In Matthew 15 verse 30, great multitudes came to him having with them the lame, blind, mute, the maimed, many others. They laid them down at Jesus' feet and he healed them. So he performed miracles, but he didn't do so upon request. He wasn't on earth to just perform miracles, though some thought of him in that way. In Matthew 12 verses 38 through 40 some of the scribes and Pharisees answered saying, Teacher, we want to see a sign from you. But he answered and said to them, an evil and adulterous generation seeks for a sign and no sign will be given to it except the sign of the prophet Jonah. For as Jonah was three days and three nights in the belly of the great fish, so will the Son of man be three days and three nights in the heart of the earth. We want to see a sign. He said the greatest sign you're going to see is my resurrection. Now the works that Christ performed had a purpose. They were intended to bring us to faith in him as our Messiah. John will say that in chapter 20 when he says in verses 30 and 31, truly Jesus did many other signs in the presence of his disciples which are not written in this book. But these are written that you may believe that Jesus is the Christ, the Son of God and that believing you may have life in his name. And so Jesus is performing miracles and a miracle is now being asked of him. Again in verse 46, Jesus came again to Canaan of Galilee where he had made the water wine and there was a certain nobleman whose son was sick at Capernaum. So John is now giving us details concerning a second miracle recorded in his gospel, the healing of a nobleman's son. So when it speaks here concerning the city of Capernaum when you're looking at a map, Capernaum was about 20 miles to the east of the city of Canaan in the northern portion called Galilee. And it's a city that Jesus ministered in quite often. We already have noted this but let me remind you the city became Jesus' ministry headquarters when he left Nazareth. The city of Capernaum was the home of Peter, Andrew, James, John as well as Matthew. And after so many works there Jesus even went on to pronounce a curse on the city because he had performed so many works there in Capernaum. And so the city of Capernaum was a place that Jesus was residing in as his ministry headquarters if you will. And so Capernaum is noted here. It says there was in verse 46 a certain nobleman whose son was sick at Capernaum. Now when you see nobleman, the word nobleman speaks of a royal official. He might have been part of Herod Antipas' court. And so this is a royal official that's being mentioned here that's simply referred to as a certain nobleman. But his son, it says, is very sick and he's in the city of Capernaum. So verse 47, when he heard that Jesus had come out of Judea from the south into Galilee, Galilee the north, he went to him, implored him, come down, heal his son for he was at the point of death. So Jesus' fame has spread. The entire area has heard of him. And Jesus is now well known.  Those who were demon possessed, epileptics, paralytics again, and he healed them. And so his fame has spread. This man has heard of him, knows of him, is sending for him. It says in verse 47, he went to him and implored him, begged him, come down, heal my son, he's going to die. There's no chance of him recovering. And the father is beside himself with concern. Some of us as parents could understand that to some degree.  You know, as we raised four kids, there were times when they would be ill. In times when Marie and I didn't have the wherewithal to take care of them, we didn't have the finances. And now if they're ill, what are we going to do? And if they get sick, you begin to be greatly concerned. But what happens when that baby's at the point of death and when that young one is at the point of death? And that's what's happening with this noble man, his son, who he dearly loves, is very sick. He's at the point of death and he comes and begs Jesus Christ, please do something for us. So as I look at him, let me take a moment to say this father's a great example. This is a man who loved his son and this is also a man who knew who am I supposed to bring my concerns to. At a certain point, we men, we who are husbands, we who are fathers, we have to come to the point of realizing that the most important and best thing we can ever do is to bring our kids to the Lord. Whenever we have a need, we need to know who to go to. We can't solve all the problems. We can't make them well. We have to go to the Lord on their behalf. We have to learn to do that. And if you have yet to learn to do that, you need to learn to do that. We can't make them well. There are some things that our kids are doing. Even right now they may not be sick because of their physical illness, but their life isn't a healthy life. And there are things we need to learn to do. And one of the things that we need to learn to do is to pray and to take our concerns to Jesus Christ, to cast our cares on Him because He cares for us. There are things that we can do. We can put food on the table. We can put shoes on their feet, clothing on their back. We can do things like that financially. We can do a lot of things for our kids that gives to them advantages. But there are times when our kids are simply going in a direction that we would not have them to go. And there's really nothing we can do. It's not like we don't want to. It's that we run out of how can we do? And I had to learn, and I still do, by the way, even any parent knows this, but the older your kids get, always remember this, they're still your kids. They're still your kids.  Because why? That's what I would ask. Why? Because to her, I was always going to be her baby. That's the way it is. Get used to it.  