Book #0 of the Infinity Cycle
“It is really nice to know that an alumnus of IIT is penning a book. It makes me feel joyous that the book is on mathematics and that, too, in a novel form to make it more interesting. . . . It brings out the fascinating power of maths and its history and the unique role played by religion and culture in the most entertaining and enjoyable fashion.”
—–Anand Kumar, Founder, Super 30
“A beautifully written, stylish novel . . . a gripping mystery”
“Expertly weaves the multiple threads of its plot . . . into a captivating whole.”
“A gripping novel . . . a pacey thriller. . . . An enjoyable read from the very first chapter, the kind of book you cannot stop reading till the end. An impressive debut!”
—–Guillermo Martínez, Multi-award winning author of international bestsellers The Oxford Murders and The Book of Murder
“An intelligently written thriller”
—–CNN-IBN IBN Live
“Science, religion and crime – the perfect ingredients for a mystery thriller. . . . (the author) must have grilled his grey cells to invent such a page-turner.”
—–The New Indian Express
The second bullet keeled Jeffrey over and he dropped the duffel bag and car key and tumbled down the flight of steps like a sack of coals.
There was a rustle among the leafless shrubs lining the walkway to the parking lot, soon followed by the squelching of shoes in slush. His breath fogging in quick spurts in a fit of panic, he gripped the wrought-iron banister and tried to rise to his feet, but his shoes kept slipping in the snow. He continued to try nevertheless, like a hamster on a wheel, but stopped when he found himself face-to-face with a hulk in a hoodie.
The hulk ran his eyes over Jeffrey for a moment, then turned away and went nosing about the gym’s entrance. He picked up the car key lying on the steps and flung it up and away into the darkness with all his might. That done, he sprang down the steps in two hops, grabbed Jeffrey by his lapels, and pulled him up a little. Then thrust his pockmarked face menacingly close to Jeffrey and shook him.
‘Gimme your cell phone,’ he barked, spittle and snow spraying on Jeffrey’s face, ‘and your wallet.’
Jeffrey made no reply. He went limp in the thug’s hands and then started twitching a little.
The thug patted down Jeffrey’s pockets and fished out his wallet and cell phone. Shoving him away and letting him slump to the ground, he tucked the loot into his pocket and tore away towards a grey Corolla that lay in wait across the street, rumbling, ready to go.
By the time a helpless Jeffrey heard the car zipping away, he was bleeding profusely from the two puncture wounds. His head throbbed feverishly and his body winced with unbearable pain. Moments back, he was standing in front of the gym door after an invigorating workout, feeling the chill air whistling into his windpipe and diffusing in his lungs, gazing at the silent snowfall with an equally quiet admiration, marvelling at the fractal patterns that ruled all the way from the tiny snowflakes dancing in front of him to the humongous old temples he’d seen in India. And now, he was gasping for breath, a fire raging in his gut, sweat beading his brow.
It didn’t take Jeffrey long to come to the realization that he only had minutes to live. He had heard that gastric acids were safe only as long as they were confined to their sacs where they could be continuously neutralized; once they oozed out, they’d burn away the tissues to death in a matter of minutes. Whatever he had to do, he had to do within that window, with whatever strength was still left in him.
He could think of only one thing . . . pass on information to nail his killer, the real killer. At least that way, he would have his revenge, even if posthumously.
Now he could see why his cell phone had been taken along with the wallet. There was a phone inside the gym, but with his wallet gone and his ID along with it, there was no way he could open the door and get in. But he quickly remembered that there was an emergency callbox in the gym’s parking lot. It was perhaps forty yards away and there was no guarantee that he would even make it that far, let alone speak articulately. But with a definite death staring down in his face and the slush making it impossible to leave a message in blood, that seemed like the only course left to pursue.
Jeffrey was no mere fitness freak but a muscle maniac who tended to every fibre in his body with the utmost precision and care – nothing else would explain his lonely visit to the gym so early on a winter’s day. All that effort came to the fore now in his final minutes. Mustering every ounce of energy left in his fast withering body, he dragged himself towards the parking lot, tumbling and scrambling every inch of the way, the dripping blood blending with snow and congealing in crimson patches in his wake.
He collapsed at the foot of the phone post, after barely managing to slur a few incoherent words into the callbox. A strange peace swept over him as he passed out: even if the name didn’t get through properly the magic number he’d uttered would definitely drive home the message.
For someone used to going gadding about town in a limousine – when not driving his own Bentley, that is – the Expanzo was not the best mode of transport Joshua had seen. However, that was all he had at his disposal right now, replete with its soda-can shell and pet-crate leg-room.
But he wasn’t going to complain. On the contrary, he was pleased that Indian motor travel had finally graduated past those curvy creamy caravans from prehistory, running on adulterated petrol, belching out billows of jet-black smoke that threatened to asphyxiate one to death: Ambassadors. The new millennium had just dawned and India was slowly wriggling out of the clutches of the government and hobbling into the arms of the market. Joshua, an optimist who looked at the cup as half full rather than half empty, preferred to view the Expanzo, in spite of its toxic tailpipe and budget airline leg-room, as a positive indication of that economic evolution.
