## Monday, 9 May 2016

### Play the game of life with SAP HANA

What is the game of life?
So first of all, what is the game of life? You can find rules of the game of life from Wikipedia as follows.

"The universe of the Game of Life is an infinite two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, alive or dead. Every cell interacts with its eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

1. Any live cell with fewer than two live neighbours dies, as if caused by under-population.
2. Any live cell with two or three live neighbours lives on to the next generation.
3. Any live cell with more than three live neighbours dies, as if by overcrowding.
4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.
The initial pattern constitutes the seed of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed—births and deaths occur simultaneously, and the discrete moment at which this happens is sometimes called a tick (in other words, each generation is a pure function of the preceding one). The rules continue to be applied repeatedly to create further generations."

Motivation
I still remember when I studied at TU Berlin, I had a course named parallel programming. There was an assignment about the game of life in which we need to implement it with MPI. That's the first time when I heard of the game of life. Since parallelism is one of the powerful weapons owned by SAP HANA, an idea came to my mind, why not play the game of life with SAP HANA? Let's give it a shot.

Problems
Consider the following simple 3x3 initial pattern with live cells shown in black and dead cells shown in white.

According the above rules, the first generation will look like as follows. You can do an exercise yourselves.

Now let's start to play the game of life with SAP HANA. The first thing we need to do is modeling the problem in SAP HANA. For the two-dimensional grid, we can create two axes, the horizontal x-axis and the vertical y-axis. For two statuses, we can use 1 for live and 0 for dead. So, the initial pattern will be displayed as below.

And it can be implemented with the following SQL statements.

CREATE COLUMN TABLE LIFE (
X INTEGER, -- x-axis
Y INTEGER, -- y-axis
S INTEGER, -- status
PRIMARY KEY (X, Y)
);
INSERT INTO LIFE VALUES (1, 1, 0);
INSERT INTO LIFE VALUES (1, 2, 0);
INSERT INTO LIFE VALUES (1, 3, 1);
INSERT INTO LIFE VALUES (2, 1, 1);
INSERT INTO LIFE VALUES (2, 2, 1);
INSERT INTO LIFE VALUES (2, 3, 0);
INSERT INTO LIFE VALUES (3, 1, 0);
INSERT INTO LIFE VALUES (3, 2, 1);
INSERT INTO LIFE VALUES (3, 3, 1);

The problem comes now. How can we create the first generation and more further generations repeatedly? Assume the following two points.

1. We always want to update the "LIFE" table itself instead of creating new tables.
2. We do not care the order of tuples.

Play with self-join
First of all, we may figure out the following approach/steps for this problem.

1. Calculate # of live neighbours for each cell
SELECT A.X, A.Y, A.S, SUM(B.S) - A.S N
FROM LIFE A INNER JOIN LIFE B ON ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2
GROUP BY A.X, A.Y, A.S;

The logic is very simple. We use the self-join approach.

1. We use "ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2"  to filter neighbours plus the cell itself.
2. We use "SUM(B.S) - A.S" to calculate # of live neighbours for each cell.

Since we do not care the order of tuples, the results seem unordered. You can check the correctness of the results manually as shown below. The number in the brackets means # of live neighbours for each cell.

There are also several similar alternatives with self-join, e.g.,

Alternative 1
SELECT A.X, A.Y, A.S, SUM(B.S) N
FROM LIFE A INNER JOIN LIFE B ON ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2 AND (A.X <> B.X OR A.Y <> B.Y)
GROUP BY A.X, A.Y, A.S;

Alternative 2
SELECT A.X, A.Y, A.S, SUM(B.S) N
FROM LIFE A INNER JOIN LIFE B ON (A.X = B.X AND ABS(A.Y - B.Y) = 1) OR (A.Y = B.Y AND ABS(A.X - B.X) = 1) OR (ABS(A.X - B.X) = 1 AND ABS(A.Y - B.Y) = 1)
GROUP BY A.X, A.Y, A.S;

2. Apply the rules to get the results of the next generation
SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT A.X, A.Y, A.S, SUM(B.S) - A.S N
FROM LIFE A INNER JOIN LIFE B ON ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2
GROUP BY A.X, A.Y, A.S);

Based on the results of step 1, we apply the following simplified rules to calculate the next generation.

1) The current status of the cell is dead
If there are exactly three live neighbours, the next status will be live; Otherwise, the next status will be still dead.

2) The current status of the cell is live
If there are two or three live neighbours, the next status will be still live; Otherwise, the next status will be dead.

You can also check the results manually.

3. Update the "LIFE" table with the results in step 2
UPDATE LIFE SET S = NEXTGEN.S FROM LIFE,
(SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT A.X, A.Y, A.S, SUM(B.S) - A.S N
FROM LIFE A INNER JOIN LIFE B ON ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2
GROUP BY A.X, A.Y, A.S)) NEXTGEN
WHERE LIFE.X = NEXTGEN.X AND LIFE.Y = NEXTGEN.Y;

Perhaps you are not familiar with the special syntax of "UPDATE" here, don't worry. You can find the example at the bottom of UPDATE - SAP HANA SQL and System Views Reference - SAP Library. So far, we've created the first generation successfully.

You can repeat step 3 to play further.

Play with window function
Now let's play with window function. First of all, in order to compare with the self-join approach, we need to reset the "LIFE" table to the same initial pattern.

