## Monday, November 03, 2014

### Functional and Reactive Domain Modeling

Manning has launched the MEAP of my upcoming book on Domain Modeling.

The first time I was formally introduced to the topic was way back when I played around with Erik Evans' awesome text on the subject of Domain Driven Design. In the book he discusses various object lifecycle patterns like the Factory, Aggregate or Repository that help separation of concerns when you are implementing the various interactions between the elements of the domain model. Entities are artifacts with identities, value objects are pure values while services model the coarse level use cases of the model components.

Traditionally we followed the recommendations of Erik in our real world implementations and used the object oriented paradigm for modeling all interactions. We started talking about rich domain models and anemic domain models. The rich model espoused a richer agglomeration of state and behavior within the model, while the anemic model preferred to keep them decoupled. Martin Fowler sums this up in his post on Anemic Domain Models ..
"The basic symptom of an Anemic Domain Model is that at first blush it looks like the real thing. There are objects, many named after the nouns in the domain space, and these objects are connected with the rich relationships and structure that true domain models have. The catch comes when you look at the behavior, and you realize that there is hardly any behavior on these objects, making them little more than bags of getters and setters. Indeed often these models come with design rules that say that you are not to put any domain logic in the the domain objects. Instead there are a set of service objects which capture all the domain logic. These services live on top of the domain model and use the domain model for data."

## Go Functional

In Functional and Reactive Domain Modeling I look at the problem with a different lens. The primary focus of the book is to encourage building domain models using the principles of functional programming. It's a completely orthogonal approach than OO and focuses on verbs first (as opposed to nouns first in OO), algebra first (as opposed to objects in OO), function composition first (as opposed to object composition in OO), lightweight objects as ADTs (instead of rich class models).

The book starts with the basics of functional programming principles and discusses the virtues of purity and the advantages of keeping side-effects decoupled from the core business logic. The book uses Scala as the programming language and does an extensive discussion on why the OO and functional features of Scala are a perfect fit for modelling complex domains. Chapter 3 starts the core subject of functional domain modeling with real world examples illustrating how we can make good use of patterns like smart constructors, monads and monoids in implementing your domain model. The main virtue that these patterns bring to your model is genericity - they help you extract generic algebra from domain specific logic into parametric functions which are far more reusable and less error prone. Chapter 4 focuses on advanced usages like typeclass based design and patterns like monad transformers, kleislis and other forms of compositional idioms of functional programming. One of the primary focus of the book is an emphasis on algebraic API design and to develop an appreciation towards ability to reason about your model.

## Go Reactive

The term "reactive" has recently become quite popular in describing systems that are responsive, scalable and adaptive. In implementing complex domain models, we find many areas which can be made more responsive by implementing them as non blocking operations instead of the standard blocking function calls. Using higher level concurrency primitives like actors, futures and data flow based computing we can compose asynchronous operations and increase the net throughput of your model. The second part of the book discusses how you can combine the principles of functional programming with the reactive way of implementing behaviors. The paradigm of application design using the principles of functional programming together with an asynchronous non-blocking mode of communication between the participating entities promises to be a potent combination towards developing performant systems that are relatively easy to manage, maintain and evolve. But designing and implementing such models need a different way of thinking. The behaviors that you implement have to be composable using pure functions and they can form building blocks of bigger abstractions that communicate between them using non-blocking asynchronous message passing.

## Functional meets Reactive

Functional and Reactive Domain Modeling takes you through the steps teaching you how to think of the domain model in terms of pure functions and how to compose them to build larger abstractions. You will start learning with the basics of functional programming principles and gradually progress to the advanced concepts and patterns that you need to know to implement complex domain models. The book demonstrates how advanced FP patterns like algebraic data types, typeclass based design and isolation of side-effects can make your model compose for readability and verifiability. On the subject of reactive modeling, the book focuses on the higher order concurrency patterns like actors and futures. It uses the Akka framework as the reference implementation and demonstrates how advanced architectural patterns like event sourcing and command-query-responsibility-segregation can be put to great use while implementing scalable models. You will learn techniques that are radically different from the standard RDBMS based applications that are based on mutation of records. You’ll also pick up important patterns like using asynchronous messaging for interaction based on non blocking concurrency and model persistence, which delivers the speed of in-memory processing along with suitable guarantees of reliability.