He could not extend a similar courtesy to the surging traffic on the road, embrace it as a sign of progress and prosperity. When his boxy little vehicle ended its misery and came to a juddering halt at Gemini Circle, he anxiously peered out through the windshield and asked his chauffer, ‘Traffic jam at this time of day?’
‘Must be some police holdup, sir,’ Durai Raj said. ‘Probably some bloody VIP passing. But no need to worry, sir. It should clear in a couple of minutes. Traffic is not usually bad around this time.’
Joshua was no stranger to India but he hadn’t come across a chauffeur like Durai Raj. Durai spoke English with a rare degree of fluency for an Indian car driver, give or take a staccato phrase or splintered sentence. It had gone a long way towards establishing a good rapport between them. Durai’s facility with the language and his polished and neutral accent had intrigued Joshua from the start. But only now, with barely hours to go before he took off on a flight to Boston via London, could Joshua break out of his inhibition and ask him something that had been on his mind ever since he met him.
‘Do you mind if I ask you something, Durai?’ Joshua said.
‘Why don’t you go work for a call centre? You speak so well. And you are so polite and helpful – and have experience interacting with foreigners . . .’
‘No sir!’ Durai said, the shock showing on his face.
‘Why, what’s the problem?’
‘Not one, sir. Many.’
‘One, I don’t have a college degree, sir. Only an auto mechanic diploma–’
‘You need a college degree to work as a customer service rep?’
‘Yes sir. There are so many unemployed graduates here. When they’re willing to do these jobs, why will they choose a worthless diploma like me? Also, I am over thirty-five, sir. Too old for the job. They take only young people in the twenty-plus age group. Anyway, even if they give me a job, I won’t take it up, sir. That’s a different matter. I am happy to be a driver, sir.’
‘That’s what I was asking. Why wouldn’t you work at a call centre?’
‘Call centres are like Chinese factories, sir,’ Durai said. ‘My cousin worked at a call centre for one year and is suffering already. He has to take calls one after another without a gap, sir. Says he can never leave the chair without the team leader’s permission. Has to get permission even to go . . . you-know-where, sir.’ He sheepishly lifted up his little finger and made that universal gesture.
‘You mean . . . use the toilet?’ Joshua asked him, just to make sure.
‘Yes sir. If calls are coming the team leader won’t even give permission to . . . to use the toilet, sir. Phone call always gets priority over nature’s call. That’s what happened to my cousin, sir. By controlling it for a long time he got kidney stone in one year. Very painful, sir.’
‘That’s a shame,’ Joshua said.
The logjam started clearing and Joshua let Durai manoeuvre the vehicle out of the Gemini interchange onto Mount Road and set it on a more or less steady course to the airport.
Joshua spent the rest of the drive thinking about the immediate tasks that lay in wait for him in Boston. Like most people at his level he had left himself completely at the mercy of his assistant Nancy these six weeks, reading and responding only to those few mails she had flagged for his immediate attention. Though Nancy had been holding the fort well, managing his office in his absence, he was going to have his hands full upon return.
As soon as the car pulled up by the kerb at the airport, Joshua fished out whatever Indian rupees remained in his wallet. Durai’s face lit up as he caught sight of the crispy bills in the rear-view mirror. This was the moment he had been waiting for. The cab fare had already been paid upfront at the hotel. Whatever Joshua gave him off the record would be his exclusive tip, a well-deserved award for a service with a smile.
With each passenger bringing along a small tribe of relatives to see them off, the departure area was teeming with hundreds of people who spilled out onto the road and choked the traffic, slowing it down to a crawl. Currency touts were doing their rounds, weaving in and out through the crowd fanning a bundle of five-hundred rupee bills – some minted in Karachi, some in Sivakasi. Durai, inspired by the impending tip, sprang out of the car with a fresh burst of energy, keeping one furtive eye on the hawala touts in case Joshua dished out any US dollars. The shiny exterior of the Expanzo drew the attention of two nosy onlookers standing nearby. They edged closer to Durai and tried to take a peek inside the vehicle, but Durai shooed them away and held the door open reverentially for Joshua.
Joshua stepped off the car and slipped on his jacket, feeling the breast pocket with some apprehension to make sure his passport and ticket were still there. He slung his laptop onto his shoulder. It housed the only copy of all the notes he’d made in India and he was not going to take any chances with it.
‘Please wait here, sir. I’ll go get a trolley,’ Durai said and dissolved into the crowd. He materialized in no time with a trolley in tow and started transferring the luggage from the boot.
Joshua knew he’d brought only three pieces of luggage with him, laptop included, but that didn’t stop him from taking one final peek into the boot to make sure nothing was left behind. He managed to get only a fleeting glance before an oblivious Durai banged the boot door down, but that was enough for Joshua to catch a glimpse of what lay inside: a tattered dog-eared placard that chauffeurs held aloft to receive visitors at the airport. It carried some title scrawled out by an unsteady hand in blue capital letters with large, uneven gaps between them:
M R W I L L A I M J
‘Thank you very much, sir. Have a nice flight,’ Durai locked the boot and said in eager anticipation of his tip.
