by M. N. KRISH
His schoolteachers used to like him a lot. He would help his school headmaster with writing timetables.
He scrolled further down and there was another.
Many people in Kumbakonam used to know him. Someone or the other would always come looking for him. If they ask him to give tuitions to children, he would help them. If they ask him how to go on a pilgrimage, he would help also.
‘I think some of the things she said got lost in translation, sir,’ Divya said. ‘It’s much clearer in the audio-file.’
Joshua finally stirred in his seat. ‘What!’ he said, flinching. ‘You’ve listened to the interview?’
‘Yes sir, I listened to it this morning, that’s why I was late,’ Divya said. ‘It’s already uploaded on your homepage. That’s what helped me convince myself that I was seeing what I was seeing.’
Lakshman looked stunned for a second and then reached out to the mouse and keyboard. Soon the old frail voice of Janaki Ammal quivered out of the speakers and filled the air.
Lakshman turned up the volume to the max.
It was an old recording and there was a persistent hiss in the background punctuated by occasional grating and squeaking, but there was no mistaking what she was saying. Lakshman and Divya listened intently.
Janaki Ammal spoke a brahmin dialect of Tamil, not just any brahmin dialect but that of Iyengars, one suffused with baroque Tamil phrases in addition to Sanskrit terms. Lakshman, who was never fully at ease with the brahmin idiom or dialect, especially as spoken by someone from the old school, had to think hard to puzzle out what she was saying. He was better now than he was back in the Eighties, but even so, he couldn’t grasp some of the nuances in her speech.
‘Here, sir, it’s coming,’ Divya alerted Lakshman when the relevant segment was approaching.
Lakshman summoned every bit of concentration he was capable of and listened.
A lot of people in Kumbakonam knew about him. So many people would come to the house to visit him. Some people would ask him to give private tuitions to their children. He would gladly oblige. Some Vaishnavites came home once and asked him to tell them a shortcut to go on a pilgrimage to all the Divya Desam temples. He spent a lot of time helping them.
Lakshman asked her to play back the segment once again. There was no mistaking what she was saying.
‘What is it?’ asked Joshua.
‘Sloppy translation. My mistake,’ Lakshman said and explained it to him.
‘But still,’ Lakshman said, ‘108 temples, 106 on earth, two up in the skies somewhere . . . I can’t imagine how Jeffrey managed to make sense of all this. Even I am hearing about it for the first time. Did you know all this before, Divya?’
‘No sir,’ Divya said.
‘Then how did you figure it out now?’ Lakshman asked. ‘I don’t think your programs are that smart.’
‘No sir, it wasn’t the programs,’ Divya said. ‘It was a guy in Civil who told me: Venu Sampath. I talked to him when I was having trouble making sense of the output files.’
‘My God!’ Lakshman said. So that’s how Jeffrey figured it out? It hit Lakshman like a bolt of lightning. Narasimhan Thathachari. The very name said it all. It couldn’t get any more Iyengar than that.
‘Were you able to reverse engineer the algorithm?’ Lakshman asked Divya.
‘No sir, that’s impossible, for me at least,’ Divya said, a touch embarrassed. ‘We don’t have any sense of the termination criteria, sir; the last pages are missing. The programs couldn’t go further without them. Those people knew what pages to remove, sir; they’ve taken away the crucial sections, so even if someone else found the notebook, they can’t do much with it.’
Lakshman still didn’t seem to have gotten it and Joshua decided to step in. ‘Okay, you don’t have the algorithm, but could you figure out which mathematical problem it’s aimed at solving?’ he asked Divya.
This was the moment she had been waiting for. The revelation about Divya Desams, she owed entirely to Venus. But this one was her own.
‘I think so, sir,’ Divya said. ‘The pilgrims want to start from Kumbakonam, travel to nineteen other cities to worship at all the Divya Desam temples and finally return to Kumbakonam. They want to know what sequence to follow so they can complete the pilgrimage as quickly as possible. Since there are nineteen cities to visit, there are nineteen factorial or trillions of possibilities; the problem is figuring out which of them is the best. This is nothing but the classical Travelling Salesman Problem, sir. You explained it to me after your presentation last week; it has no mathematical solution and even the world’s most powerful computers cannot handle that kind of exponential growth in complexity.’
