Thinking in SQL系列之一：棋盘摆放问题

原创： 牛超   2017-02-08   微信：ncfire     QQ10867910

说到SQL，很多朋友都认为无非就是增删改查，简单，不值得投入过多的精力。于是乎遇到复杂的数据处理，便借助PLSQL、T-SQL、JAVA等语言们来实现主体功能，逐渐出现弱化SQL的趋势。

本人从业以来，经历过大大小小的一些项目，而上述情况非常普遍。实在看不下去SQL被拆解得支离破碎的现状，个人感觉数据处理，SQL为王。

如何将问题转为SQL问题，即以SQL来思考问题就非常关键，这也是本文的目的。

SQL处理的对象是集合，即SET BY SET，区别于其它语言的ROW BY ROW甚至POINT BY POINT，处理能力以及性能不可等视。因此，如何合理的构造集合是SQL处理问题的关键。

用一个例子来说明：

在5X5的方格棋盘中（如图），每行、列、斜线（斜线不仅仅包括对角线）最多可以放两个球，如何摆放才能放置最多的球，这样的摆法总共有几种？输出所有的摆法。 

要求： 用一句SQL实现。

输出格式： 从方格棋盘第一行至第5行，每行从第一列到第5列依次输出，0表示不放球，1表示放球。例如：1001000000000000000000000。

一行输出一个行号和一个解，按解所在的列字符串顺序从大到小排序。

个人的环境是ORACLE，因不能使用PLSQL和自定义函数，既然构造集合是关键，那就头脑风暴一下：

1. 构造所有的单行集合

2. 摆列组合，为防止集合过大，可以提前行冲突检查

3. 冲突检查，包括纵向与斜线检查，斜线冲突采用移位法

解法一：最直接的方式，也是最容易想到的，但不灵活，不易扩展，

WITH BIN_LIST AS --构造单行所有可能的摆位
(  
SELECT 
CASE WHEN ROWNUM >= 16 THEN 1 ELSE 0 END ||
CASE WHEN MOD(ROWNUM,16) >= 8 THEN 1 ELSE 0 END ||  
CASE WHEN MOD(MOD(ROWNUM,16),8) >= 4 THEN 1 ELSE 0 END ||  
CASE WHEN MOD(MOD(MOD(ROWNUM,16),8), 4) >= 2 THEN 1 ELSE 0 END || 
MOD(ROWNUM,2) BIN
FROM DUAL
CONNECT BY ROWNUM <= POWER(2,5)-1
) ,
LINES AS (--最多可以放两个球--
SELECT BIN , ROWNUM NUM --BIN的索引号--
FROM BIN_LIST
WHERE REGEXP_COUNT(BIN,'1') <= 2
),
LINES_MOVE AS --移位准备--
 (SELECT LINES.*, ROWNUM - (NUM - 1) * 5 RNUM
    FROM LINES 
   START WITH NUM = 1
  CONNECT BY ROWNUM <= NUM * 5),
BIN_NUM_ALL AS --移位后集合，用于斜线冲突计算--
 (SELECT NUM,
         BIN SBIN,--原BIN--
         RPAD(BIN, 5 + (RNUM-1), '0') BINL, --左移，升序补0--
         RPAD(BIN, 5 + 4 - (RNUM-1), '0') BINR, --右移，降序补0--
         RNUM --1~5移位--
    FROM LINES_MOVE 
   ORDER BY BIN, RNUM) ,
REPORT AS (
SELECT LN1.BIN BIN1,
       LN1.NUM NUM1,
       LN2.BIN BIN2,
       LN2.NUM NUM2,
       LN3.BIN BIN3,
       LN3.NUM NUM3,
       LN4.BIN BIN3,
       LN4.NUM NUM3,
       LN5.BIN BIN3,
       LN5.NUM NUM3,
       LN1.BIN || LN2.BIN || LN3.BIN || LN4.BIN || LN5.BIN BIN_LIST,
       TO_NUMBER(LN1.BIN) + TO_NUMBER(LN2.BIN) + TO_NUMBER(LN3.BIN) + TO_NUMBER(LN4.BIN) + TO_NUMBER(LN5.BIN) SUM_VERTICAL, --纵向合计--
       TO_NUMBER(MLN1.BINL) + TO_NUMBER(MLN2.BINL) + TO_NUMBER(MLN3.BINL) + TO_NUMBER(MLN4.BINL) + TO_NUMBER(MLN5.BINL) SUM_LEFT, --斜角 左下->右上 合计--
       TO_NUMBER(MLN1.BINR) + TO_NUMBER(MLN2.BINR) + TO_NUMBER(MLN3.BINR) + TO_NUMBER(MLN4.BINR) + TO_NUMBER(MLN5.BINR) SUM_RIGHT --斜角 左上->右下 合计--
  FROM LINES LN1 --L1 
  JOIN BIN_NUM_ALL MLN1
    ON MLN1.NUM = LN1.NUM
   AND MLN1.RNUM = 1
 CROSS JOIN LINES LN2 --L2 
  JOIN BIN_NUM_ALL MLN2
    ON MLN2.NUM = LN2.NUM
   AND MLN2.RNUM = 2
  JOIN LINES LN3 --L3
    ON TO_CHAR(TO_NUMBER(LN1.BIN) + TO_NUMBER(LN2.BIN) + TO_NUMBER(LN3.BIN)) NOT LIKE '%3%'--纵向冲突检查--
  JOIN BIN_NUM_ALL MLN3
    ON MLN3.NUM = LN3.NUM
   AND MLN3.RNUM = 3
   AND TO_CHAR(TO_NUMBER(MLN1.BINL) + TO_NUMBER(MLN2.BINL) + TO_NUMBER(MLN3.BINL)) NOT LIKE '%3%'--斜向冲突检查--
   AND TO_CHAR(TO_NUMBER(MLN1.BINR) + TO_NUMBER(MLN2.BINR) + TO_NUMBER(MLN3.BINR)) NOT LIKE '%3%'--斜向冲突检查--
  JOIN LINES LN4 --L4
    ON TO_CHAR(TO_NUMBER(LN1.BIN) + TO_NUMBER(LN2.BIN) + TO_NUMBER(LN3.BIN) + TO_NUMBER(LN4.BIN)) NOT LIKE '%3%'
  JOIN BIN_NUM_ALL MLN4
    ON MLN4.NUM = LN4.NUM
   AND MLN4.RNUM = 4
   AND TO_CHAR(TO_NUMBER(MLN1.BINL) + TO_NUMBER(MLN2.BINL) + TO_NUMBER(MLN3.BINL) + TO_NUMBER(MLN4.BINL)) NOT LIKE '%3%'
   AND TO_CHAR(TO_NUMBER(MLN1.BINR) + TO_NUMBER(MLN2.BINR) + TO_NUMBER(MLN3.BINR) + TO_NUMBER(MLN4.BINR)) NOT LIKE '%3%'
  JOIN LINES LN5 --L5
    ON TO_CHAR(TO_NUMBER(LN1.BIN) + TO_NUMBER(LN2.BIN) + TO_NUMBER(LN3.BIN) + TO_NUMBER(LN4.BIN) + TO_NUMBER(LN5.BIN)) NOT LIKE '%3%'
  JOIN BIN_NUM_ALL MLN5
    ON MLN5.NUM = LN5.NUM
   AND MLN5.RNUM = 5
   AND TO_CHAR(TO_NUMBER(MLN1.BINL) + TO_NUMBER(MLN2.BINL) + TO_NUMBER(MLN3.BINL) + TO_NUMBER(MLN4.BINL) + TO_NUMBER(MLN5.BINL)) NOT LIKE '%3%'
   AND TO_CHAR(TO_NUMBER(MLN1.BINR) + TO_NUMBER(MLN2.BINR) + TO_NUMBER(MLN3.BINR) + TO_NUMBER(MLN4.BINR) + TO_NUMBER(MLN5.BINR)) NOT LIKE '%3%'
)
SELECT ROWNUM, BIN_LIST
FROM (
SELECT BIN_LIST ,RANK() OVER(ORDER BY REGEXP_COUNT(BIN_LIST,'1') DESC ) RK
FROM REPORT ORDER BY BIN_LIST
)
WHERE RK = 1 ;--取摆球最多的集合--

