Generalized communication cost efficient multi-way spatial join: revisiting the curse of the last reducer

Abstract

With the huge increase in usage of smart mobiles, social media and sensors, large volumes of location-based data is available. Location based data carries important signals pertaining to user intensive information as well as population characteristics. The key analytical tool for location based analysis is multi-way spatial join. Unlike the conventional join strategies, multi-way join using map-reduce offers a scalable, distributed computational paradigm and efficient implementation through communication cost reduction strategies. Controlled Replicate (C-Rep) is a useful strategy used in the literature to perform the multi-way spatial join efficiently. Though C-Rep performance is superior compared to naive sequential join, careful analysis of its performance reveals that such a strategy is plagued by the curse of the last reducer, wherein the skew inherently present in the datasets and the skew introduced by replication operation, causes some of the reducers to take much longer time compared to others. In this work, we design an algorithm GEMS (G eneralized Communication cost E fficient M ulti-Way S patial Join) to address the skewness inherent in the connectivity of spatial objects while performing a multi-way join. We analysed all the algorithms concerned, in terms of I/O and communication costs. We prove that the communication cost of GEMS approach is better than that of C-Rep by a factor O(α) where α is the number of reducers in a single row/column of a grid of reducers. Our experimental results on different datasets indicate that GEMS approach is three times superior(in terms of turn around time) compared to C-Rep.

Appendices

Appendix

A Asymptotic complexity of GEMS vs Diagonal based Replication Schemes

The proof for Theorem 1 is provided below:

Proof

Let the grid structure be as given in Fig. 16. Let the number of rows /columns be α. Let the spatial dataset of size D be uniformly distirbuted. Each cell gets $P = \frac {D}{\alpha ^{2}}$ of the spatial data. Using lemma 1, we can see that the diagonal based replication schemes have a replication cost of O(α⁴ ∗ P). Using lemma 2, we can also see that the GEMS approach has communication cost of O(α³ ∗ P). As a consequence we can observe that the GEMS approach improves over diagonal based replication schemes by O(α) □

Fig. 16

Diagonal based replication schemes

Lemma 1

The communication cost of diagonal based replication schemes is O(α⁴ ∗ P)

Proof

The diagonal based replication schemes perform the replication along the diagonal of the grid structure. Let (i,i) be a cell on the diagonal where i is counted from top right corner of the grid structure as depicted in Fig. 16. Consider the replication cost of all the cells of the grid structure which are on i’th row to the right of cell (i,i). Let j be the index varying from 1 to i-1 where j indicates the position of the cell counted from the right end of the grid structure. The replication cost of j’th cell on the i’th row is j*i*P as marked by the hashed vertical rectangle. Here, P is the amount of data to be replicated. The replication cost of all the cells on the i’th row to the right of cell (i,i) is ${\sum }_{j=1}^{i-1} i*j*P$. The replication cost of cells on the i’th column which are on the top of cell (i,i) can be similarly computed as ${\sum }_{j=1}^{i-1} i*j*P$. The replication cost (C_d) of all the cells in the grid structure can be computed by varying the position of i from 1 to α. Such computation is summarized as:

$$ \begin{array}{@{}rcl@{}} C_{d} &=& \left( {\sum}_{i=1}^{\alpha} (i^{2} + {\sum}_{j=1}^{i-1} 2*i*j) \right)* P \\ &=& \left( {\sum}_{i=1}^{\alpha} (i^{2} + 2 * i* {\sum}_{j=1}^{i-1} j) \right)* P \\ &=& \left( {\sum}_{i=1}^{\alpha} i^{2} + 2 * i* i*(i-1)/2 \right) * P \\ &=& \left( {\sum}_{i=1}^{\alpha} i^{2} + i^{3} - i^{2} \right) * P \\ &=& \left( {\sum}_{i=1}^{\alpha} i^{3} \right) * P \\ &=& \left( \frac{\alpha*(\alpha+1)}{2}\right)^{2} * P = O(\alpha^{4}*P) \end{array} $$

(2)

The first term on right hand side of the Eq. 2 refers to the communication cost of cells on the diagonal of the grid structure which is (i² ∗ P). □

Lemma 2

The communication cost of GEMS based replication scheme is O(α³ ∗ P)

Proof

There are altogether α² cells in a grid of α rows/columns. If a cell is present in i’th row and j’th column of the grid (as marked in Fig. 17), the replication will be done to all the cells in i’th row and j’th column. Hence, each cell is replicated to 2 ∗ α − 1 cells using GEMS approach. So the total replication cost of GEMS approach is

$$ \begin{array}{@{}rcl@{}} C_{g} = \alpha^{2}*(2*\alpha-1)*P \end{array} $$

(3)

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Fig. 17

GEMS based replication