You know, I'm old mom, come on. Get used to it. You're always going to be the baby. Always. My kids know that.  I always do. I've got a 41-year-old baby. My Korean. But my 41-year-old baby is going to have another baby. She's going to have another baby, oh my... Oh my goodness. 41-years-old. Stop. Just before Thanksgiving. So we'll have a Thanksgiving baby. Anyway, so they're always... They're always going to be your baby. Excuse me, and you can still. And you still, as I do, and Marie, my wife does, you still go to Jesus when you're concerned for him. It's the first place we need to go, guys. And in this band's case, there's nothing he can do. He's at the point of death. Come and help me. I need your help. He says it. He came out of Judea into Galilee, meaning he had come from the south and he was now close by. He went to him and begged him, implored him. Come down and heal his son. For he's at the point of death. Now notice Jesus' response in verse 48. Jesus said to him, unless you people see signs and wonders, you will by no means believe. Now the man had basically done what the psalmist had taught him to do. When the psalmist in Psalm 50 verse 15 said, call upon me in the day of trouble. I will deliver you. You shall glorify me. And the man had done that. He had taken the words of the psalmist, will say to heart. In Psalm 145, 18, the Lord is near to all who call upon him, to all who call upon him in truth. He did what he was supposed to do. He implored him. He came to him. He said, help me. And it appears almost as if he's a little rude, doesn't it? Unless you, unless you people see signs and wonders, you will by no means believe. That's kind of stern. Now Jesus had just ministered to Samaritans and many had believed him. But here notice with me, Jesus is speaking to Jews and he's rebuking their hardened hearts. He's already been performing miracles. We've seen them mentioned in chapter two, but they're desiring to see this. Again, at first it seems like a harsh, even an unfeeling rebuke. It's like he's saying, if you're interested in my immediate works, that's one thing. But you're not interested in the knowledge of who I am. You want to see works, but you're not interested in me. You want to see me do something for you, but you're not interested in who I am. In 1 Corinthians chapter one, verses 22 and 23, Paul said, Jews request a sign. Greeks seek after wisdom. But we preach Christ crucified. To the Jews a stumbling block. To the Greeks, foolishness. But to those who are called, both Jews and Greeks, Christ the power of God and the wisdom of God. Jews request a sign and Greeks are looking for wisdom. You see, some people believe in miracles. They do, but they don't know the one who performs them. Some can speak of wonders that they've seen, but they can't speak of the one who performs these wonders. One of my favorite scriptures is found in Psalm 103 verse 7. It simply reads, he made known his ways to Moses, his acts to the people of Israel. His ways and his acts, he made his ways known to Moses, his acts to the children of Israel. Ways and acts. There are people, a little illustrated like this, there are people who know Pastor David will say, by my activities or my actions or my acts.  And they say, the response is, I go to Calvary Chino Valley. Oh, you know Pastor David? Oh yeah, he teaches on Wednesday, he teaches on Sunday. Yeah, I know him. How well do you know? I just know him. I see him up there. They know my acts. But if you ask my children or you ask my wife, do you know Pastor? They'll say, you mean dad or my husband? They not only know, my kids and my wife not only know my activities, but they know my ways. I understand the word ways in the biblical sense. My modo. That's a nice Spanish word I learned, it's one of the few I know. My modo. They know my ways. They know not what I'm doing, but why I do those things. They know me, not just by standing behind a pulpit, but they can tell you why I stand behind that pulpit. See, there's a difference between knowing the activities and knowing the reasons. The people of Israel knew that God did certain things, but Moses knew why God did certain things. Do you get it? He showed his ways to Moses, but his acts to the children of Israel. So the children of Israel could say that God had done a work with the Red Sea and that he had slaughtered Egyptians and they knew his acts. Moses knew why. And there's a difference between knowing that God can do something and knowing that God who does that thing. And there's a lot of people today who believe that there is a God who does miracles. And then there are those who know why God does miracles. And this man knows that Jesus can do miracles. And these people had already seen that he does miracles. But he still says, unless you people see signs and wonders, you will by no means believe. Because the miracles have not brought you to faith than to the one who is performing them. You know that I can do things. You don't know why I'm doing those things. And so it's a rebuke, yes. But what he is doing here, he's actually directing the man's faith and breaking him down so that he'll trust in Jesus. Because Jesus is about to heal. But I want you to note, he's going to heal at a distance. He's going to do a work at a distance. Notice when the man, verse 49 says, Sir, come down before my child dies.  