Looking forward to an exciting journey with the book. I am sure you will also find interest in the topics that I discuss there. And feel free to jump on to AuthorOnline and fire questions that we all can discuss. I am sure this will also lead to an overall improvement in the quality of the book.

## Monday, May 12, 2014

### Functional Patterns in Domain Modeling - Anemic Models and Compositional Domain Behaviors

I was looking at the presentation that Dean Wampler made recently regarding domain driven design, anemic domain models and how using functional programming principles help ameliorate some of the problems there. There are some statements that he made which, I am sure made many OO practitioners chuckle. They contradict popular beliefs that encourage OOP as the primary way of modeling using DDD principles.

One statement that resonates a lot with my thought is "DDD encourages understanding of the domain, but don't implement the models". DDD does a great job in encouraging developers to understand the underlying domain model and ensuring a uniform vocabulary throughout the lifecycle of design and implementation. This is what design patterns also do by giving you a vocabulary that you can heartily exchange with your fellow developers without influencing any bit of implementation of the underlying pattern.

On the flip side of it, trying to implement DDD concepts using standard techniques of OO with joined state and behavior often gives you a muddled mutable model. The model may be rich from the point of view that you will find all concepts related to the particular domain abstraction baked in the class you are modeling. But it makes the class fragile as well since the abstraction becomes more locally focused losing the global perspective of reusability and composability. As a result when you try to compose multiple abstractions within the domain service layer, it becomes too much polluted with glue code that resolves the impedance mismatch between class boundaries.

So when Dean claims "Models should be anemic", I think he means to avoid this bundling of state and behavior within the domain object that gives you the false sense of security of richness of the model. He encourages the practice that builds domain objects to have the state only while you model behaviors using standalone functions.

One other strawman argument that I come across very frequently is that bundling state and behavior by modeling the latter as methods of the class increases encapsulation. If you are still a believer of this school of thought, have a look at Scott Meyer's excellent article which he wrote as early as 2000. He eschews the view that a class is the right level of modularization and encourages more powerful module systems as better containers of your domain behaviors.

As continuation of my series on functional domain modeling, we continue with the example of the earlier posts and explore the theme that Dean discusses ..

Here's the anemic domain model of the Order abstraction ..

case class Order(orderNo: String, orderDate: Date, customer: Customer,
lineItems: Vector[LineItem], shipTo: ShipTo,
netOrderValue: Option[BigDecimal] = None, status: OrderStatus = Placed)

In the earlier posts we discussed how to implement the Specification and Aggregate Patterns of DDD using functional programming principles. We also discussed how to do functional updates of aggregates using data structures like Lens. In this post we will use these as the building blocks, use more functional patterns and build larger behaviors that model the ubiquitous language of the domain. After all, one of the basic principles behind DDD is to lift the domain model vocabulary into your implementation so that the functionality becomes apparent to the developer maintaining your model.

The core idea is to validate the assumption that building domain behaviors as standalone functions leads to an effective realization of the domain model according to the principles of DDD. The base classes of the model contain only the states that can be mutated functionally. All domain behaviors are modeled through functions that reside within the module that represents the aggregate.

Functions compose and that's precisely how we will chain sequence of domain behaviors to build bigger abstractions out of smaller ones. Here's a small function that values an Order. Note it returns a Kleisli, which essentially gives us a composition over monadic functions. So instead of composing a -> b and b -> c, which we do with normal function composition, we can do the same over a -> m b and b -> m c, where m is a monad. Composition with effects if you may say so.

def valueOrder = Kleisli[ProcessingStatus, Order, Order] {order =>
val o = orderLineItems.set(
order,
setLineItemValues(order.lineItems)
)
o.lineItems.map(_.value).sequenceU match {
case Some(_) => right(o)
case _ => left("Missing value for items")
}
}

But what does that buy us ? What exactly do we gain from these functional patterns ? It's the power to abstract over families of similar abstractions like applicatives and monads. Well, that may sound a bit rhetoric and it needs a separate post to justify their use. Stated simply, they encapsulate effects and side-effects of your computation so that you can focus on the domain behavior itself. Have a look at the process function below - it's actually a composition of monadic functions in action. But all the machinery that does the processing of effects and side-effects are abstracted within the Kleisli itself so that the user level implementation is simple and concise.

With Kleisli it's the power to compose over monadic functions. Every domain behavior has a chance of failure, which we model using the Either monad - here ProcessingStatus is just a type alias for this .. type ProcessingStatus[S] = \/[String, S]. Using the Kleisli, we don't have to write any code for handling failures. As you will see below, the composition is just like the normal functions - the design pattern takes care of alternate flows.

