In continuation of our discussion on how to apply linear regression in an interative manner to build it as an when the data becomes available, let us know look at GLMnet algorithm,
As we had noted earlier, the GLMNet algorithm can work when we have fewer data because it gives methods to
1) determine if we are overfitting and
2) hedge our answers if we are overfitting
With ordinary least squares regression, objective is to minimize the sum squared error between actual values and our linear approximation. Coefficient shrinkage methods add a penalty on coefficients.
It yields a family of regression solutions - from completely insensitive to input data to unconstrained ordinary least squares.
First we build a Regularized Regression. We want to eliminate over-fitting because it tunes the model and it gives best performance on held out data. In order to avoid this overfitting, we constrain the regression with penalties such as coefficient shrinkage. Two types of coefficient shrinkage are most frequently used : sum of squared coefficients and sum of absolute value of coefficients.GLMNet algorithm incorporates ElasticNet penalty which is a weighted combination of sum squares and sum of absolute values.
We now look at how to use this in an incremental manner.
First, the existing data already has a regularized expression and a fitting line, shaped with coefficient shrinkage. Either the new data affects the existing regression or a new regression is created for the new data and the two fitting lines are replaced by a common one.
Both methods require slightly different computation from the first iteration because we adjust existing line/s instead of fitting a new line. Again we can divide this into non-overlapping data and overlapping data. For the non-overlapping data, a new regression is the same as the first iteration. For the overlapping data, we compute the shrinkage directly against the existing line. The use of shrinkage here directly gives the corrections applied. Although the shrinkage was originally used to eliminate overfitting, it also seems to give an indication of the correction to be applied to the line. The  higher the number of data points and the larger the shrinkages, the more the correction applied to the line. In this way we start from a completely data insensitive line to one that fits it best. For the sake of convenience, we use the existing line from previous regression.  This is a slight different computation we introduce.
If we don't want to introduce a different computation as above, we can determine a GLMNet regression for the new points exclusively and then proceed to combining the regressions from new and existing datasets
Combining regression is a computation over and on top of the GLMNet computation for any dataset.  While GLMNet works well for small dataset, this step enables the algorithm to be applied repeatedly as the data becomes available more and more. This will also be required when we combine regression lines such as for non-overlapping data.

Previously we discussed using computations that don’t span the entire data all at once albeit in parallel but compute based on as and when data is made available. We took an example of computing mean this way.  

Let us take an example of the summation form to compute the mean 

For example if we take 3,1,5, 7, 9  the mean is 25/5 = 5 

If we were to add 10,15,17,19,24, the mean would now become 85  + 25 / 10 = 11 

We could arrive at this by adding 85 /5  = 17 and taking the sum (5 + 17) / 2 for two equal ranges otherwise we take the weights based on their element count = 11 

In this case we are reusing the previous computation of the data and we don't have to revisit that data again. We make summarized information already which we adjust based on new data. In the case the summary from the first data set is the result of the computation from the first data set. 

Now let we can generalize this technique to be applied to all summation forms. 

In the summation form, the data is already chunked into subgroups and then the computation is performed. The reducer merely aggregates the results. In other words, the data doesn’t overlap so it is easier to aggregate the results.  In a streaming mode for the data when it becomes available the same computations can be done as and when the non-overlapping data arrives. Therefore the summary from each data set now need to be combined and since they represent non-overlapping data, it becomes easier to combine since each summary will contribute independently to the overall picture and the reducer in this case will merely applying smoothing of the aggregated results.  