Joshua had seen it coming minutes after Divya walked into the room. But Lakshman had not been so quick on the uptake. Divya drove the point home in plain and simple English.
Ramanujan had an algorithm for the Travelling Salesman Problem!
Given that it was about twenty years before the coinage Travelling Salesman Problem took hold in the 1930s he would have probably called it the Vishnu Pilgrim Problem or something close to it.
It descended upon Lakshman like a ton of bricks, leaving him reeling.
59
The fan squeaked, a monkey screeched, but not a whimper escaped anybody for a long time.
Divya had swept both Joshua and Lakshman off their feet and she knew it. She could now count on a reco from both of them and was tingling with excitement, but tried to keep a poker face.
‘Are you saying Ramanujan had an algorithm for TSP and Jeffrey of all the people figured it out and plagiarized it?’ Lakshman asked when he found his voice.
‘I think so, sir,’ Divya said.
‘My God! Does this mean Ramanujan had a solution to the P versus NP question?’ Lakshman said.
Solving combinatorial problems like the Travelling Salesman Problem involved finding the best solution among trillions of possibilities. It was like searching for a black cat in a dark room full of cats. If you had some way of checking in the dark whether a cat you had in hand was black, then did it make it easy for you to catch the black cat? That was roughly what lay at the heart of the P versus NP problem: If the solution to a problem could be verified easily, then could the solution be found easily? If the answer was yes, then P was equal to NP. Otherwise it was not. No one had been able to answer this question so far.
‘The P, NP terminology is more recent,’ Joshua said, ‘but Ramanujan could’ve had an answer in his own language, just like he did for so many things. Remember the P versus NP question crops up in the context of partitioning where he’d made quite a few discoveries.’
‘But I somehow find it hard to believe or digest that he even had an algorithm for TSP,’ Lakshman said.
‘Perish the doubt,’ Joshua said. ‘He had the algorithm for sure and Jeffrey ripped it off. That’s the bottom-line.’
‘How are you so sure?’ Lakshman asked sharply. ‘And why did Jeffrey get killed?’
‘Say a little-known patent clerk in Switzerland tells a crook E equals mc2 and why. What would the crook do?’ Joshua asked.
Lakshman looked flummoxed for a moment before he replied: ‘You mean kill the clerk and steal the idea?’
‘Keep going,’ Joshua said. ‘Steal the idea and do what?’
‘Pass it off as his own, publish it in his name, claim all the credit for himself?’
‘Possible, but the stakes are still low. Raise them a little.’
‘Oh my God!’ Lakshman said. ‘Jeffrey wanted to claim the million-dollar prize from the Clay Institute?’
‘That’s for the P versus NP problem – we don’t know if he got his claws around that solution. I’m talking about TSP and I’m talking about a king among crooks, not just any crook. A million dollars is small potatoes for him.’
A million dollars and small potatoes
? Lakshman knitted his brows and chewed on it for a bit.
Joshua turned to Divya. ‘What do you think, young lady? What would he do with E equals mc2?’
‘Build a bomb, sir?’ Divya said, hesitantly.
‘Bingo!’ Joshua said.
Divya was as puzzled as Lakshman now.
‘What do you mean?’ Lakshman asked.
‘I’ll tell you what I mean, no, better yet, I’ll show you what I mean,’ Joshua said. ‘Do you have the Saturday’s paper we bought in Kumbakonam? I remember dumping it in your office before we went for the movie.’
Divya eyed Joshua with confusion. She’d seen every newspaper on Saturday, trying to pin down a good movie to watch with Venus. She didn’t remember reading anything remotely connected to this.
Saturday’s Hindu was still in the bin. Lakshman fished it out and handed it to Joshua, who opened it and placed it on the table.