He's healing at a distance. He's not even present. And that requires a tremendous amount of faith. There are other places in Scripture where Jesus actually heals at a distance. Remember the incident with a woman who is referred to as a syrophoenician woman. It's recorded in Mark chapter 7, 24 through 30. How she, as a woman of syrophoenician, she's not a Jew. And Jesus is ministering and she approaches him and she says, I need your help. I have a daughter who is demonized. She's demon-possessed. And I need you to do a work on my behalf. And Jesus speaks to her. And he says, it's not right to take the children's bread and to give it to dogs. That's not a nice thing to say. It's not like today when somebody sees his friend and says, what's happening, dog? That's not what Jesus was saying. And she said, well, that's true. She said, but even the little dogs eat the crumbs from the master's table. You need to know that when Jesus and she are speaking, the word dog is an important word to note because when she responds, she uses a word that speaks of a puppy or a house dog. There are two different words used at that time to describe what we call a dog. One of the words spoke of the dogs that were wild, that ran in packs and that would go to the garbage heaps and would scavenge. And they were ferocious. And that is a word that was translated dog. But there's a second word. And the second word is the word that this woman and Jesus are using when they're speaking concerning the dog. And that's the house dog. That's a puppy, a puppy that eats from the table, the scraps that are... And she says, well, even the little dogs eat the scraps from the master's table. And Jesus' response there is, hmm. Hmm. This is faith that's being revealed. And as a result, he tells her, you have your request, it's done. Jesus didn't go to the woman's house. He just at a distance delivered this little girl who had been demon-possessed. And there's another time that we see it in Matthew chapter 8 verses 5 through 13. There's a centurion who has a young servant that's just a young boy. That means a lot to him. And he comes and he says to Jesus, I have a servant that's very ill and I need you to heal him. And Jesus says to this man, I'll come with you. And this is the real famous statement. The man says, Lord, no. He says, I am not worthy that you should come under my roof. But just speak a word and he'll be healed. And Jesus marvels at this man's faith. He says, I haven't seen faith like this even in Israel. And from a distance, Jesus performs that miracle. So he does it with a syrophoenician's daughter. He does it with a centurion's servant and he's about to do it here in the case of this noble man's son. Now this man has a faith because he trusts in Jesus' promise. But we have to realize it would have been difficult to just listen to what he's saying and to believe. So we need to remember that faith is often refined through testing. And in the testing of our faith the impurities are exposed and are discarded. In Mark 9 verses 22 through 24 this man is speaking of his son and a demon that is abusing him. And it reads, often he has thrown him both into the fire and into the water to destroy him. But if you can do anything have compassion on us and help us. And Jesus said to him, if you can believe, all things are possible to him who believes. And immediately the father of the child cried out and said with tears, Lord I believe, help my unbelief. Do you ever have unbelief? Do you ever have that? I do believe. It's not the things I believe I'm concerned with. It's the unbelief that seems to surround and overpower my belief. And Jesus says that the work is going to be done. It's one thing to say it. It's another thing to receive it. And in the midst of all of this this man's faith is being stretched. Now think about it. This man has already shown faith. He made the effort to come to Jesus. Again that's at least a day's journey away. It's about 20 miles. And so many humbled himself and he came to a carpenter. He recognized that he himself had no power to help his son and he came to the one he believed could. And he would not be discouraged. He remained persistent even when Jesus at first seems to be rebuking him and those around him. And I've discovered as you have that love and concern for someone has a way of reducing us to humility. Doesn't matter to me. I just want you to do something. I know you can. I pray that you will. So the man's response again in verse 49 Sir come down before my child dies. Notice he didn't argue. Notice he didn't defend himself. And notice he didn't get insulted. He simply pleaded for his son. And when he says my child my child is an expression of tenderness and affection. It's another way of simply saying this is my baby.  I need your help. I believe with all of my heart I know this to be a fact scripturally and experientially that the Lord has a way of reducing us to just bare basics so often. He removes all the props and the things that I use to suspend myself or hold myself up so that I have nothing to rest on except for him. And I have to tell you that's not a comfortable place for me. I'm a person who likes to have an escape plan. Plan B, plan C. But he has a way of taking those props, removing those crutches so that the only thing I can be propped up with is him. Removing all of the things that I can trust in until I can trust only in him. Have you discovered that too? Have you found that to be true? It is true. He'll remove things that you so held fast to. Sometimes the things that you felt you had learned from him in the past then he has a way of upsetting even those things so that you have a more refined faith. I am poor and needy yet the Lord thinks upon me you are my help and my deliverer do not delay, oh my God.  You're my help. You're my deliverer. But don't delay. I need you now. And that's what this man is doing.  