Problem: Given a graph $\backslash ($ G$=($V$,$ E$)\backslash ),$ our goal is to find the PageRank values of all nodes in $\backslash ($ V $\backslash ).$ In addition, we include the random jump factor $\backslash (\backslash$alpha $\backslash )($which is a command$-$line parameter$)$ and redistribute all missing PageRank mass $\backslash ($ m $\backslash )$ due to dangling nodes. For dangling nodes, we set $\backslash ($ p$=0\backslash )$ during the map phase $($i$.$e$.,$ during the distribution of PageRank mass$)$ of the PageRank algorithm, and its PageRank mass is considered to be missed. The missing PageRank mass $\backslash ($ m $\backslash )($from all dangling nodes$)$ will be added back based on the following equation:

$\backslash [$

p$^\{\backslash$prime$\}=\backslash$alpha$\backslash$left$(\backslash$frac$\{1\}\{|$G$|\}\backslash$right$)+(1-\backslash$alpha$)\backslash$left$(\backslash$frac$\{$m$\}\{|$G$|\}+$p$\backslash$right$)$

$\backslash ]$

Input Format: Each line contains a tuple of $($nodeID$,$ nodeID, weight$),$ separated by spaces. Each tuple indicates a directed edge from the former node to the latter node. We ignore the edge weights in this problem, as they are not needed by the PageRank algorithm that we taught in class.

Output Format: Each line contains a tuple of $($nodeID$,$ PageRank value$),$ separated by spaces. We only output tuples for nodes whose PageRank values are above certain threshold, where the threshold value $($between $0$ and $1)$ is given as a command$-$line argument.

Sample Command:

$```$

hadoop jar $[.$jar file$]$ PageRank $[$alpha$][$iteration$][$threshold$][$infile$][$outdir$]$

$```$

Notes:

$-$ Your program $($call it PageRank.java$)$ will execute the PageRank algorithm over a fixed number of iterations. The program will take a command$-$line argument Iterations to indicate the number of MapReduce iterations needed to be executed. We assume that Iterations is at least one.

$-$ You need to write a MapReduce program $($call it PRAdjust.java$)$ to adjust the PageRank values due to random jumps and dangling nodes after each iteration of the PageRank algorithm.

$-$ We need to implement a class of node structure $($call it PRNodeWritable.java$)$ and write a MapReduce program to convert the input files into adjacency list format $($call it PRPreProcess.java$).$