For example in a linear regression, the lines may be drawn in chunks across different data as they arrive and then the reducer merely has to sort the summaries and apply a generalized line across the data. In this case it can take the beginning and the end points of the pieces to draw the line or find one that matches most of the slopes. Thus it is possible to aggregate results on non-overlapping data 

For the overlapping data however, the summaries are not trivial to combine because the previous summary may be completely overwritten. However, we look at the datapoints that are new in the current summary and calculate their affect on the previous slope. Since the new data points may be scattered over and beyond the previous data points, it should be easy to determine whether they affect the previous slope uniformly or non-uniformly. For the uniform case, there may not need to be a correction as in a change of the gradient and may at most require a displacement of the regression line. For the non-uniform case, depending on which end has more of the newer data points assuming all points contribute equally, a clockwise or anticlockwise rotation may be performed to get the correction to the previous slope. Thus we have an algorithm where we use the previous summary to have created the current summary. 

Furthermore, if there are two data point blocks that have a partial overlap, then they are split into two non-overlapping and one overlapping regions The regression line for the non-overlapping regions is preserved. The regression lines from the two contributing blocks in the overlap can merely be replaced by a line that is equidistant from both and passing through their intersection. In other words, summaries are additive too.  

Since we already mentioned that summation forms are all applicable this way, and now with the summaries being additive too, we have expanded the kind of algorithms that can be written this way. 

Previously we discussed using computations that don’t span the entire data all at once albeit in parallel but compute based on as and when data is made available. We took an example of computing mean this way.  

Let us take an example of the summation form to compute the mean 

For example if we take 3,1,5, 7, 9  the mean is 25/5 = 5 

If we were to add 10,15,17,19,24, the mean would now become 85  + 25 / 10 = 11 

We could arrive at this by adding 85 /5  = 17 and taking the sum (5 + 17) / 2 for two equal ranges otherwise we take the weights based on their element count = 11 

In this case we are reusing the previous computation of the data and we don't have to revisit that data again. We make summarized information already which we adjust based on new data. In the case the summary from the first data set is the result of the computation from the first data set. 

Now let we can generalize this technique to be applied to all summation forms. 

In the summation form, the data is already chunked into subgroups and then the computation is performed. The reducer merely aggregates the results. In other words, the data doesn’t overlap so it is easier to aggregate the results.  In a streaming mode for the data when it becomes available the same computations can be done as and when the non-overlapping data arrives. Therefore the summary from each data set now need to be combined and since they represent non-overlapping data, it becomes easier to combine since each summary will contribute independently to the overall picture and the reducer in this case will merely applying smoothing of the aggregated results.  

For example in a linear regression, the lines may be drawn in chunks across different data as they arrive and then the reducer merely has to sort the summaries and apply a generalized line across the data. In this case it can take the beginning and the end points of the pieces to draw the line or find one that matches most of the slopes. Thus it is possible to aggregate results on non-overlapping data 

For the overlapping data however, the summaries are not trivial to combine because the previous summary may be completely overwritten. However, we look at the datapoints that are new in the current summary and calculate their affect on the previous slope. Since the new data points may be scattered over and beyond the previous data points, it should be easy to determine whether they affect the previous slope uniformly or non-uniformly. For the uniform case, there may not need to be a correction as in a change of the gradient and may at most require a displacement of the regression line. For the non-uniform case, depending on which end has more of the newer data points assuming all points contribute equally, a clockwise or anticlockwise rotation may be performed to get the correction to the previous slope. Thus we have an algorithm where we use the previous summary to have created the current summary. 

Furthermore, if there are two data point blocks that have a partial overlap, then they are split into two non-overlapping and one overlapping regions The regression line for the non-overlapping regions is preserved. The regression lines from the two contributing blocks in the overlap can merely be replaced by a line that is equidistant from both and passing through their intersection. In other words, summaries are additive too.  

Since we already mentioned that summation forms are all applicable this way, and now with the summaries being additive too, we have expanded the kind of algorithms that can be written this way. 

Previously we discussed using computations that don’t span the entire data all at once albeit in parallel but compute based on as and when data is made available. We took an example of computing mean this way.  