1. Calculate # of live neighbours for each cell
In the window function approach, we have two sub-steps as follows.

1) Calculate # of live "vertical" neighbours for each cell
SELECT X, Y, S, (S + IFNULL(LEAD(S) OVER (PARTITION BY X ORDER BY Y), 0) + IFNULL(LAG(S) OVER (PARTITION BY X ORDER BY Y), 0)) N FROM LIFE;

In this sub-step, we just calculate N(North) + C(Center) + S(South) for each C(Center). We partition the "LIFE" table by X vertically

For instance, # of live "vertical" neighbours of cell (2, 2) is 2.

2) Calculate # of live neighbours for each cell
Based on sub-step 1), we can calculate the final result by partitioning the "LIFE" table horizontally. In this sub-step, we partition the table by Y.

SELECT X, Y, S, (N + IFNULL(LEAD(N) OVER (PARTITION BY Y ORDER BY X), 0) + IFNULL(LAG(N) OVER (PARTITION BY Y ORDER BY X), 0) - S) N
FROM (SELECT X, Y, S, (S + IFNULL(LEAD(S) OVER (PARTITION BY X ORDER BY Y), 0) + IFNULL(LAG(S) OVER (PARTITION BY X ORDER BY Y), 0)) N FROM LIFE);

2. Apply the rules to get the results of the next generation
SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT X, Y, S, (N + IFNULL(LEAD(N) OVER (PARTITION BY Y ORDER BY X), 0) + IFNULL(LAG(N) OVER (PARTITION BY Y ORDER BY X), 0) - S) N
FROM (SELECT X, Y, S, (S + IFNULL(LEAD(S) OVER (PARTITION BY X ORDER BY Y), 0) + IFNULL(LAG(S) OVER (PARTITION BY X ORDER BY Y), 0)) N FROM LIFE));

3. Update the "LIFE" table with the results in step 2
UPDATE LIFE SET S = NEXTGEN.S FROM LIFE,
(SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT X, Y, S, (N + IFNULL(LEAD(N) OVER (PARTITION BY Y ORDER BY X), 0) + IFNULL(LAG(N) OVER (PARTITION BY Y ORDER BY X), 0) - S) N
FROM (SELECT X, Y, S, (S + IFNULL(LEAD(S) OVER (PARTITION BY X ORDER BY Y), 0) + IFNULL(LAG(S) OVER (PARTITION BY X ORDER BY Y), 0)) N FROM LIFE))) NEXTGEN
WHERE LIFE.X = NEXTGEN.X AND LIFE.Y = NEXTGEN.Y;

Self-join vs. window function
Since we have only played the very small 3x3 game of life with SAP HANA, we cannot compare the performance between self-join and window function. In order to compare the performance, we need to generate a bigger grid. We can first create a stored procedure which enables us to generate a NxN grid.

CREATE PROCEDURE GENERATE_LIFE(IN X INTEGER) LANGUAGE SQLSCRIPT AS
i INTEGER;
j INTEGER;
BEGIN
DELETE FROM LIFE;
FOR i IN 1 .. X DO
FOR j IN 1 .. X DO
INSERT INTO LIFE VALUES (i, j, ROUND(RAND()));
END FOR;
END FOR;
END;

Then we can call the above stored procedure to generate the initial pattern, e.g., a 100x100 grid.

CALL GENERATE_LIFE(100);

Now let's do some comparisons. Here we just compare step 2 between self-join and window function since step 3 is just the update operation. Hence, we compare the following two SELECT statements.

Self-join
SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT A.X, A.Y, A.S, SUM(B.S) - A.S N
FROM LIFE A INNER JOIN LIFE B ON ABS(A.X - B.X) < 2 AND ABS(A.Y - B.Y) < 2
GROUP BY A.X, A.Y, A.S);

Window function
SELECT X, Y, CASE S WHEN 0 THEN (CASE N WHEN 3 THEN 1 ELSE 0 END) ELSE (CASE N WHEN 2 THEN 1 WHEN 3 THEN 1 ELSE 0 END) END S
FROM (SELECT X, Y, S, (N + IFNULL(LEAD(N) OVER (PARTITION BY Y ORDER BY X), 0) + IFNULL(LAG(N) OVER (PARTITION BY Y ORDER BY X), 0) - S) N
FROM (SELECT X, Y, S, (S + IFNULL(LEAD(S) OVER (PARTITION BY X ORDER BY Y), 0) + IFNULL(LAG(S) OVER (PARTITION BY X ORDER BY Y), 0)) N FROM LIFE));
• Test environment:
• SAP HANA SPS08 Rev. 80
• CPU: 8 cores
• Memory: 64GB
• Disk: 200GB
 N=100 N=400 N = 100 N = 400 Time with self-join J~40s ~1400s Time with window function ~30ms ~120ms # of cells 100 * 100 = 10,000 400 * 400 = 160,000
From the above table, we can find the following results:

1. The performance of window function is much better than self-join.
2. The time with window function seems a linear function of N.
3. The time with self-join seems a exponential growth with N.

We can find some reasons from the following two visualization plans. It takes about 99.99% time for the large column search node in self-join to make the non equi join and do the filter. And the join result has 25,600,000,000 rows! Meanwhile, the visualization plan of window function shows that two sequential window nodes just need to consume # of cells.
Visualization plan of self-join (N = 400)

Visualization plan of window function (N = 400)

Source: scn.sap.com