Before Joshua could respond, a wail of sirens Dopplering in rent the air, plunging the crowd into a hush for a moment. Within seconds, a large fleet of police cars and jeeps, as if in a ministerial motorcade, sailed into view. Horns blaring, lights flashing, they homed in on the departure area and screeched to a halt by the kerb, screaming onlookers scattering like coins on a carrom board to get out of the way. Joshua froze in his spot, stunned by the abruptness of it all.
Police officers quickly popped out of the vehicles and hurried towards the main entrance, cleaving through the crowd gathered in front, craning their necks, scanning people’s faces at random.
One of the officers who had a thick but neatly trimmed moustache lurched forward and came sprinting towards Joshua, whose towering build and alabaster complexion made him stick out conspicuously from the rest of the pack. ‘Are you Dr Joeshua Yeskeel from America?’
Joshua was so taken aback he didn’t even point out that the officer was mangling his name. ‘Yeah?’ he said.
‘Can I see your passport?’ The officer stretched out a brawny hand.
Joshua fished out his passport from his breast pocket and handed it over. The officer flipped through the pages and tallied the photograph with Joshua’s face. Satisfied, he closed the booklet and tucked it into his shirt pocket. Then he gave a thumbs-up sign to another cop standing near a car. Joshua could hear the officers whistling or yelling to one another, relaying thumbs-up signs and calling off their search.
‘You need to come with us,’ the officer said to Joshua.
Before Joshua could open his mouth, four other policemen trooped in and circled him. It didn’t feel like he had much choice in the matter; the officer’s tone had a certain finality to it and the ring of four cops removed whatever doubt remained.
Joshua tucked the cash in his hand back into the wallet and followed the officer. As he wheeled around, he could see another policeman pointing at his luggage and saying something with a stern face to Durai Raj who was leaning forward with his arms folded and nodding deferentially – a typical Indian shake of the head, more of a roll than a pitch or yaw, whose meaning Joshua always had a hard time decoding.
One hand giveth but the other taketh away. Lakshman had staggered into his Bonn Avenue home in buoyant spirits after the binge with Joshua, looking forward to the night ahead. But he wound up pacing the reading room with frown on his face, fury in his stride and intense outrage in the pit of his stomach.
He’d made quite an effort to fall asleep, trying every possible posture, every mental exercise known to him. But he was unable to turn off the thoughts that kept bouncing around in his head and he lay tossing and turning in bed, waking up Urmila in the process.
Urmila never took kindly to Lakshman’s occasional indulgence in alcohol. She had chosen to look the other way the previous day as he was meeting his old friend after a long while. But she did not mince her words or pull her punches tonight when his inebriation also brought home a contagious insomnia as accompaniment. It was the holy month of Margali and she had to wake up early in the morning, a few hours before sunrise, and light an oil lamp in front of the house. There was no way she was going to be able to do that if Lakshman kept disrupting her sleep, treating the bed as some kind of trampoline in a theme park. She was left with no choice but to banish him from the bedroom for the rest of the night.
The reading room on the upper level had a spare bed and Lakshman retreated there, tail tucked between legs, hoping that the change of place would do him some good. But no such luck; sleep was a no-show even upstairs. So he decided to walk up and down the room and tire himself out, involuntarily mulling over the new assignment that had suddenly landed on his plate out of nowhere.
As it often happened, it had started with a seemingly innocuous phone call. He had just returned home from the bar and settled down in front of the late night news analysis on the market downturn when the phone rang.
‘I hope I’m not disturbing you, calling so late, Professor Lakshmana Raman,’ the familiar voice said, as always preferring the formal even in private.
‘Not at all, sir,’ Lakshman said and switched off the TV. ‘I’d gone out and just returned home.’
‘Oh, good,’ the director said. ‘I know I could catch you tomorrow, but thought we could talk in peace now without getting disturbed by others.’
Please, please, don’t ask me to come down to your bungalow, Lakshman prayed. I’m too sloshed to present myself to anybody, least of all you. ‘What’s the matter, sir?’
‘Two things. First, I wanted to congratulate you on the success of the conference in November. I didn’t get the chance to tell you earlier, but I have been getting only good feedback about it. Looks like everyone found it quite worthwhile. Congratulations.’
Lakshman had proven his organizational mettle barely weeks ago by successfully conducting an international conference at the Institute. Though it was planned at short notice, it was a roaring success, attracting scholars from all parts of the world and eliciting excellent reviews from them. Lakshman had picked up the cue from the director and masterminded the entire event from start to finish. So he could jolly well take the credit he deserved for his work and he did that with a safe touch of humility. ‘Thank you, sir,’ he said. ‘It wouldn’t have happened without your unflinching guidance and institutional backing.’
‘Well, well, why do you have to be so modest always?’ the director said.