His money, his influence, his power, his position, his prestige. Nothing that he had can help. When it all came down he had advantages. But with all of his advantages he was still helpless. He didn't have the resources for healing his son. There's nothing that he can do. And he's at that place when Jesus in verse 50 says go your way your son lives. The man believed the word that Jesus spoke to him and went his way. I'm not coming home with you but you can go home in peace. Your child lives. You need to take me at my word. I don't have to come with you. I can heal him from here. And the man believed it says. The man believed the word that Jesus spoke to him and he went his way. And he went his way. Well in verse 51 it says as he was now going down his servants met him and told him saying your son lives. He inquired of them the hour when he got better and they said to him yesterday at the seventh hour the fever left him. So the father knew that it was at the same hour in which Jesus said to him your son lives. And he himself believed in his whole household. And so after receiving Jesus's word he trusted him and he remained overnight. You see that because in verse 52 when he's inquiring about the hour they say yesterday. And so that means he spent the night. The journey home from where he was at was 20 miles. That's what the Jews would call what the Jews would say that you could cover in one day a 20 mile thing. So naturally he had gone to see Jesus Jesus had spoken to him. He spent the night and the next day he's on his way making this 20 mile journey. And as he's going down his servants meet him according to verse 51. And they're saying to him his son's okay. Now when you think about that that means that when Jesus spoke the day before he was able to sleep in peace and then the next day began his journey.  Have you ever been so upset and so concerned that you were praying your heart out maybe into the evening even late at night and finally you have a sense that the Lord is giving you peace and you put your head on your pillow and you sleep like a child knowing that God is in control I've wrestled many times like I said I have four kids I've wrestled a lot over the years when my children were going through their teen years especially because you know what they may have a pastor for a for a father but they have a sin nature of their own they got it from their mom they have what is called the Adamic nature they got it from me from their dad and though they've been raised like yours with devotions and prayer I can't remember a time in their early life that we didn't pray for when they went to school that we didn't pray for them at night when they went to bed that we didn't say grace with them at every meal we ever had they had devotions five nights out of the week for years I would read them biographies or autobiographies of missionaries I would read them the Bible they would hold hands as small children I'd have them seated excuse me seated in front of me on the front room floor when they began to be able to read I'd have their little Bibles in front of them I would ask them what do you want to pray for I still remember when I was teaching them about Solomon I still remember when my kids were all young Anna I think at that age she was about at that time was about four or five and I I still remember we're going through Solomon where God says Solomon ask what is it that you want anything and I'll give it to you Solomon and I remember wanting to make this point with my children and so I stopped my reading in that dramatic pastoral fashion and I said wow God says you can have anything anything that you want anything what would you want the oldest Korean tries to be spiritual and says some spiritual thing and David my son tries to out do her with some spiritual thing and Joseph says a spiritual thing and then I come to Anna the baby and what would you ask for if you could have anything if you could have God give you anything what would you want gum gum I still think of that because I think I'm that way too ask and it shall be given you what do you want gum I can do the same kind of thing my sights are set so low but we poured into our children with tears and patient love and yet sometimes as they got into their teen years and their later teens it was one of those things where you held on for dear life I am so serious parents know what I mean you hold on for dear life so my advice has always been to parents because there's some parents with small perfect kids the kids are so perfect they embarrass you they're that good and they don't understand now you could have kids who are perfect too you need to use my way of raising kids and I just say that's unusual that's really neat and I bless the Lord for that but I didn't have that I had ordinary kids and they were tough and they could be heartbreakers and there were a lot of tears and a lot of there was a lot of pain but it's all worth it in the end because all four of my kids serve the Lord now thank God so you hold on and you go to the Lord on your face and you implore him and you say save my child he is near to death and that's what this man is doing save my boy he's going to die and that's what he's doing and Jesus says he's alright and I've done that and I put my head on my pillow at night and I remember Isaiah 26.3 you will keep him in perfect peace whose mind has stayed on you because he trusts in you and I've been able to put my head on my pillow and sleep and he says the peace of God which surpasses all understanding will guard your hearts and minds through Christ Jesus so you cast your cares on him and so they come to him and they say to him in verse 51 your son lives so he inquires of them the hour when he got better they said to him yesterday at the seventh hour the fever left him at one in the afternoon that's what the seventh hour represents now this may be a way of testing to see whether it was a coincidence or a miracle so let me say this very briefly but there is obviously nothing wrong with