Let us take an example of the summation form to compute the mean 

For example if we take 3,1,5, 7, 9  the mean is 25/5 = 5 

If we were to add 10,15,17,19,24, the mean would now become 85  + 25 / 10 = 11 

We could arrive at this by adding 85 /5  = 17 and taking the sum (5 + 17) / 2 for two equal ranges otherwise we take the weights based on their element count = 11 

In this case we are reusing the previous computation of the data and we don't have to revisit that data again. We make summarized information already which we adjust based on new data. In the case the summary from the first data set is the result of the computation from the first data set. 

Now let we can generalize this technique to be applied to all summation forms. 

In the summation form, the data is already chunked into subgroups and then the computation is performed. The reducer merely aggregates the results. In other words, the data doesn’t overlap so it is easier to aggregate the results.  In a streaming mode for the data when it becomes available the same computations can be done as and when the non-overlapping data arrives. Therefore the summary from each data set now need to be combined and since they represent non-overlapping data, it becomes easier to combine since each summary will contribute independently to the overall picture and the reducer in this case will merely applying smoothing of the aggregated results.  

For example in a linear regression, the lines may be drawn in chunks across different data as they arrive and then the reducer merely has to sort the summaries and apply a generalized line across the data. In this case it can take the beginning and the end points of the pieces to draw the line or find one that matches most of the slopes. Thus it is possible to aggregate results on non-overlapping data 

For the overlapping data however, the summaries are not trivial to combine because the previous summary may be completely overwritten. However, we look at the datapoints that are new in the current summary and calculate their affect on the previous slope. Since the new data points may be scattered over and beyond the previous data points, it should be easy to determine whether they affect the previous slope uniformly or non-uniformly. For the uniform case, there may not need to be a correction as in a change of the gradient and may at most require a displacement of the regression line. For the non-uniform case, depending on which end has more of the newer data points assuming all points contribute equally, a clockwise or anticlockwise rotation may be performed to get the correction to the previous slope. Thus we have an algorithm where we use the previous summary to have created the current summary. 

Furthermore, if there are two data point blocks that have a partial overlap, then they are split into two non-overlapping and one overlapping regions The regression line for the non-overlapping regions is preserved. The regression lines from the two contributing blocks in the overlap can merely be replaced by a line that is equidistant from both and passing through their intersection. In other words, summaries are additive too.  

Since we already mentioned that summation forms are all applicable this way, and now with the summaries being additive too, we have expanded the kind of algorithms that can be written this way. 

#codingquestion
List of stations and distances between them are given and find all pairs shortest distance.
Solution:
A straightforward implementation could be Floyd Warshall algorithm. This algorithm computes the shortest path weights in a bottom-up manner. It exploits the relationship between a pair of intermediary vertices and the shortest paths that pass through them. If there is no intermediary vertex, then such a path has at most one edge and the weight of the edge is the minimum. Otherwise, the minimum weight is the minimum of the path from I to j or the path from I to k and k to j. Thus this algorithm iterates for each of the intermediary vertices for each of the given input of an N*N matrix to compute the shortest path weight.
Algorithm Warshall(A[1..n, 1..n])
{
   R-0  is initialized with the adjacency matrix ; // R-0 is the node to itself.
   for k = 1  to n do
      for i = 1 to n do
        for j = 1 to n do
            R-k[i, j] = R-(k-1) [i, j] OR
                             (R-(k-1) [i, k] AND  R-(k-1) [k,j])
  return R-(n)

}
Another method could be to use the Bellman Ford algorithm to find the shortest pair distance between two vertices and also to find all paths.
      public static bool GetShortestPath(int[,] graph, int numVertex, int start, ref List<int> distances, ref List<int> parents)
        {
            // initialize Single Source
            for (int i = 0; i < numVertex; i++)
            {
                distances[i] = int.MaxValue;
                parents[i] = -1;
            }

            distances[start] = 0;