The canny political animal that he was, Lakshman knew that the director was only flattering to deceive. The Supreme Being on the Fifth Floor was one of those people who would pat a donkey’s back only as a prelude to weighing it down with a massive payload of dirty laundry and administering a painful prod in the ribs with a whip-handle. So Lakshman pricked up his ears and listened.
The director continued: ‘It’s this rare combination of ability and modesty that brings me to you time and again.’
Lakshman held his beer breath. Oh my God!
‘The Institute is in need of your services once again, Professor Lakshman. We need to organize another important event.’
‘Another conference, sir?’
‘No, not a conference, but a ceremony to award an honorary doctorate.’
‘Can’t we club it with the next convocation in August, sir? It’ll save us so much trouble.’
‘I knew you would suggest that, but no, we can’t afford to wait till then. We need to get it done by January.’
‘So soon, sir?’
‘Yeah. I know the time is short, that’s why I’m calling you so late in the day. If there is anybody capable of making it happen, it is you.’
‘Anyone coming from Germany, sir?’
‘No, not from Germany. From Bombay. Mumbai, I should say.’
‘Bombay?’ Lakshman asked, somewhat puzzled.
‘Who’s coming from Bombay, sir?’
The director sighed. So heavily that it went whooshing into Lakshman’s ears like a cyclone and made him shudder.
‘One Mr Pomonia, Mr Chiman Pomonia.’
‘The . . . the . . . the . . . industrialist?’ Lakshman stammered.
‘If you wish to call him that, yes, the same Mr Pomonia. Our brief is to promote him to Dr Pomonia.’
‘WHAT!’ Lakshman gasped. He did not normally lose his composure with the director, but he was too rattled to observe the usual rules of engagement.
‘Do you mind coming over to my bungalow for a few minutes? I’ll be waiting for you in the garden. It’s better to discuss certain things in person rather than over the phone. Besides, it’ll also give me the opportunity to introduce you to the masala milk our cook makes; it’s the best digestive I have come across in thirty years.’
Lakshman could not believe that the Institute had chosen to bestow an honorary doctorate upon Pomonia of all people. Conferring honorary doctorates was all part of the rough and tumble of academic life at the Institute and Lakshman hardly expected to have any say in who got chosen for the award, but it was always someone he’d found acceptable or at least tolerable. Dignitaries from Germany could usually look forward to red-carpet ceremonies and honorary doctorates at the Institute by the sheer virtue of that nation’s contributions to it, as could be evidenced from street names like Berlin Avenue and Bonn Avenue on the campus. According a similar treatment to Chiman Pomonia rankled Lakshman no end.
Pomonia was, of course, a big fish, a billionaire, many times over. All self-made pelf, every single paisa of it. From a little boy in a village who was forced by his father’s sudden demise to drop out of high school and take up milk vending on a rented bicycle, he grew into a bigwig who sat at the head of a conglomerate that oversaw hundreds of companies around the world. His economic empire cut a broad swath across a multitude of industries, including dairy, textiles, IT, finance, retail, energy, media and entertainment, and contributed – or controlled, take your pick – the single biggest slice of the Indian GDP pie. Suffice to say, if Pomonia heaved a sigh, all market indices on Dalal Street would tremble in tune, like aspen in high wind.
Smitten by Pomonia’s phenomenal success, an American university had recently decided to invite him as their commencement speaker and award him a medallion of honour during a ceremony the following summer. The domestic media, always on alert to celebrate or shame anyone and everyone with the slightest soupçon of melanin in their skin, lost no time blowing the thing out of proportion. The deafening buzz threw all Pomonia loyalists in the government into a tizzy. There was of course the little matter of ten million dollars Pomonia had pledged to the university for the establishment of a Chair in his name, but somehow it never managed to get much attention. There was only one question echoing everywhere: How could Indian universities sit tight in the sidelines when a foreign university was paying such rich tributes to an enterprising son of Mother India?
Pomonia’s lapdogs in the government got into action before long and decided to launch pre-emptive strike on America. Their strategy: Get a premier Indian institution to deck up Pomonia in royal regalia and dole out an honorary degree at a glittering function before he flew off to graciously accept his medallion in the US. The minister for human resources, a true son of the Madras soil who lorded over institutions of higher learning in the country, passed an unwritten decree that his hometown would be the one to do the honours to Pomonia and it would do so by the end of January. By the time Lakshman was done sipping his masala milk in the director’s garden, the project had become his primary responsibility.
Lakshman couldn’t digest the fact that the Institute was stooping so low, choosing a businessman who revelled in operating on the margins of the legal and the illegal, moral and the immoral. Being made the ringmaster of the whole charade rubbed it in even further, bringing him down from the high with a thud and leaving him not just sleepless but in a state of ferment. Exiled from the bedroom by Urmila, his T-shirt and veshti drenched in sweat, he continued to pace the reading room furiously, cursing himself: Why did I even get promoted?
