## The best leader election in the world!

As the world watches with intense interest the quadrennial spectacle of the US presidential election and the endless sparring and debating, we discuss the most efficient way to elect a leader!

Television viewers are getting ready for another Trumpillary debate. The media has been licking its lips at skyrocketing TRP ratings supported by all the sparring and salacious stories. Since I’ve now got you here like those media channels, possibly against your will, let me tell you that this post is about electing a leader in a distributed network and is in, all likelihood, going to be much more boring than the US presidential election. This is not surprising at all, since the candidates in the distributed network are all honest, non-malicious,  rational, ‘think’ in exactly the same manner, and really and truly interested in electing the best candidate!

Moreover, their  desirability/’fitness’ is immediately comparable (e.g. given by a unique ID such as a mac address). Unfortunately, for the Trump(H)illary contest  this would be like  looking at the following picture and assiming the hand mudras define the candidates telling who will win e.g. will Trump’s Gyan mudra (i.e. knowledge hand gesture, which looks like F in the one hand sign language) trump Hillary’s Abhay mudra? (i.e. Fearlessness hand gesture, hmm, I hope you already see the contradictions!).

Clinton and Trump try to win using hand mudras? (from https://goo.gl/gRMsQt)

OK, now to the technical part of this post. Leader Election is one of the most fundamental problems in distributed systems. The idea is simply to select one of the nodes in the network as a leader and let it solve the critical problem of the day and then just convey the solution to the rest of the network. Electing a leader can often be the first step in many systems and was formally introduced as a rigorous problem in the context of token ring networks by Le Lann (1977).

We were fortunate to derive some very interesting results for this classic problem in the past few years. This is in the form of the following two papers –

The first paper has some very interesting lower bounds (which were assumed to be folklore but never proven). One aim of the paper was to achieve algorithms which can meet the lower bounds- that is leader election algorithms for arbitrary topologies must need O(diameter) time and O(#edges) messages simulteneously. Algorithms which meet either O(diameter) time or O(#edges) messages ignoring the other parameter have been long known but achieving both simultaneously has been very challenging. The paper has randomised algorithms which (almost) achieve that. The paper also has what we consider the best determinisitic leader election algorithm in terms of achieiving both parameters simultaneously. The result is an O(D.logn) time and O(m.logn) messages algorithm called The Double-Win Growing Kingdom Algorithm (Here, D: diameter, n: #nodes, m:#edges). Yes, it is more bloody than Game of Thrones! (Imdb rating; 9/5/10, wow).

I tell the high level idea and show some (almost) cool animations by one of my students. Our algorithm turned out to be similiar to Abu-Amara and Kanevsky’s (ICCI 1993) algorithm which they erroneously thought to work in O(D) time and O(m + log n) messages. Here is an informal outline of the algorithm:

(Note that the network here is depicted as a graph with the computers/agents as nodes and interconnections between them as edges. The agents can only communicate by sending messages to their neighbours along these edges. So, we are designing a graph theoretical algorithm.)

Initially every node (each with a unique ID) in the network wants to be a leader i.e. is a candidate and begins with radius=1. Ultimately, the node with the highest ID will win. Each node tries to win over other nodes and expand their kingdom by conquering other nodes! The algorithm terminates when the sole candidate left cannot grow its kingdom anymore.

In each round, for each node:

2. If you get a message from a higher ranked node than you, drop out and join that node’s kingdom!
3. Collision: You get hit by messages from multiple candidates. Choose the highest rank one and join its kingdom. Do not further the winner messages!
4. Inform the winners of their boundary of conquest by sending back acknowledegments of victory along the path the conquest messages came (actually, a broadcast tree)

… and so on…

The above idea is fairly straightforward but the main challenge is to keep the messages and time low – I am skipping the technical details except mentioning that each round in itself has four phases. This shows up in the following cool simulations done by my student Christopher (Thanks Christopher!):

Questions:

The ideal algorithm will be O(D) time + O(m) messages – there is a substantial gap here:

• Can a determistic algorithm achieve these bounds or get closer to it than the Double-Win Growing Kingdom algorithm?
• Is there a better lower bound than O(D) time + O(m) messages? i.e. Is the problem even more difficult than we think?