testing to see whether a miracle actually occurred and in our day there are many who profess to be miracle workers and it seems that that number is increasing on my Facebook page I'm still old so I still use Facebook there'll be people who are putting on my feed by my book and you'll know how to be a prophet and you'll know how to prophesy and perform miracles and all you gotta do is come to my seminar you guys if you're on social network with me I know all of you more than likely are you know that there are feeds that come and request for this book and come to this conference that I read those every day I'll teach you how to break the 200 barrier and you know there's obviously some algorithm that's involved so that they send me spiritual things all the time about pastors conferences and this and that so I get these things all the time how to perform miracles come to our miracle or our prophetic school we'll teach you the seven steps of being a prophet things of that nature I get those all the time perhaps some of you may see those on your own feed but I get those all the time there's nothing wrong with testing to see whether these things are true or not as a matter of fact it's a wise thing to do it's a command in the Old Testament in the book of Deuteronomy in chapter 13 listen to what it says if there arises among you a prophet or a dreamer of dreams and he gives you a sign or a wonder and the sign or the wonder comes to pass of which he spoke to you saying let us go after other gods which you have not known let us serve them you shall not listen to the words of that prophet or that dreamer of dreams for the Lord your God is testing you to know whether you love the Lord your God with all your heart and with all your soul if he says even if it's miracle or his word comes to pass he says there's this new way to go he says no don't follow him in the Old Testament you have then in the new you have the same 2nd Peter 2 verse 1 there were false prophets among the people even as there will be false teachers among you who will secretly bring in destructive heresies even denying the Lord who bought them and bring on themselves with destruction the church is to put the words and the works of these people to a test 1st Thessalonians 521 test all things hold fast what is good 1st John 4 1 beloved do not believe every spirit test the spirits whether they are of God because many false prophets have gone out into the world and what is the test does what is being said Isaiah 820 to the law and to the testimony if they speak not according to this word it is because there's no light in them somebody said the law here is a reference to the law of Moses the first 5 books of the Old Testament which contains also the 10 commandments the testimony is a reference to the rest of the Old Testament and by extension includes the New Testament which contains the testimony of Jesus Christ Revelation 12 17 when joined together this explains what the law of Moses means in mankind's practical experience how do I know that this is from the Lord it has to line up with Scripture and if it's taken me away from Christ and taken me to a new way a different way it's not true so it's wise and proper to test any miracle to see whether it's from God does that work bring glory to God did that work bring us closer to Jesus does it line up with the Bible you see he already had trusted Jesus's word but it would seem that he needed to verify the timing and this question served to deepen his faith in the promise Jesus made because it validated what he already was beginning to believe and so he inquired again in verse 52 and they had said he says when did this happen they said yesterday at the seventh hour the fever left him it happened at one o'clock in the afternoon the exact time that he was speaking to Jesus the fever left at once he was made perfectly well immediately I'll say one more thing and move on there are so many who are referring to themselves as healers they used to be referred to as faith healers now they simply say they have a healing ministry it's the same thing and there are those who have prayed for somebody and they've said confess in faith and receive and you are healed by stripes you are made whole and then that person doesn't get well and I've seen him where the false teacher will say well you are well that's just a lying symptom well in this particular case he was healed immediately immediately no lying symptoms involved at all so verse 53 the father knew that it was at the same hour in which Jesus said to him your son lives and he himself believed he had a deeper sense of faith in Christ and his whole household not only did he trust in the Lord completely but his whole house did too his whole household was converted and the sickness of the child became the means of salvation for the household so very often what it first appears to be the means of our greatest hurt and be turned to our greatest joy and that's because faith is often refined in the furnace of affliction in Psalm 119 verse 67 the psalmist said and this is so so deep before I was afflicted I went astray but now I keep your word that's interesting but it's true let's see if I can say this briefly I can't say anything briefly but I'll try Job when you read the first two chapters and God makes a declaration concerning Job remember that God speaks of Job as being a righteous man who hates evil and Job is portrayed in the first two chapters of the book of Job as the most righteous man on the face of the earth we all know that Satan appeals to God that he might have Job to test him when you look at the conversation the sons of God appeared before the Lord and Satan came also God speaks to Satan where have you been he asks him when I taught the book of