The masala milk had joined the beer now and they swirled around each other like yin and yang, the ensuing vortex sending the tikkas on a spin. Individually, each of them was an epitome of gastronomic delight, but together, they were a recipe for gastrological disaster. All that frenetic walking up and down only made matters worse.
Lakshman stopped pacing and took a deep breath, but the churning in his belly only got wilder like a simmering volcano. He could feel the hot bile bubbling up to his throat. Figuring it was only a matter of time before it erupted, he decided to pre-empt the process. He ducked into the toilet and started retching. As quietly as he possibly could. Urmila had fallen sleep with much difficulty and he couldn’t afford to wake her up with his theatrics.
He was just about done washing up when the phone rang, breaking the stillness of the night. Thinking of Urmila yet again, he hauled up his veshti and dashed out of the bathroom wondering who on earth was calling at this ungodly hour.
Joshua had been in good form earlier that evening. But he hadn’t reckoned that he was going to meet his match soon, least of all in the shape and form of the petite girl seated inconspicuously in the back rows.
If the reigning movie star could be taken as a touchstone of progress in any place, Joshua had told Lakshman, then I can state confidently that Madras has made no progress at all. It was Rajnikanth whose manifold Technicolor avatars were painted over the city walls when Joshua had visited back in the Eighties, and it was Rajnikanth who ruled the roost even now, pixellated and Photoshopped. So Joshua had no trouble tapping into popular sentiment to drive home the idea of his new shortest path algorithm.
‘Suppose point A is this auditorium right here and point B is a movie theatre in the city where a Rajnikanth movie is playing, how do you get from here to there as quickly as possible? That’s the Shortest Path Problem in a nutshell,’ he said, his eyes sparkling with mischief as he surveyed the audience for their reaction.
The audience hadn’t expected an American like Joshua to be so tuned in to the zeitgeist in this part of the world, and they roared in approval, much to his delight. Lakshman too broke into a handsome smile in his seat, suddenly realizing why Joshua had earlier made that comment out of the blue. But then that was vintage Joshua. He knew when to kiss, when to bow and when to shake hands; he knew what it took to elevate an academic presentation into a performance in any corner of the world.
‘The Shortest Path Problem has been researchers’ favourite for several decades now, ever since Dijkstra published his algorithm in 1959. But I must point out that it was not Dijkstra who first invented the shortest path algorithm. While it is true that he came up with it independently, it was the guys at RAND Corporation who had it first. But they kept their stuff tightly under wraps, whereas Dijkstra published his work in Europe a few years later and set the ball rolling for all of us. Dozens of shortest path algorithms have been developed by researchers since then. These algorithms are all fantastic in their own way, but – there is always a but – none of them runs as fast as this new algorithm here. This baby uses what we call an asymmetric radix bucket data structure which speeds up the computations to unprecedented levels. Let me show you how it works and you can see for yourself where it gets its mojo from,’ Joshua said and moved towards his laptop.
Ably supported by thirty slides, most of them densely packed with Greek symbols and other mathematical exotica, he illustrated the intricacies of his new algorithm. Most of the students assembled in front of him looked as if they were at the screening of a movie in an alien language without the benefit of subtitles. Joshua tried hard to make the technicalities as accessible as possible, but he began losing his audience at a drastic rate, starting at slide three. Lakshman dropped off at slide fifteen and by slide twenty, there was only one person in the entire auditorium who was still with Joshua, furiously scribbling down notes on the scrap papers spread on the desk. Joshua was on a tight schedule – he had just enough time to have dinner with Lakshman, check out of the Oceanic and make his way to the airport, and he’d planned his presentation for under an hour. But he slowed down a little when he saw the clueless faces in front of him and took an extra ten minutes to run through the slides.
A palpable wave of relief swept through the audience when Joshua was done. Lakshman prompted the crowd for a round of applause and threw the floor open for questions. Seeing Joshua sneaking a peek at his double-dial Rolex and getting a little restless, he added, ‘Professor Ezekiel’s running out of time, so we have to keep it brief.’
‘Sir,’ a slender hand decorated with a red wristband shot up in the air from the second row.
‘Yes, young man,’ Joshua said.
A stubbly chap in his mid-twenties clad in an electric yellow T-shirt rose from his seat twirling his wristband almost as if readying for a fight. A wireless mike passed from hand to hand and made its way to him. He grabbed it and said: ‘Sir, your algorithm looks very good, in fact, fantastic in its own way, but – there is always a but – like you said in the beginning, people have been working on the Shortest Path Problem for so many decades now and dozens of algorithms are already available, so what is the real necessity to develop yet another algorithm now, sir? What is the point? Why keep re-inventing the wheel? Why can’t people use something already available and just move on with life?’
Joshua felt his face flush with shock and rage. Such rudeness! That too in India! ‘May I know your name, young man?’ he said, summoning as much politeness as he could.
‘Pathivendaparuppu Adianthaprabhu Bommuluri Bhagavatharao Vivekandanda, sir. But you can call me Vivek.’
‘What do you do, Vivek?’
‘Doing my PhD, sir.’
‘In . . .?’