Once the Order is valued, we need to apply discounts to qualifying items. It's another behavior that follows the same pattern of implementation as valueOrder.

def applyDiscounts = Kleisli[ProcessingStatus, Order, Order] {order =>
val o = orderLineItems.set(
order,
setLineItemValues(order.lineItems)
)
o.lineItems.map(_.discount).sequenceU match {
case Some(_) => right(o)
case _ => left("Missing discount for items")
}
}

Finally we check out the Order ..

def checkOut = Kleisli[ProcessingStatus, Order, Order] {order =>
val netOrderValue = order.lineItems.foldLeft(BigDecimal(0).some) {(s, i) =>
s |+| (i.value |+| i.discount.map(d => Tags.Multiplication(BigDecimal(-1)) |+| Tags.Multiplication(d)))
}
right(orderNetValue.set(order, netOrderValue))
}

And here's the service method that composes all of the above domain behaviors into the big abstraction. We don't have any object to instantiate. Just plain function composition that results in an expression modeling the entire flow of events. And it's the cleanliness of abstraction that makes the code readable and succinct.

def process(order: Order) = {
(valueOrder andThen
applyDiscounts andThen checkOut) =<< right(orderStatus.set(order, Validated))
}
In case you are interested in the full source code of this small example, feel free to take a peek at my github repo.

## Sunday, April 06, 2014

### Functional Patterns in Domain Modeling - Immutable Aggregates and Functional Updates

In the last post I looked at a pattern that enforces constraints to ensure domain objects honor the domain rules. But what exactly is a domain object ? What should be the granularity of an object that my solution model should expose so that it makes sense to a domain user ? After all, the domain model should speak the language of the domain. We may have a cluster of entities modeling various concepts of the domain. But only some of them can be published as abstractions to the user of the model. The others can be treated as implementation artifacts and are best hidden under the covers of the published ones.

An aggregate in domain driven design is a published abstraction that provides a single point of interaction to a specific domain concept. Considering the classes I introduced in the last post, an Order is an aggregate. It encapsulates the details that an Order is composed of in the real world (well, only barely in this example, which is only for illustration purposes :-)).

Note an aggregate can consist of other aggregates - e.g. we have a Customer instance within an Order. Eric Evans in his book on Domain Driven Design provides an excellent discussion of what constitutes an Aggregate.

# Functional Updates of Aggregates with Lens

This is not a post about Aggregates and how they fit in the realm of domain driven design. In this post I will talk about how to use some patterns to build immutable aggregates. Immutable data structures offer a lot of advantages, so build your aggregates ground up as immutable objects. Algebraic Data Type (ADT) is one of the patterns to build immutable aggregate objects. Primarily coming from the domain of functional programming, ADTs offer powerful techniques of pattern matching that help you match values against patterns and bind variables to successful matches. In Scala we use case classes as ADTs that give immutable objects out of the box ..

case class Order(orderNo: String, orderDate: Date, customer: Customer,
lineItems: Vector[LineItem], shipTo: ShipTo, netOrderValue: Option[BigDecimal] = None,
status: OrderStatus = Placed)


Like all good aggregates, we need to provide a single point of interaction to users. Of course we can access all properties using accessors of case classes. But what about updates ? We can update the orderNo of an order like this ..

val o = Order( .. )
o.copy(orderNo = newOrderNo)

which gives us a copy of the original order with the new order no. We don't mutate the original order. But anybody having some knowledge of Scala will realize that this becomes pretty clunky when we have to deal with nested object updation. e.g in the above case, ShipTo is defined as follows ..

case class Address(number: String, street: String, city: String, zip: String)
case class ShipTo(name: String, address: Address)

So, here you go in order to update the zip code of a ShipTo ..

val s = ShipTo("abc", Address("123", "Monroe Street", "Denver", "80233"))
s.copy(address = s.address.copy(zip = "80231"))

Not really pleasing and can go off bounds in comprehensibility pretty soon.