Job many years ago now I pointed out that Jesus rather God is interrogating him what he's saying is tell me what you've been up to and he says I've been going to and fro throughout the earth that infers I've been up to no good and that's why God says have you become acquainted with my servant Job if you read the scriptures as they're really portraying that conversation you can almost see a look in the face of the enemy he says of course I have but you put a hedge about him I can't touch him now God already knows that he's been looking for a weakness in Job he just simply asks him or tells him he's really commanding him to give an account it's a commanding officer speaking to a lesser give an account yeah I've been there I've looked around yes I've seen him but you put a hedge about him which gives us the inference that he tried to get to him and couldn't so when you begin to read Job you see that God says to him and he says Satan says you know but if you give me an opportunity he'll curse you to your face and he says you can do this and that but you can't take his life what's he do well he takes his family he takes his wealth and then finally in the second chapter he takes his health skin for skin all that a man owns he he will give for his skin if you let me touch him he will curse you to your face because Satan knows that when we're sick we have a tendency of blaming God why won't you help me I'm in need I'm crying out I can't walk I'm hurt we have a way of thinking God has rejected us and he knew that because he had been doing that with man since man was on the face of the earth God says you can touch you can touch him but you can't take his life and so you see throughout the book of Job the afflictions that were upon this righteous man a man that at one time well he says it I walk into a room and all the elders stand to give me honor and now I've become the song of children who deride me in the street and my own wife is asking me how long will you hold to your integrity curse God and die thanks honey appreciate your encouragement and all along he wouldn't accuse the Lord to understand God's ways and the affliction that he went through he was able to say I know that my Redeemer lives I know that my Redeemer lives how I wish I had a daiseman I wish I had a mediator someone to stand before me and God between us so I could plead my cause he was broken and then finally at the end in the book of Job is where the Lord is speaking with your loudness and your complaints and God begins he says equip yourself as a man and listen I'm going to ask you some questions you've been asking all these questions all this time and then God starts asking him questions where were you when I when I put the stars in place when I told the sea to go this far and go no further where were you I didn't ask you for any advice you've got questions and you see it it's a very powerful rebuke that the Lord gives to this very righteous man and then Job finally says I repent and dust Sackloth and Ashes he says I heard of you with the hearing of my ear but now I see you with the seeing of my eye and I abhor myself I've come to realize then all of this affliction I've learned something of you now I'm not saying go home and say God make me like Job I want to sit with a broken piece of pottery and scrape off the scabs now I'm not saying that but I am saying that in the midst of his affliction as he cried out he learned something that he wouldn't have known any other way and I'm not one who will ever write books on suffering because frankly I'm the one who hates it the most but I can tell you that it has been in my quiet hours in my brokenness and my tears and saying God help that I've seen that my God never leaves me nor forsakes me I've discovered that I've discovered that when I've been broken that he lifts me and heals me and I have seen that he's able to do that and I have and so have you I have learned things about God in the furnace of affliction things that would not have known about and not only that but I've discovered that those are the things that I've brought into my ministry for God so when I speak to people who are hurt and broken I can say to them understand God is with you He will not forsake you He never leaves you He will deliver you and you will rejoice I know my God is able because that's what God does and this man had a little boy that he loved with all of his heart and he was dying and he came to the only person he thought could help and Jesus at first seems to rebuke him but finally says go your way he lives and he starts making his way home and his men come and say your son's alive what time? one o'clock that's when Jesus had said that and he believed and he told his family this is what happened tell your children about the wonderful works of God so that they too might believe and that's what we see in this story here he was able to influence his family to follow Messiah Jesus and then finally verse 54 this again is the second sign Jesus did when he came out of Judea into Galilee this is the sign the second one mentioned in the Gospel of John a sign marked his first mention in Galilee and this sign marks the second interestingly the first miracle was performed without a request but the second miracle was drawn from him by the imperfect faith and agonizing pleading of a father when he did the work in Cana and made the water into wine that's a different kind of miracle because nobody asked him to do that his mother simply said they're out of wine and that's when Jesus had said what do I have to do with you and that's when she said to the men whatever he says to you, you do it this time he's asked and this time he performs a little bit different but it's the second miracle mentioned in John's Gospel