‘Economics, sir,’ the fellow said proudly and sat back down.
‘No wonder,’ Joshua said, trying hard not to snarl. He turned towards Lakshman and gave him a fleeting but piercing glare. You let these people in for my talk?
Lakshman threw his hands up in the air and shook his head. I didn’t know.
‘Well, some of you, like our economist friend Vivek,’ Joshua said sharply, making it sound like an insult, ‘may feel tempted into thinking that there is no justification for new research on the Shortest Path Problem, but as I have tried to show you, the new algorithm runs much faster than all other algorithms available in the market. I’m not really re-inventing a wheel here; I am, in fact, all for dispensing with the wheel, because what this algorithm has got is wings to fly. When you have a big sprawling urban area like Madras with thousands of streets and intersections, hundreds of origins and hundreds of destinations, and tens of thousands of origin-destination pairs, finding all the shortest paths is not easy. There’re way too many possibilities to consider – even the best algorithms out there could sometimes take minutes to run on state-of-the-art workstations. But this radix bucket data structure makes it easy to solve thousands and thousands of such Shortest Path Problems almost instantly. If you want the quickest route to the movie theatre, it’ll be there in front of you before you’re even done clicking the mouse.’
‘I think sites like MapQuest do pretty much the same thing, sir,’ Vivek sprang up from his seat and shot back into the mike. ‘Also, if we want to go from here to the movie theatre, we don’t really need to save time. It’s all right to take a longer but more convenient route and reach a few minutes late. We don’t really need a highly complex algorithm like this one. Other than trivial situations like this, is there any real use for your algorithm?’
Vivek’s abrasiveness set off a flutter in the audience.
Joshua had spent several months hammering the algorithm into shape and there was no way he could take it lying down. He stared at the student inscrutably for a few seconds and then said, ‘Well Mr Economist, let me try a different tack and see if it washes with you. Let’s take the same network as before,’ he clicked back to the only slide with a picture, ‘but with a little twist. The nodes you see here are not traffic intersections but currencies of nations. The links connecting them are not lengths of roads but exchange-rates between currencies. Say, node A is your Indian rupee, node B is the Malaysian ringgit, node C is the US dollar and so on. If you want to go from A to B, that is, convert rupees into ringgit, there are many ways you could do it. You could buy ringgit directly using the exchange rate, or you could first buy US dollars and then re-convert the dollars into ringgit, or you could go through a long chain of conversions to other currencies before winding up with the ringgit. The shortest path algorithm can help you figure out the best way to do it, the one with the least exchange cost, the one that gives the most bang for the buck. . . . Is this a trivial situation? Yeah sure, if you’re trying to change the cash in your wallet. Do you need a complex algorithm that helps you save a few pennies? I don’t think so. But all that changes if you scale things up a bit.’
‘To what level, sir?’ Vivek asked.
‘You tell me, you’re the economist,’ Joshua trained his gimlet eyes on Vivek. ‘What’s the turnover like in forex markets?’
‘Oh, it’s in trillions of dollars, sir.’
‘No sir. Every day.’
‘Every day? Are you sure?’ Joshua asked, incredulity writ large on his face.
‘Yes sir,’ Vivek said with a confident smirk. ‘People buy and sell billions in various currencies in the blink of an eye; the trades add up to trillions by the end of the day.’
‘People like who?’
‘Banks, hedge funds, national treasuries . . .’
‘If you walk up to them with a new currency trading algorithm that can run very fast, say, in the blink of an eye, are they going to listen to you or turn you away?’
‘They’ll listen to me, sir,’ Vivek said confidently.
‘What makes you say that?’
‘Exchange rates keep fluctuating, sir. They could go up or down in a second, so decisions to buy and sell forex have to be made very very fast.’
‘Yes sir. Even milliseconds and decimal points matter in the world of economics and finance; that’s why finance companies lay their own special optical fibre cables between trading centres, they don’t even want a millisecond delay in transactions if they can help it. If I have a fast algorithm, it could be very useful in high-frequency trading. Even if banks save only a small fraction of their exchange costs, it would add up to millions of dollars given the sheer volume of transactions.’
‘Thank you, sir,’ Joshua said to Vivek with exaggerated politeness, even bowing a little, one hand on his tummy. Then turning to the audience, ‘See, I keep my friends close but enemies closer. This smart young man here is now my new best friend.’
The audience broke into guffaws. Vivek looked around, puzzled for a moment before realizing how Joshua had played him. He passed the mike over and sank back into his seat swearing never to open his mouth again.
‘It’s not just road traffic or economics; shortest path algorithms come in handy in genomics as well. It’s useful in Sequence Alignment, for comparing long strings of DNA, RNA or proteins, like it’s done with evidence collected from crime scenes. Anyone specializing in this area here with us today?’ Joshua said and paused to survey the audience.
Seeing no hands go up, he did the honours himself, spending a few minutes explaining the idea of Sequence Alignment with an example on the whiteboard and driving home the importance of having an efficient algorithm like his in the arsenal.
