Swami Buaji was supposed to be able to blow the conch for
hours on end continuously, having mastered the art of circular breathing, or
inhaling and exhaling at the same time.
I personally witnessed him mark the opening of the interfaith service at
St. Bartholomew’s Church in New York City, before the start of the U.N. General
Assembly in 2002, by blowing his conch nonstop for a few minutes.
It occurs to me that we may all be capable of circular
breathing, and may already be doing it in different ways in life. Let me elaborate.
On Guru Purnima day last July, I took a yoga class with the
incomparable Regina Grassia. She picked spanda, or pulsation, as the class theme. You can find spanda all around you in nature: in the day-night cycle, the seasons, the waves of the ocean, the phases of the moon and so on. In a yoga pose, you practice spanda by gathering your energy towards the core with the inhalation, and directing it outward into a stretch or a bend or a twist with the exhalation. Although spanda is something that is done in every practice, when we do it with particular attention and focus, it is a beautiful thing.
For Guru Purnima, I had decided to dedicate my practice to
the teacher who had the most far-reaching impact on my life: my thesis advisor
Prof. Sudhakar Reddy, who led, persuaded, cajoled and challenged me to become a
researcher.
We are doing a series
of standing poses. In the triangle pose, we are drawing our feet towards each
other isometrically with the inhalation, and expanding like a five-pointed
star outward with our exhalation and even bending backwards while
extending. My interactions with
Prof. Reddy start coming back to me.
My first serious encounter with Reddy occurred early in graduate school. While my
research interest was in computer networks, Reddy kept pushing me to prove the
fault tolerance of a class of graphs that he showed were denser than the best
known dense graphs in the literature.
That is, for a given number of neighboring nodes, and a given diameter (maximum
distance between any pair of nodes), you could pack more nodes
into the graph than other known graphs, especially as the diameter kept
growing. I was more interested in practical
problems like designing communication protocols, and kept resisting him for the
whole semester. On the eve of the last day of the semester, we had a showdown with me trashing the theoretical nature of the problem he gave me, and he defending the practical significance of the problem, and then saying, “You should have known this on day one. Maybe you are not capable of solving
this kind of problems. Maybe you
should just do a project and get out with a Master’s.” That stung, and I set out to prove
that I could solve this problem.
Dense graphs are desirable for interconnecting computer
nodes because they reduce the number of hops in passing messages, reduce
transit traffic load, and can result in higher performance computer networks. Fault tolerance is important for a
computer network to continue to operate when things go wrong. A computer network can tolerate any
combination of d-1 faults if there
are d disjoint paths between any pair
of nodes. I started solving the fault
tolerance problem in two parts: by
showing that a special subset of more than d
nodes have d disjoint paths between
any pair of nodes within that subset; and then showing that any arbitrary node
in the graph has disjoint paths to d
distinct nodes in that special subset.
Thus, if up to d-1 nodes are
knocked out, there still exists a path from any node to at least one node in
the special subset, and between any pair of surviving nodes in that subset,
guaranteeing a path from any node to any other node in the graph.
I solved the first part of the problem under a tree at
Pentacrest, on the corner of Washington and Clinton streets in Iowa City, on
the last afternoon of that spring semester, oblivious to other students milling
around and getting out of town. I
rushed to Reddy who scrutinized my proof, nodded approvingly, and said to go
get the second half of the proof, which I did in short order.
Each meeting with Reddy was an inhalation, which would energize me and end
in a plan of action. My effort
between the meetings was exhalation and expansion, generalizing the proof to
other graphs, coming up with distributed routing and local rerouting schemes
and so on. By the time I wrote up my Master's thesis, I was in love with the problem Reddy gave me.
Regina is now having
us do Veerabhadrasana II, body facing sideways, front knee bent ninety degrees
and the other leg straight, drawing energy towards the core during inhalation,
and shooting it out the front knee, the outer edge of the rear foot, and both
outstretched arms while lowering your pelvis during the exhalation.
Reddy turned up the intensity for my Ph.D. thesis. He kept pushing me to go deeper into
the dense graph problem. I tried
to keep him happy since he was paying me, but I was getting interested in a
different type of networks called multistage networks. One day, I asked him to give me an hour
to describe the new problem I was studying. That one hour became an entire afternoon. Most multistage networks being studied
were based on building block switches which had two inputs and two outputs, and
had unique paths between source-destination pairs, thus lacking fault
tolerance. One network which could
tolerate single faults was based on 5 x 5 nodes, which we thought could be
improved on. At the end of the
meeting, we decided to look at reducing the size of the nodes to 4 x 4, as a
first step.
I was elated that I didn’t have to work on two different research
problems at the same time anymore.
That night, I went to watch a basketball match between Iowa and Michigan, and celebrated at a friend’s
house.
Next morning at the office, I found a note from Reddy:
“Vijay, please see me.” He
had a whole bunch of notes on solving the problem we identified. He had a basic scheme to assign a
conjugate node for every one of the original 2x2 nodes, replacing the nodes
with 4x4 nodes, and then making extra connections to the conjugate nodes in
such a way that single faults could be bypassed. I was amazed at the amount of interest he was taking in my
problem, and was congratulating myself on my good fortune. When he was done, he closed his note
pad and said: “Now, show me what
you have done.”
He was no longer leading me by the hand. He was sparring with me in the
ring. Soon I showed an algorithmic
way of designing such networks, and he an equivalent one, and so it went.
Regina eventually has
the class doing back bends. Lengthen
the back with every inhalation, and bend deeper with every exhalation.
My inhalations of advising sessions with Reddy, and
exhalations of individual effort were not always regular. Sometimes, I would delay going to see
him either because I was stuck and too proud to admit it to him, or because I
was embarrassed that I didn’t put in enough effort. This was akin to holding my breath in a yoga pose. Other times, however, I would delay
going to see Reddy because I was in the flow and didn’t want to interrupt my
work. In this case, it was like a
long exhalation and getting deeper into a pose. The yoga teacher may want to move on to another pose, but
sometimes I feel unfulfilled and want to stay with a pose a little longer and
get deeper.
Eventually, I came up with a class of networks called
Augmented Shuffle Exchange Networks (ASEN) which were single fault tolerant and
based on 3x3 nodes, and characterized their performance and reliability. My exhalations were getting
longer. Reddy gave me a list of
things I had to do to graduate, and soon sent me packing.
As Regina’s class gets into shavasana, it becomes apparent to
me that my entire graduate school experience was an inhalation on a larger
scale. For when I joined Bell Labs, I built the chips for ASEN, which
led to a series of networking projects. Each project
was an exhalation that followed the previous project, while simultaneously
being an inhalation that preceded the next project. It was circular breathing.
I submit that this kind of spanda and circular breathing are not limited to research or learning. For
instance, there is a form of this process taking place in relationships,
parenting, and adventures. And, if you believe in reincarnation, over successive
lifetimes.