            var allEdges = GetAllEdges(graph, numVertex);
            for (int k = 0; k < numVertex - 1; k++)
            {
                for (int i = 0; i < allEdges.Count; i++)
                {
                    // relax
                    int sum = distances[allEdges[i].Item1] == int.MaxValue ?
                        distances[allEdges[i].Item1] :
                        distances[allEdges[i].Item1] + allEdges[i].Item3;
                    if (distances[allEdges[i].Item2] > sum)
                    {
                        distances[allEdges[i].Item2] = distances[allEdges[i].Item1] + allEdges[i].Item3;
                        parents[allEdges[i].Item2] = allEdges[i].Item1;
                    }
                }
            }

            for (int i = 0; i < allEdges.Count; i++)
            {
                if (distances[allEdges[i].Item2] > distances[allEdges[i].Item1] + allEdges[i].Item3)
                {
                    return false; // cycle exists;
                }
            }

            return true;
        }
        public static void GetAllPaths(int[,] graph, int numVertex, int source, int destination, int threshold, ref List<int> candidatePath, ref List<int> candidateDist, ref List<List<int>> pathList, ref List<List<int>> distanceList)
        {
            if (candidatePath.Count > threshold) return;
         
            var path = new List<int>();
            var distances = new List<int>();
            GetOutboundEdges(graph, numVertex, source, ref path, ref distances);
            if (path.Contains(destination) && (candidatePath.Count == 0  || (candidatePath.Count > 0 && candidatePath.Last() != destination)))
            {
                candidatePath.Add(destination);
                candidateDist.Add(distances[path.IndexOf(destination)]);
                if (pathList.Contains(candidatePath) == false)
                    pathList.Add(new List<int>(candidatePath));
                if (distanceList.Contains(candidateDist) == false)
                    distanceList.Add(new List<int>(candidateDist));
                candidatePath.RemoveAt(candidatePath.Count - 1);
                candidateDist.RemoveAt(candidateDist.Count - 1);
            }

            for (int i = 1; i < path.Count; i++)
            {
                // if (i != source)
                {
                    candidatePath.Add(path[i]);
                    candidateDist.Add(distances[i]);
                    GetAllPaths(graph, numVertex, path[i], destination, threshold, ref candidatePath, ref candidateDist, ref pathList, ref distanceList);
                    candidatePath.RemoveAt(candidatePath.Count - 1);
                    candidateDist.RemoveAt(candidateDist.Count - 1);
                }
            }

        }

Parallel and Concurrent Programming in NoSQL databases.

In NoSQL databases, the preferred method of parallelizing tasks is the MapReduce algorithm. It is a simple scatter gather programming model that enables all mappers to behave the same on their respective localized subgroups of data and a reducer that can aggregate the data from all the mappers. This applies well to algorithms  that calculate sums over the entire data range.

However, we look at dataflow parallelism that is not as strict or simplistic and enables even more algorithms to be applied to Big Data.  In Dataflow processing ,  there is a foundation for message passing and parallelizing CPU intensive and I/O intensive applications that have high throughput and low latency. It also gives you explicit control over how data is buffered and moved around the system While traditionally, we required callbacks and synchronization objects, such as locks, to coordinate tasks and access to shared data, by using this programming model, we can create data flow objects that process the data as they become available.  You only declare how the data is handled when it becomes available and also any dependencies between the data. Requirements to synchronize the shared data and other maintenance activities are common across algorithms and can be facilitated with a library. Moreover, this library can schedule asynchronous arrival of data, so tthroughput and latency can be improved. This kind of stream processing enables algorithms to be applied that are difficult to be expressed in Summation Form.