## Belfast to Mexico City via Self-healing Compact Routing!

Thanks to the Newton Fund and the Mexican Academy of Science (AMS) I get to spend six weeks in Mexico city visiting my colleague at UNAM, researching, what else but Compact Self-Healing Routing… and authentic Mexican food 😉

The first part of this post is by way of thanks to the Newton Fund  , the UK academies (say, The Royal Society) and the Mexican Academy of Sciences (AMC) for a Newton mobility grant  which will allow me to visit my colleague Armando Castaneda at UNAM for six weeks in August and September this year. The call funds upto three months of ‘foreign activity’ so I could have possibly asked for more time but I was unsure of being able to get away for so long. Armando managed to visit me at Belfast some time ago so this can be even thought of as a return visit! He even managed to time his visit to coincide with the Belfast Marathon (while he’s training for a marathon) – curiosier and curioser’, as Alice would say!

So, what’s this resilient compact routing? About three years ago, I was fortunate to be an  I-CORE postdoc with Danny Dolev  and Armando was a postdoc at Technion. We started discussing ideas around routing and possibly due to my long line of work with self-healing algorithms (on which many blog posts may follow!) we started to gravitate towards the question: Can we route messages despite failures in the network?  At about the same time, Shiri Chechik bested our Leader Election paper (On the complexity of universal leader election) with her paper Compact routing schemes with improved stretch for the best paper award at PODC 2013. With many life-changing events in-between (such as getting faculty positions and moving to many degrees drop in average temperature and many degrees more of precipitation!), the first paper in this line has just managed to struggle over in early 2016. Compact Routing messages in self-healing trees (Arxivwas a finalist for the best paper award in ICDCN 2016.

So, what’s this resilient compact routing (take 2)? Routing is a very important primitive’ for networks – the ability of the network to take a message from a source node and deliver it to a target. We encounter it every time we get onto a network- as soon as we connect to a router, to a website, send an email, make a skype/voip call etc.. In practice, the most used protocols for routing are based on well known standard graph distance finding algorithms such as Djikstra’s and Bellman-Ford, which itself is a testament to the longevity of these algorithms and to the power of graph algorithms, in general. If a node x gets a packet which started at node a and needs to end at node b, node x will refer to it’s routing table – a table which tells it which of its neighbours to send its packet to.

Often,  a routing table will contain an entry for every node in the network telling where to forward a message addressed to that node. Now, this means the table can be really really huge depending on the size of the network. In practice, there are ways around this. One way, which makes for some nice theory is to do some preprocessing on the network e.g. build spanning trees, do DFS traversal, maybe some renaming and port changes, to reduce the size of the routing tables and the packet header. The crux of many of these schemes seems to be (the now seemingly simple) idea of interval routing (which by itself may not be compact) introduced by Santaro and Khatib in 1982. The idea may be summarised as follows:

• Starting from a particular node, do a Depth First Search (DFS) traversal and construct the corresponding DFS tree. Also, give each node a label that is that node’s DFS number (ID) (say, the time step at which that node was first encountered in the DFS traversal).
• Now, at each node, if you store the ‘largest’ ID of its subtree and the IDs of its children, you can now get intervals – which tell you which neighbour in the tree a node should send a packet addressed to another node. Hence, the name Interval Routing.

This spawned an active and productive field of research for the last few decades particularly in the effort of reducing both the size of the labels and tables while reducing the necessary tradeoff to be paid in terms of the distance. It is well known that if you reduce the space you use for routing, you cannot use the shortest paths and must pay in some measure by using a longer path (this measure is called stretch).