In our domain model we use an abstraction called a Lens for updating Aggregates. In very layman's terms, a lens is an encapsulated get and set combination. The get extracts a small part from a larger whole, while the set transforms the larger abstraction with a smaller part taken as a parameter.

case class Lens[A, B](get: A => B, set: (A, B) => A)

This is a naive definition of a Lens in Scala. Sophisticated lens designs go a long way to ensure proper abstraction and composition. scalaz provides one such implementation out of the box that exploits the similarity in structure between the get and the set to generalize the lens definition in terms of another abstraction named Store. As it happens so often in functional programming, Store happens to abstract yet another pattern called the Comonad. You can think of a Comonad as the inverse of a Monad. But in case you are more curious, and have wondered how lenses form "the Coalgebras for the Store Comonad", have a look at the 2 papers here and here.

Anyway for us mere domain modelers, we will use the Lens implementation as in scalaz .. here's a lens that helps us update the OrderStatus within an Order ..

val orderStatus = Lens.lensu[Order, OrderStatus] (
(o, value) => o.copy(status = value),
_.status
)

and use it as follows ..

val o = Order( .. )
orderStatus.set(o, Placed)

will change the status field of the Order to Placed. Let's have a look at some of the compositional properties of a lens which help us write readable code for functionally updating nested structures.

# Composition of Lenses

First let's define some individual lenses ..

// lens for updating a ShipTo of an Order
val orderShipTo = Lens.lensu[Order, ShipTo] (
(o, sh) => o.copy(shipTo = sh),
_.shipTo
)

// lens for updating an address of a ShipTo
)

// lens for updating a city of an address
_.city
)

And now we compose them to define a lens that directly updates the city of a ShipTo belonging to an Order ..

// compositionality FTW
def orderShipToCity = orderShipTo andThen shipToAddress andThen addressToCity

Now updating a city of a ShipTo in an Order is as simple and expressive as ..

val o = Order( .. )
orderShipToCity.set(o, "London")

The best part of using such compositional data structures is that it makes your domain model implementation readable and expressive to the users of your API. And yet your aggregate remains immutable.

Let's look at another use case when the nested object is a collection. scalaz offers partial lenses that you can use for such composition. Here's an example where we build a lens that updates the value member within a LineItem of an Order. A LineItem is defined as ..

case class LineItem(item: Item, quantity: BigDecimal, value: Option[BigDecimal] = None,
discount: Option[BigDecimal] = None)


and an Order has a collection of LineItems. Let's define a lens that updates the value within a LineItem ..

val lineItemValue = Lens.lensu[LineItem, Option[BigDecimal]] (
(l, v) => l.copy(value = v),
_.value
)

and then compose it with a partial lens that helps us update a specific item within a vector. Note how we convert our lineItemValue lens to a partial lens using the unary operator ~ ..

// a lens that updates the value in a specific LineItem within an Order
def lineItemValues(i: Int) = ~lineItemValue compose vectorNthPLens(i)

Now we can use this composite lens to functionally update the value field of each of the items in a Vector of LineItems using some specific business rules ..

(0 to lis.length - 1).foldLeft(lis) {(s, i) =>
val li = lis(i)
lineItemValues(i).set(s, unitPrice(li.item).map(_ * li.quantity)).getOrElse(s)
}

In this post we saw how we can handle aggregates functionally and without any in-place mutation. This keeps the model pure and helps us implement domain models that has sane behavior even in concurrent settings without any explicit use of locks and semaphores. In the next post we will take a look at how we can use such compositional structures to make the domain model speak the ubiquitous language of the domain - another pattern recommended by Eric Evans in domain driven design.

## Monday, March 31, 2014

### Functional Patterns in Domain Modeling - The Specification Pattern

When you model a domain, you model its entities and behaviors. As Eric Evans mentions in his book Domain Driven Design, the focus is on the domain itself. The model that you design and implement must speak the ubiquitous language so that the essence of the domain is not lost in the myriads of incidental complexities that your implementation enforces. While being expressive the model needs to be extensible too. And when we talk about extensibility, one related attribute is compositionality.

Functions compose more naturally than objects and In this post I will use functional programming idioms to implement one of the patterns that form the core of domain driven design - the Specification pattern, whose most common use case is to implement domain validation. Eric's book on DDD says regarding the Specification pattern ..
It has multiple uses, but one that conveys the most basic concept is that a SPECIFICATION can test any object to see if it satisfies the specified criteria.
A specification is defined as a predicate, whereby business rules can be combined by chaining them together using boolean logic. So there's a concept of composition and we can talk about Composite Specification when we talk about this pattern. Various literature on DDD implement this using the Composite design pattern so commonly implemented using class hierarchies and composition. In this post we will use function composition instead.