A few more questions mostly of the mundane kind followed next and he cleared them one by one with his customary skill and ease. But his comrade-of-old who had invited him to give this talk was getting tetchy in his seat, appalled by the sort of questions being raised by the students. If that fellow Vivek was abrasive, the others were inane and inconsequential. Lakshman edged forward in his chair, twisted around a little and started craning his neck: Was there no one who could show a little bit of class in asking questions? A little bit of spark?
Where was Biju John? He was supposed to be doing his PhD in this area but was he even attending this talk? What about his most favourite student? Did she manage to come?
Row by row, Lakshman carefully panned the audience and before long, she came into view.
Clad in a cream salwar-kameez, an olive-green dupatta draped across her body like a shawl, she was in the back rows, busy scribbling something, paying no attention to the Q&A whatsoever.
Lakshman moved past her in search of other students from his stable. When he grew tired of rubbernecking, he slumped back in his seat and resumed his focus on Joshua. Soon a mousy squeak made him straighten up like a ramrod: ‘Sir?’
‘Yes, young lady,’ Joshua said and the mike made its way to her.
She grabbed it and said: ‘May I use the board, sir? It’ll be easier for me that way.’
‘Well, that’s a first,’ Joshua said. ‘Sure, come on down.’
Lakshman turned around with a sudden rush of energy and chimed in, motioning her towards the podium. ‘Come down, come down.’
She put the mike down, stole one quick glance at the crumpled sheets of paper on her desk and briskly marched down the steps, her olive-green dupatta fluttering in the wake.
There were two whiteboards in the auditorium, one on each flank of the screen. Joshua had used the right one for his genomics example. The one on the left remained untouched, and she picked up a marker from the desk and homed in on it. She sketched out a modified version of the radix bucket data structure on one frame of the whiteboard. On the next frame, she scrawled out a series of mathematical expressions with so many exotic symbols that they held Joshua in a hypnotic thrall.
‘Sir, it seems to me, if you change your radix bucket data structure like this,’ she pointed at the first frame, ‘and apply a non-Robinsonian push-pop slash index operation like this,’ she pointed at the second frame, ‘the algorithm could be speeded up further. . . . Is that correct, or am I going wrong somewhere?’
Joshua stood frozen, staring at the whiteboard like it were a piece of sculpture.
Some of the students, getting a tad impatient, not in the least because they couldn’t make head or tail of what the crazy girl had scribbled on the board, started slinking out of the hall through the rear exits.
Lakshman noticed the silent exodus and looked at the clock. Dinnertime was approaching and if he waited any longer, only a sprinkling of the audience, if any at all, would be left to applaud when he delivered the vote of thanks and handed over a memento to Joshua. Not wanting to be caught skimping on formalities, he decided to step in and expedite the proceedings.
‘Why don’t we take this offline? It’s getting late,’ he said and walked down to the pit.
Frowning, the girl returned to her seat.
Lakshman made a little speech thanking Joshua for taking the trouble to come and deliver the talk on campus, mouthing the usual platitudes about how students would have found his talk interesting and useful and stimulating and things to that effect. He handed over the memento and the audience dispersed after obliging with a final round of applause.
The rituals over with, Joshua turned to the whiteboard and resumed his owlish stare. The girl hesitantly made her way back to the podium, having consulted her scrap-work once again. Two minutes of pin-drop silence followed, with Joshua goggling at the whiteboard, Lakshman at Joshua and she at both of them, brimming with anticipation.
Neither Lakshman nor Joshua showed any sign of snapping out of their silence. Unable to bear it anymore, she stepped in and broke the silence herself. ‘Sir, I think we can try comparing the number of steps before and after the change. I can try counting them if you want,’ she said and picked up the marker.
Joshua finally stirred. ‘No need for that, young lady. You’re absolutely right. I wasn’t sceptical about what you said. I was only wondering why I hadn’t thought of it myself . . . God,’ he said with a sigh of resignation, ‘I must be getting old. What’s your name again?’
‘Pleasure to meet you,’ Joshua said and shook hands with her. ‘What do you study? Not economics, I’m sure.’
‘No sir, I’m in computer science.’
‘Why doesn’t that surprise me,’ said Joshua. ‘Which year are you in?’
‘Second year, sir.’
‘For some strange reason, we call that the sophomore year in the US.’
‘Yes sir, I’ve heard that.’
‘Don’t tell me you work for this fellow here,’ Joshua pointed an accusing finger at Lakshman.
‘Sorry Josh, she’s guilty as charged,’ Lakshman said.
‘In that case,’ Joshua said and turned to Divya, ‘you wouldn’t mind doing me a favour?’
‘Yes sir?’ Divya said.
‘Could you type all this up in LaTeX,’ Joshua said, circling a finger over her handiwork on the board, ‘and email it to me? I’m in the process of submitting the paper after the second round of revisions. I’ll now rejig the whole thing with your non-Robinsonian push-pop operation before sending it. You will of course be my co-author if that’s all right with you.’