Looking at the above description, one will notice that developing these data structures seem to rely heavily on the initial DFS traversal. This implies that one may need to do a lot of recomputation if the network changes. In this spirit, there are some (but not many) works on `resilient’ compact routing i.e. compact routing that can handle changes to the network. Ours is one such attempt – in which we show how to do the well know Thorup-Zwick routing (over trees for now) in the self-healing model. In brief, the self-healing model is a responsive model of resilience where an adversary chooses a node/processor  to take down or even insert (presumably to do the most damage in either case) and the neighbours of the attacked node/processor react by adding some edges/connections to the graph/network. We show how to both the compact routing and self-healing in low-memory (O(log^n) at most, where n is the number of nodes in the network).

I will defer the technical details of both self-healing and our compact self-healing routing to another post but needless to say, I am quite excited about continuing this work!

## Discovering the Distributed Algorithms behind Biological Cell Communication

What are the distributed algorithms behind cell communication? Stuck in a sandpit, I and colleagues at QUB gather up some ideas which will hopefully also find some applications.

Ever been in a sandpit? When I was asked to be in one (called the Applied Maths Sandpit at our new EPS faculty.) I was not sure what it would be. It could be an innocent fun activity in the sand as in the first picture, an unlikely dream holiday at a golf course (as in the second picture), or, most likely, a grueling day which I was not forewarned of (third picture).  As it turns out, it was rather nice and, ultimately, useful. There was no sand around, of course! (and neither was there much warm Sun, unsurprisingly for here).

For a start, I met a few brilliant colleagues I did not know existed. Then, in a day of real intense cut-throat Dragon’s den like competition, we pitched our, mostly half-baked, ideas (btw, I am joking about the ‘cut-throat’, after all,  we were (applied) mathematicians, not MBA enterpreneurs at the gathering!). Amazingly, at the end of it, our emergently formed motley crew of  me, Fred Currell, Thomas Huettemann, Dermot Green and Alessandro Ferraro (All except me from QUB Maths and Physics) have been given resources to recruit and spoil a PhD student for a proposal that makes up the ideas of this post.

So, here comes a very high level,  sparse, ambitious, and rough sketch:

Living organisms can be thought of as clusters of cells in communication (typically at gap-junction interfaces). Within interacting communities new cells are born and old ones are removed, through (sometimes programmed) death. There is a strong environmental influence on these processes. On a much smaller scale, quantum-mechanical processes are at work within cells, complicating the picture further. We think real life processes are efficient, fault-tolerant, self-healing and scalable, leading us to hypothise that there must be powerful distributed algorithms somewhere in these networks of cells waiting to be discovered.

Networks  are often modelled as a graphs: the cells are nodes and a common surface between two cells facilitates communication. In biological systems, things change and this dynamism in networks is often addressed by failure models, including adversarial and accidental (random) death. The network can react to these changes in various ways and we seek a mathematical framework in which to formulate and analyse the various questions arising.

The team thought of three systems which could be of interest (some of the team already work on these though I know little about them at the moment):

1. Volvocine green algae — These capture the evolutionary emergence of multicellularity, including what is believed to be the simplest multi-cellular eukaryotic organism: Tetrabaena socialis

Rough outline of phylogenetic relationships in volvocine green algae.

2. Light harvesting in photosynthetic organisms — the mechanism whereby living organisms harvest energy from light is believed to be one of the clearest biological systems countering the view that life is too “warm and wet” for quantum phenomena to be relevant.

A quantum machine for efficient light-energy harvesting (from the paper)

3. The tumour spheroid — a simple multicellular mimic of a tumour, this system is amenable to direct laboratory study and is known to show many of the hallmarks of cancer, including the lack of growth regulation mechanisms, meaning it seeks to grow avidly.

A study showing effects on size, shape and growth rate of tumours (from the paper)

The PhD advertisement is here (if you know of suitable candidates): http://www.qub.ac.uk/schools/eeecs/Research/PhDStudy/PhD-2016-17-65/

Talking of Maths sandpits, somebody is already working putting them to work: MathsSandPit.co.uk

Question: What’s the best way discover algorithms that nature uses? (Of course, this is a very old question!).