The headlines on the front page read:
TURMOIL IN MARKETS!
SENSEX NOSEDIVES! SPOOKED FOREIGN INVESTORS FLEE!
RUPEE HITS ALL-TIME LOW! PLUNGES PAST THE 80-MARK AGAINST THE DOLLAR!
60
Lakshman’s brain blades had turned a bit rusty from disuse. Sure, he used to give Joshua a run for his money at Georgia Tech, elbowing him out when it came to awards and fellowships. But all that changed irreversibly after his return to India. ‘I’m not sure it makes sense, Josh,’ he said. ‘Do you mean to say the stock market crashed because of Jeffrey and the rupee fell because the market crashed?’
‘No, it’s the other way round,’ Joshua said. ‘The main problem is the rupee. It collapsed because of Jeffrey and that triggered the stock market crash.’
‘Still doesn’t make sense, Josh. What does the TSP have to do with it?’
Joshua turned to Divya and said, ‘What about you, young lady? Does it make sense to you?’
Divya’s little grey cells had been in an overdrive since last afternoon. She hadn’t seen this particular thing coming, but once Joshua gave the hint, everything fell into place. ‘I think so, sir,’ she said. ‘I think it’s related to what you were saying the other day. Going from city A to city B is like changing currency A to currency B. In the Travelling Salesman Problem, people want to start from one city, visit several other cities and come back to the starting point as quickly as possible, just like the pilgrims in Kumbakonam. In forex trading, people want to start with one currency, keep converting it to other currencies in a sequence and return to the original currency making as much profit as possible. In TSP we measure the distance and try to find the best sequence of cities, in the forex trading problem we measure the exchange rates and try to find the best sequence of currency transactions.’
‘Exactly,’ Joshua said. ‘TSP and the forex trader problem are essentially the same mathematical problem. Not just similar, but same. If you can solve one, you can solve the other.’
Similar, yes. But same? Divya wasn’t so sure. ‘But sir,’ she said, ‘the distances in TSP are additive. We go from A to B and then from B to C, we just sum the distances from A to B and B to C. But exchange rates are not like that, they’re multiplicative. If we change currency A to B and then B to C we have to multiply the exchange rates between A and B and between B and C. How can the two be the same?’
‘That’s easy, I bet you know it yourself,’ Joshua said. ‘How do you convert a multiplication operation into a simple addition?’
Divya nearly slapped her forehead: Why did I even ask the question? ‘Using a log transformation, sir.’
‘Logarithmic transformation, exactly,’ Joshua said. ‘Once you apply that on exchange rates, the two problems become identical, mathematically speaking: same combinatorial nature, exponential growth in complexity and no quick solution. NP-hard.’
Lakshman finally saw it. ‘So what you’re saying is, Jeffrey took Ramanujan’s algorithm to milk the forex markets with arbitrage trading? Start with a currency, go through a cycle of conversions to other currencies, and end up with a profit almost instantly?’
‘Exactly,’ Joshua said. ‘No one could do that till now to this level of sophistication. The mathematical problem was as such hard enough and the fact that currency rates could change in a matter of seconds made it even harder. There was no algorithm that could run fast enough to spot such long arbitrage cycles in real-time. Say you start with the dollar and trade in ten other currencies in your basket; you’ll have ten factorial or millions of combinations to consider. By the time the computer cranks through them all, the exchange rates are no longer what they used to be when you started out. Just like salami slicing, this is a decimal point game; you deal in basis points, so even a slight tick in the rates will throw everything haywire. But if you have a fast algorithm that runs in real-time you essentially have a money machine. You could be minting money every minute, out of thin air. The best part is it’s all fully legal.’
‘Really, sir?’ Divya said.
‘Absolutely. In the world of finance and economics there’s a thin line that separates investing and plundering. This is one of those things that sits smack on that boundary. Whether it’s here or there depends on your perspective, where your parallax is.’
‘I still find it hard to believe that he would’ve seen the connection between TSP and forex trading,’ Lakshman said. ‘Even I didn’t, till now.’