Divya looked at him, stunned. Co-author of a paper? With Joshua? Joshua Ezekiel? The Joshua Ezekiel? The Alfred P. Sloan Chair Professor at MIT? The world-renowned authority on algorithms? She couldn’t believe it. ‘This, in a paper, sir?’ she said, pointing at the whiteboard.
‘Yeah, this,’ said Joshua with a smile. ‘Why do you look so surprised? You know what you’ve done? This fellow won’t let me tell you, but I’m going to anyway. You’ve pulled off a major coup under our very noses, that’s what you’ve done.’
Divya, beginning to feel awkward, blushed.
Lakshman beamed at her, proud beyond words. ‘She won a gold medal at the math Olympiad, Josh,’ he said. ‘So you shouldn’t be too surprised.’
‘I’ll try,’ Joshua said and turned to Divya. ‘I’ve got to go now, but we’ll be in touch on email. Here’s my card, and I’m sure I can reach you through this guy here.’
‘Yes sir,’ Divya nodded, her face all a-flush. Oh my God! I’m writing a paper with Joshua Ezekiel! ‘I also have another question, sir, if you don’t mind.’
‘Sure, if you don’t mind me packing up my things as we speak,’ Joshua began unplugging the wires from his laptop.
‘You said the RAND Corporation had the shortest path algorithm before Dijkstra. Why didn’t they publish it first, sir? Why did they keep it secret?’
‘Know anything about RAND?’
‘No sir, never heard of them.’
‘Look them up. They’re a military research organization, though these days they’re also into policy analysis and stuff to make up for all the bad press they got during the Cold War. If I have to guess, the army was using the algorithm for logistics, for planning the movements of troops, weapons and supplies; it was classified information. Today it’s a silly algorithm you find in every textbook, but fifty years ago it was a powerful secret that determined the course of a war.’
‘Why do you have to guess, Josh?’ Lakshman butted in. ‘If I remember right, you went to work for RAND right after your PhD.’
‘Shh,’ Joshua said, bringing a finger to his lips. ‘Don’t go raking up my past. Every saint has a past, every sinner has a future.’
Divya laughed. ‘I also have another question, sir. Not really a question but a doubt in my mind.’
‘As long as it has nothing to do with RAND or what I did for them,’ Joshua said, stuffing the laptop into a bag and strapping it in place.
‘No sir. It’s something do with your algorithm,’ Divya said. ‘You talked about how it can find the shortest paths between many origin-destination pairs. Can we also use it in a situation where we don’t want to just go from A to B but want to start at A, visit some other nodes, B, C, D, E and so on, and then finally come back to the starting point A as quickly as possible?’
‘Aha,’ said Joshua with a twinkle in his eyes.
Lakshman smiled as well.
‘Well, I can assure you that this algorithm will never be able to handle the problem you describe. For that matter, no algorithm on earth can,’ Joshua said. ‘Because you’re no longer dealing with the warm and fuzzy domain of the Shortest Path Problem, you are now in the whole squalid world of the Travelling Salesman Problem. Heard of it?’
‘No sir,’ Divya said apologetically.
‘That’s surprising,’ Joshua said.
‘Come on, cut her some slack, Josh. She’s only in her third semester,’ Lakshman said.
‘I was kidding,’ Joshua said. ‘The complexity of the Travelling Salesman Problem goes up steeply with the size of the problem, sort of like the old Indian grains-on-the-chessboard story; no algorithm can rein that in. If you have ten nodes to visit you have ten factorial, or millions of possible routings to consider; if you have fifteen nodes, they run into the billions; by twenty you have trillions. Even the world’s most powerful computers can’t cope with that kind of exponential growth in complexity, so it makes no sense for you to waste your time on it. Just send me what you’ve got here,’ he pointed at the board, ‘and I’ll take it from there.’
‘When you email me, do me a favour and cc it to my personal ID as well – it’s on the back of the card I gave you. I have tons of emails sitting in my MIT inbox I still need to plough through; I don’t want yours to get lost in the deluge.’
‘Okay sir,’ Divya said. ‘I’ll send it to both IDs.’
Divya took leave of Joshua and Lakshman and bounded up the steps, her face glowing like a beetroot.
‘Well, Lax,’ Joshua said once Divya was out of earshot, ‘my last day in India has turned out to be the best. I wasn’t too keen on this talk to start with, but you really made it worth it. I owe you one.’
‘Hey, I didn’t do anything,’ Lakshman said.
‘We can squabble about it over beer. The dinner plan’s still on, right?’
‘Any plan that involves beer is always on,’ Lakshman said. They wiped the boards clean and made their way out to execute it.
My last day in India has turned out to be the best.
Joshua was going to regret crowing like that before long. Wheels were already in motion to prove him wrong on both counts: the day was neither going to be his last in India nor his very best.
M.N. Krish is a predictive analytics expert and graduated from IIT Madras in 1995 with a B.Tech in Civil Engineering. The Steradian Trail can be ordered at leading bookstores online via www.mnkrish.com/buy.html.
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