# Specification - Where ?

One of the very common confusions that we have when we design a model is where to keep the validation code of an aggregate root or any entity, for that matter.
• Should we have the validation as part of the entity ? No, it makes the entity bloated. Also validations may vary based on some context, while the core of the entity remains the same.
• Should we have validations as part of the interface ? May be we consume JSON and build entities out of it. Indeed some validations can belong to the interface and don't hesitate to put them there.
• But the most interesting validations are those that belong to the domain layer. They are business validations (or specifications), which Eric Evans defines as something that "states a constraint on the state of another object". They are business rules which the entity needs to honor in order to proceed to the next stage of processing.
We consider the following simple example. We take an Order entity and the model identifies the following domain "specifications" that a new Order must satisfy before being thrown into the processing pipeline:

1. it must be a valid order obeying the constraints that the domain requires e.g. valid date, valid no of line items etc.
2. it must be approved by a valid approving authority - only then it proceeds to the next stage of the pipeline
3. customer status check must be passed to ensure that the customer is not black-listed
4. the line items from the order must be checked against inventory to see if the order can be fulfilled
These are the separate steps that are to be done in sequence by the order processing pipeline as pre-order checks before the actual order is ready for fulfilment. A failure in any of them takes the order out of the pipeline and the process stops there. So the model that we will design needs to honor the sequence as well as check all constraints that need to be done as part of every step.

An important point to note here is that none of the above steps mutate the order - so every specification gets a copy of the original Order object as input, on which it checks some domain rules and determines if it's suitable to be passed to the next step of the pipeline.

# Jumping on to the implementation ..

Let's take down some implementation notes from what we learnt above ..

• The Order can be an immutable entity at least for this sequence of operations
• Every specification needs an order, can we can pull some trick out of our hat which prevents this cluttering of API by passing an Order instance to every specification in the sequence ?
• Since we plan to use functional programming principles, how can we model the above sequence as an expression so that our final result still remains composable with the next process of order fulfilment (which we will discuss in a future post) ?
• All these functions look like having similar signatures - we need to make them compose with each other
Before I present more of any explanation or theory, here are the basic building blocks which will implement the notes that we took after talking to the domain experts ..

type ValidationStatus[S] = \/[String, S]

type ReaderTStatus[A, S] = ReaderT[ValidationStatus, A, S]

object ReaderTStatus extends KleisliInstances with KleisliFunctions {
def apply[A, S](f: A => ValidationStatus[S]): ReaderTStatus[A, S] = kleisli(f)
}

ValidationStatus defines the type that we will return from each of the functions. It's either some status S or an error string that explains what went wrong. It's actually an Either type (right biased) as implemented in scalaz.

One of the things which we thought will be cool is to avoid repeating the Order parameter for every method when we invoke the sequence. And one of the idioamtic ways of doing it is to use the Reader monad. But here we already have a monad - \/ is a monad. So we need to stack them together using a monad transformer. ReaderT does this job and ReaderTStatus defines the type that somehow makes our life easier by combining the two of them.

The next step is an implementation of ReaderTStatus, which we do in terms of another abstraction called Kleisli. We will use the scalaz library for this, which implements ReaderT in terms of Kleisli. I will not go into the details of this implementation - in case you are curious, refer to this excellent piece by Eugene.

So, how does one sample specification look like ?

Before going into that, here are some basic abstractions, grossly simplified only for illustration purposes ..

// the base abstraction
sealed trait Item {
def itemCode: String
}

// sample implementations
case class ItemA(itemCode: String, desc: Option[String],
minPurchaseUnit: Int) extends Item
case class ItemB(itemCode: String, desc: Option[String],
nutritionInfo: String) extends Item

case class LineItem(item: Item, quantity: Int)

case class Customer(custId: String, name: String, category: Int)

// a skeleton order
case class Order(orderNo: String, orderDate: Date, customer: Customer,
lineItems: List[LineItem])

And here's a specification that checks some of the constraints on the Order object ..

// a basic validation
private def validate = ReaderTStatus[Order, Boolean] {order =>
if (order.lineItems isEmpty) left(s"Validation failed for order \$order")
else right(true)
}

It's just for illustration and does not contain much domain rules. The important part is how we use the above defined types to implement the function. Order is not an explicit argument to the function - it's curried. The function returns a ReaderTStatus, which itself is a monad and hence allows us to sequence in the pipeline with other specifications. So we get the requirement of sequencing without breaking out of the expression oriented programming style.