This framework consists of dataflow blocks which are data structures that buffer and process data. There are three kinds of dataflow blocks. Source blocks, target blocks and propagator blocks. A source block acts as a  source of data and can be read from. A target block acts as both a source block and a target block, and can be read from and written to. Interfaces can be defined separate for sources, targets and propagators. Propagator interface should inherit from both source and target. By implementing the interfaces, we define dataflow blocks that form a pipeline which are linear sequences of dataflow blocks or networks which are graphs of dataflow blocks. A pipeline is a simple case of a network and one where a source asynchronously propagates data to targets.  As more and more data blocks are added and removed, the blocks handle all thread safety operations of linking and unlinking. Just like protocol handlers, blocks can choose to accept or reject message as in a filtering mechanism. This enables explicit data blocks to be earmarked for processing

This programming model  uses the concept of message passing where independent  components of a program communicate with one another by sending messages. One way to propagate messages is to call say a post method which acts asynchronously. Source blocks offer data to target blocks by calling the OfferMessage method. The target block responds to an offered message returning either an Accepted, Declined or Deferred status. When the target block requires the message, it calls ConsumeMessage or calls ReleaseReservation when it defers a  message.

Dataflow blocks also support the concept of completion.  A dataflow block that is in the completed state does not perform any further work. Each dataflow block has an associated task  and it represents the completion status of the block. Because we can wait for a task to finish, we can wait for the terminal nodes of a dataflow network to finish. Tasks can throw an exception or return an error on failures thus denoting success otherwise. As with most tasks, dataflow blocks can be paused, resumed or canceled thus enabling robustness and reliability.

Some dataflow blocks can be predefined – for example we can have buffering blocks, execution blocks and grouping blocks which have now relaxed the count of mappers and reducers in the conventional way of using MapReduce algorithms. The degree of parallelism can also be explicitly specified for these dataflow blocks or it can be handled automatically.

Let us take an example of the summation form to compute the mean
For example if we take 3,1,5, 7, 9  the mean is 25/5 = 5
If we were to add 10,15,17,19,24, the mean would now become 85  + 25 / 10 = 11
We could arrive at this by adding 85 /5  = 17
and taking the sum (5 + 17) / 2 for two equal ranges otherwise we take the weights based on their element count = 11
In this case we are reusing the previous computation of the data and we don't have to revisit that data again. 

Let us look at applying GLMNet algorithm to big data using MapReduce in continuation of our discussion of previous arguments.
This is one of the popular algorithms for Machine Learning. As with most regression analysis, more data is better. But this algorithm can work even when we have fewer data because it gives methods to 1) determine if we are overfitting and
2) hedge our answers if we are overfitting.
With ordinary least squares regression, objective is to minimize the sum squared error between actual values and our linear approximation. Coefficient shrinkage methods add a penalty on coefficients.
It yields a family of regression solutions - from completely insensitive to input data to unconstrained ordinary least squares.
First we build a Regularized Regression. We want to eliminate over-fitting because it tunes the model and it gives best performance on held out data.
In this step in order to avoid over-fitting, we  have to exert control over degrees of freedom in regression. We cut back on attributes when we make a subset selection  and we penalize regression coefficients with coefficient shrinkage, ridge regression and lasso regression.
With coefficient shrinkage, different penalties give solutions with different properties.
With more than one attribute the coefficient penalty function has important effects on solutions. Some choices of coefficient penalty give control over sparseness of the solutions. Two types of co-efficient penalty are most frequently used: sum of squared coefficients and sum of absolute value of coefficients. GLMNet algorithm incorporates ElasticNet penalty which is a weighted combination of sum squares and sum of absolute values.
Courtesy : Michael Bowles on applying GLMNet to MapReduce and PBWorks for GLMNet.
What I find interesting in these discussions it the use of a coefficient shrinkage, a somewhat complicating factor,  which seems to improve an otherwise simple technique of regression. Similarly 'summation form' lends itself to applying on MapReduce because different mappers can compute those sums locally for their group and reducer can aggregate them and is inherently simple However we could achieve even more sophisticated MapReduce beyond the summation form to incorporate more algorithms based on other techniques of parallelization. Such techniques include not only partitioning data and computations but also time-slicing. One such example is to parallelize iterations. Currently only a reducer does iterations based on the aggregation of previous iteration variables from mappers but if this could be pushed down to  mappers where the reducer may aggregate not just by sum but by other functions such as max, min or count.