# Analysis of MapReduce Algorithms

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3 need be ranked bases on their relevance to the query. To achieve this we need to compute certain metrics from the data in documents. This metric is called as the TF-IDF score of the documents. TF-IDF refers to term frequency inverse document frequency. It is a metric used to quantify the relative importance of a word to a document in a collection or corpus [3]. It increases with the increase in number of occurrences of that term in the document. STEP 2: Compute total number of terms in a document. The mapper takes as input the output of step 1 reducer and outputs the term count for each of the documents. The Reducer in turn sums the individual counts of words belonging to the same document and at the same time also retains the word-specific count. The functions are as given below as discussed in [5]. Mapper: Input: ((term, document id),count) Output: (document id,(term, count)) Reducer: takes as input metrics corrponding to same document id Output: ((term,document id), (count, totalcount)) Figure 3: MapReduce for Searching [5] The term frequency for a term t in document d is the ratio of the number of occurrences of the term in the document and the total number of words in the document as given below [4]. The IDF is calculated as given below where N is the total number of documents in the corpus. STEP 3: In the strp, the goal is to generate the number of documents in the corpus that the term appears in. The mapper takes as input the reducer output of the previous step. The map task emits a one every time it finds the term in the document along with the metrics of the term. The reduces then accumulates all the ones emitted by the map task corresponding to the same term. The functions are given below as discussed in [5]. Mapper: Input: ((term, document id),(count, totalcount)) Output: (term, (document id, count, totalcount, 1)) Reducer: Output: ((term, document id), (count, totalcount, totaldocs) STEP 4: Now we have all the quantities that are needed to compute the TF-IDF. Assuming total number of documents (N) is known, TF-IDF is given by, In order to implement this using MapReduce framework, we need to compute the following quantities[4]. 1. For each document, the number of times the term t occurs: 2. For each document, the total number of terms: N(d) 3. Number of documents the term t occurs: 4. Total number of documents N For each of the above steps the map reduce implementation is given below. STEP 1: Compute term frequency The map function for this step takes as input key, the document id and the values as the text document itself. for every term the occurs in the document, emits a values of 1 with the key as the tuple of (word, document id). The reducer then accumulated the counts of all the ones for each word and output the key value pair of word and count. The mapper and reducer function input outputs are abbreviated below as discussed in [5]. Mapper: Input: ((document id, document terms)) Output: ((word, document id),1) Reducer: Accumulates the word count for document id Output: ((word, document id),count) This computation can be performed by the mapping function and we can use the results of the mapping functions without any use for reducer as the metric need not be reduced further[5]. 3.2 MapReduce Paradigm in Digital Communication Systems In wireless communication systems, the performance is evaluated with the metric Packet Error Rate. For example, to arrive at a packet error rate of 10^-3 typically requires 100/10^(-3)=10^5 simulations with random channel and noise. This is time prohibitive to run in a serial fashion. Since each of the runs is independent of each other, the time can be reduced if run in a parallel fashion. If Ts is the time taken to run one packet, the serial fashion requires NTs while distributed implementation such as MapReduce can leverage parallel processing it to reduce the running time by a factor of L where L is the number of maps. 3.3 Calculating Distributed Sum Consider the case of distributed sum. Let N be the total number of terms. If done serially, the complexity in terms of additions is. Now if this can be done parallel with M parallel units, the total number of additions in one parallel unit is. So the total number of additions, including the sum across parallel units, is

6 utilize the parallel infrastructure, the must know how to utilize the potential of such architecture and code efficiently. The MapReduce paradigm helps the programmers to achieve this by providing an abstraction to the programmer and hides the implementation details of the code. But the programmers still need to decide on which operation should be parallelized. That is, it is the duty of the programmer to decide on what the map and reduce functions do. From there on, hadoop takes care of the implementation of parallelism making it easy for developer. Dynamic programming is also another algorithmic concept that helps in solving highly complex problems. By analyzing the various MapReduce algorithms, we could realize the potential of parallelization. There are many more MapReduce algorithms solving various problems. So we need to clearly identify which problems fit into the MapReduce model and efficiently utilize the parallel infrastructure. 5. REFERENCES [1] What is MapReduce? Retrieved April 5, 2014, form, 01.ibm.com/software/data/infosphere/hadoop/MapRed uce/ [2] What id Hadoop Distributed File System (HDFS)? Retrieved April 5, 2014, from, 01.ibm.com/software/data/infosphere/hadoop/hdfs/ [3] TF-IDF. Retrieved April 10, 2014, form, [4] Buttcher, S., Clarke, C.L.A., Cormack G.V., (2010) Information Retrieval: Implementing and Evaluating Search Engines. [5] MapReduce Algorithms. Retrieved April 10, 2014, from es/algorithms.pdf [6] Lin, J., Dyer, C. (2010). Data-Intensive Text Processing with MapReduce. Retrieved February 10, 2014,from [7] Chu, C.-T., Kim, S. K., Lin, Y. A., Yu, Y., Bradski, G., Ng, A., and Olukotun, K Map-Reduce for machine learning on multicore. In Proceedings of Neural Information Processing Systems Conference (NIPS). Vancouver, Canada. [8] Retrieved April 5, 2014, form,

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