‘He knew, Lax. I’ve discussed it with him myself,’ Joshua said in despair. ‘Used to joke that if we had such an algorithm we could both become billionaires.’
‘Are you sure he could’ve built the software in such a short time?’ Lakshman asked.
‘I’m positive,’ Joshua said. ‘Not only was he an ace programmer, he also had the Sulba Sutra programs ready. He could have easily built the whole system in a week.’
‘Sulba Sutra programs? They’re part of this?’
‘Yes,’ Joshua said and eyed Divya for a second. ‘You can use them to speed up some of the number crunching in real-time, all the logarithmic and power series calculations. Because runtimes are even more crucial here, you want the computations to be completed before the next blip in the exchange rates. Like someone said, even milliseconds matter in the world of finance and economics.’
Divya had more or less ignored the Sulba Sutra paper as she was busy wrestling with the programs. It hadn’t seemed all that relevant at that time, but it suddenly began making sense now.
‘But why did the rupee drop so steeply like this?’ Lakshman asked.
Joshua mused for a bit.
‘Well, as our economist friends always say, currency trading is a zero-sum game. One person’s gain is another person’s loss. If you have a systematic mechanism for spotting arbitrage in forex markets you could systematically haemorrhage the wealth of nations. The worst affected will be second-tier currencies like the rupee, currencies that are neither too strong to hold for long term nor too weak to even bother touching; they’re exactly the sort of things you can use for playing the markets. That’s also why you guys hoard hundreds of billions in forex reserves, all in a basket of strong, safe-haven currencies.’
‘You think Jeffrey had the muscle to take on a whole nation like India by himself?’ Lakshman said.
‘Not by himself, but he could always team up with some fund,’ Joshua said.
‘Even so . . .’ Lakshman said.
‘Remember George Soros?’
‘Who is that, sir?’ Divya asked, making a mental note to look him up.
‘He’s an investor who holds the record for singlehandedly causing the collapse of two currencies,’ Joshua said. ‘British pound in the early Nineties, Malaysian ringgit in the late Nineties.’
‘Singlehandedly? You mean one guy could take on the might of a whole nation?’ Lakshman said.
‘What might? Do you know how much money rolls in the currency markets every day?’ Joshua said. Then tu
rning to Divya, ‘Do you remember?’
‘I think it’s in trillions, sir,’ Divya said.
‘Trillions every day?’ Lakshman asked, incredulous.
‘Yes. What your government trades will be in the order of a few hundred million at best, not quite enough to make a big impact, especially when things are not going their way.’
‘True, but unlike other currencies, the Indian rupee is not fully convertible,’ Lakshman said. It was time he showed off a little of his economic knowledge as well. ‘The Indian government still monitors and controls the inflow and outflow of rupees into the market. They won’t let the supply increase so much that it will pull the value down drastically.’
‘I agree. That’s something I don’t understand as well. Something just doesn’t add up,’ Joshua said.
‘Is there something we can do?’ Lakshman asked.
‘Why don’t we talk to the right people, tell them what we know and let them take it from there? That’s the only thing we can do.’
‘Right people, like who?’
‘I’ll talk to that cop Carla and you being the patriotic Indian talk to the head of your Reserve Bank,’ Joshua said.
‘Why don’t you talk to both of them yourself?’ Lakshman said. ‘Call it reverse racism, but the Reserve Bank Governor is more likely to lend his ear to an unknown American than an obscure Indian even if it’s his seat that’s on line.’
Joshua thought about it for a moment.
‘I know what you mean,’ he said. ‘Just help me get connected if you don’t mind and I’ll talk to him.’
61
There was no need to embroil Divya anymore and they sent her away with many words of appreciation.
Divya knew if any action was going to be taken, it was going to be all over the media and she would find out sooner or later. Her goal had been to impress Joshua; she had accomplished that and wanted to celebrate. She went out in search of Venus in his department, mulling over the other great puzzle that remained unsolved: What kind of surprise to give him?