Here are a few other specifications based on the domain knowledge that we have gathered ..

private def approve = ReaderTStatus[Order, Boolean] {order =>
right(true)
}

private def checkCustomerStatus(customer: Customer) = ReaderTStatus[Order, Boolean] {order =>
right(true)
}

private def checkInventory = ReaderTStatus[Order, Boolean] {order =>
right(true)
}

# Wiring them together

But how do we wire these pieces together so that we have the sequence of operations that the domain mandates and yet all goodness of compositionality in our model ? It's actually quite easy since we have already done the hard work of defining the appropriate types that compose ..

Here's the isReadyForFulfilment method that defines the composite specification and invokes all the individual specifications in sequence using for-comprehension, which, as you all know does the monadic bind in Scala and gives us the final expression that needs to be evaluated for the Order supplied.

def isReadyForFulfilment(order: Order) = {
val s = for {
_ <- validate
_ <- approve
_ <- checkCustomerStatus(order.customer)
c <- checkInventory
} yield c
s(order)
}

So we have the monadic bind implement the sequencing without breaking the compositionality of the abstractions. In the next instalment we will see how this can be composed with the downstream processing of the order that will not only read stuff from the entity but mutate it too, of course in a functional way.

## Thursday, January 23, 2014

### A Sketch as the Query Model of an EventSourced System

In my last post I discussed the count-min sketch data structure that can be used to process data streams using sub-linear space. In this post I will continue with some of my thoughts on how count-min sketches can be used in a typical event sourced application architecture. An event sourcing system typically has a query model which provides a read only view of how all the events are folded to provide a coherent view of the system. I have seen applications where the query model is typically rendered from a relational database. And the queries can take a lot of time to be successfully processed and displayed to the user if the data volume is huge. And when we are talking about Big Data, this is not a very uncommon use case.

Instead of rendering the query from the RDBMS, quite a few types of them can be rendered from a count-min sketch using sub-linear space. Consider the use case where you need to report the highest occuring user-ids in a Twitter stream. The stream is continuous, huge and non ending and you get to see each item once. So you get each item from where you parse out the user-id occurring in it and update the sketch. So each entry of the sketch contains the frequency of the user-id that hashes to that slot. And we can take the minimum of all the slots to which a user-id hashes to, in order to get the frequency of that user-id. The details of how this works can be found in my last post.

Consider the case where we need to find the heavy-hitters - those user-ids whose frequency exceeds a pre-determined threshold. For that, in addition to the sketch we can also maintain a data structure like heap or tree where we update the top-k heavy hitters. When a user-id appears, we update the sketch, get its estimated frequency from the sketch and if it exceeds the threshold, also record it in the data structure. So at any point in time we can probe this accessary data structure to get the current heavy-hitters. Spark examples contain a sample implementation of this heavy hitters query from a Twitter stream using the CountMinSketchMonoid of Algebird.

Can this be a viable approach of implementing the query model in an event sourced system if the use case fits the approximation query approach ? It can be faster, relatively cheap in space and can prove to be responsive enough to be displayed in dashboards in the form of charts or graphs.

## Sunday, January 19, 2014

### Count-Min Sketch - A Data Structure for Stream Mining Applications

In today's age of Big Data, streaming is one of the techniques for low latency computing. Besides the batch processing infrastructure of map/reduce paradigm, we are seeing a plethora of ways in which streaming data is processed at near real time to cater to some specific kinds of applications. Libraries like Storm, Samza and Spark belong to this genre and are starting to get their share of user base in the industry today.

This post is not about Spark, Storm or Samza. It's about a data structure which is one of the relatively new entrants in the domain of stream processing, which is simple to implement, but has already proved to be of immense use in serving a certain class of queries over huge streams of data. I have been doing some readings about the application of such structures and thought of sharing them with the readers of my blog.

# Using Sublinear Space

Besides data processing, these tools also support data mining over streams that include serving specialized queries over data using limited space and time. Ok, so once we store all data as they come we can always serve queries with O(n) space. But since we are talking about huge data streams, it may not even be possible to run algorithms on the full set of data - it simply will be too expensive. Even if we have the entire set of data in a data warehouse, the processing of the entire data set may take time and consume resources that we cannot afford to have, considering the fee charged under the evolving models of using the platform-as-a-service within the cloud based infrastructure. Also the fact that these algorithms will be working on data streams, there's a high likelihood that they will get to see these data only in a single pass. The bottom line is that we need to have algorithms that work on sub-linear space.

Working on sublinear space implies that we don't get to store or see all data - hence an obvious conclusion from this will be the fact that we also don't get to deliver an accurate answer to some queries. We rely on some approximation techniques and deliver an accuracy with a reasonably high probability bound. We don't store all data, instead we store a lossy compressed representation of the data and deliver user queries from this subset instead of the entire set.

One widely used technique for storing a subset of data is through Random Sampling, where the data stored is selected through some stochastic mechanism. There are various ways to determine which data we select for storing and how we build the estimator for querying the data. There are pros and cons with this approach, but it's one of the simplest ways to do approximation based queries on streaming data.

There are a few other options like Histograms and Wavelet based synopses. But one of the most interesting data structures that have been developed in recent times is the Sketch, which uses summary based techniques for delivering approximation queries, gets around the typical problems that sampling techniques have and are highly parallelizable in practice.

An important class of sketch is one where the sketch vector (which is the summary information) is a linear transform of the input vector. So if we model the input as a vector we can multiply it by a sketch matrix to obtain the sketch vector that contains the synopses data that we can use for serving approximation queries. Here's a diagrammatic representation of the sketch as a linear transform of the input data.

# Count-Min Sketch

One of the most popular forms of the sketch data structure is the Count-Min Sketch introduced by Muthukrishnan and Cormode in 2003. The idea is quite simple and the data structure is based on probabilistic algorithms to serve various types of queries on streaming data. The data structure is parameterized by two factors - ε and δ, where the error in answering the query is within a factor of ε with probability δ. So you can tune these parameters based on the space that you can afford and accordingly amortize the accuracy of results that the data structure can serve you.

Consider this situation where you have a stream of data (typically modeled as a vector) like updates to stock quotes in a financial processing system arriving continuously that you need to process and report statistical queries on a real time basis.

• We model the data stream as a vector a[1 .. n] and the updates received at time t are of the form (it, ct) which mean that the stock quote for a[it] has been incremented by ct. There are various models in which this update can appear as discussed in Data Streams: Algorithms and Applications by Muthukrishnan which includes negative updates as well and the data structure can be tuned to handle each of these variants.

• The core of the data structure is a 2 dimensional array count[w, d] that stores the synopses of the original vector and which is used to report results of queries using approximation techniques. Hence the total space requirement of the data structure is (w * d). We can bound each of w and d in terms of our parameters ε and δ and control the level of accuracy that we want our data structure to serve.

• The data structure uses hashing techniques to process these updates and report queries using sublinear space. So assume we have d pairwise-independent hash functions {h1 .. hd} that hash each of our inputs to the range (1 .. w). Just for the more curious mind, pairwise independence is a method to construct a universal hash family, a technique that ensures lower number of collisions in the hash implementation.

• When an update (it, ct) comes for the stream, we hash a[it] through each of the hash functions h1 .. hd and increment each of the w entries in the array that they hash to.

• for i = 1 to d
v = h(i)(a[it]) // v is between 1 and w
count[i, v] += ct // increment the cell count by ct
end

At any point in time if we want to know the approximate value of an element a[i] of the vector a, we can get it from computing the minimum of all values in each of the d cells of count where i hashes to. This can be proved formally. But the general intuition is that since we are using hash functions there's always a possibility of multiple i's colliding on to the same cell and contributing additively to the value of the cell. Hence the minimum among all hash values is the closest candidate to give us the correct result for the query.

The figure above shows the processing of the updates in a Count-Min sketch. This is typically called the Point Query that returns an approximation of a[i]. Similarly we can use a Count-Min sketch to get approximation queries for ranges which is typically a summation over multiple point queries. Another interesting application is to serve inner product queries where the data structure is used to query inner products of 2 vectors, a typical application of this being the estimation of join sizes in relational query processing. The paper Statistical Analysis of Sketch Estimators gives all details of how to use sketching as a technique for this.

Count-Min sketches have some great properties which make them a very useful data structure when processing distributed streams. They have associativity properties and can be modelled as monoids and hence terribly performant in a distributed environment where you can parallelize sketch operations. In a future post I will discuss some implementation techniques and how we can use count-min sketches to serve some useful applications over data streams. Meanwhile Twitter's algebird and ClearSpring's stream-lib offer implementations of Count-Min sketch and various other data structures applicable for stream mining applications.