解法二：通过对功能重新梳理，可以采用正则表达式与标准的XML函数来实现简单的表达式累加运算，同时引入参数化，灵活易扩展，但速度非常慢（因排列组合生成的笛卡尔集无法及时过滤无用行）

WITH
PARAMS AS (--参数化
SELECT 2 BOLLS_LIMIT ,
5 MATRIX_LINE_LIMIT
FROM DUAL 
) ,
TESTM AS (
SELECT RPAD('1',BOLLS_LIMIT,'1') EN,
RPAD( RPAD('1',BOLLS_LIMIT,'1'),MATRIX_LINE_LIMIT,'0') LN1 , BOLLS_LIMIT, MATRIX_LINE_LIMIT
FROM PARAMS
) , TA AS (
SELECT ROWNUM RNUM , EN,
LN1, 
BOLLS_LIMIT,
MATRIX_LINE_LIMIT,
LPAD(LN1,LENGTH(LN1)+BOLLS_LIMIT*(ROWNUM-1),'0') LNS,
MOD(BOLLS_LIMIT * (ROWNUM-1),MATRIX_LINE_LIMIT) LIDX
FROM TESTM
CONNECT BY ROWNUM <= MATRIX_LINE_LIMIT
) , TB AS (
SELECT BOLLS_LIMIT, MATRIX_LINE_LIMIT , LISTAGG(
RPAD( RPAD(SUBSTR( LPAD( LN1,LENGTH(LN1)+LIDX), MATRIX_LINE_LIMIT+1 , BOLLS_LIMIT),LIDX,'0') || EN ,MATRIX_LINE_LIMIT,'0')
) WITHIN GROUP (ORDER BY RNUM) LN
FROM TA
GROUP BY BOLLS_LIMIT, MATRIX_LINE_LIMIT 
), 
TEST_RESULT AS (--提前测试计算出每行的最大球数
SELECT LN , REGEXP_COUNT(LN,'1') F_CNT ,
BOLLS_LIMIT, MATRIX_LINE_LIMIT , 
REGEXP_COUNT(LN,'1') / MATRIX_LINE_LIMIT  BOLLS_LINE_MAX
FROM TB
) ,
BIN_LIST AS ( --构造摆放集合，换一种方式
SELECT REPLACE(SYS_CONNECT_BY_PATH(BIT_NUM,','),',') BIN ,BOLLS_LIMIT , MATRIX_LINE_LIMIT,BOLLS_LINE_MAX
FROM (
SELECT ROWNUM -1  BIT_NUM , BOLLS_LIMIT , MATRIX_LINE_LIMIT,BOLLS_LINE_MAX
FROM TEST_RESULT 
CONNECT BY ROWNUM <= 2
)
WHERE LEVEL = MATRIX_LINE_LIMIT
CONNECT BY LEVEL <= MATRIX_LINE_LIMIT
) ,
LINES AS ( --根据测试的最大球数构造行集合--
SELECT /*+MATERIALIZE*/
BIN_LIST.BIN , ROWNUM NUM, --BIN的索引号--
BOLLS_LIMIT , MATRIX_LINE_LIMIT
FROM BIN_LIST 
WHERE REGEXP_COUNT(BIN_LIST.BIN,'1') = BOLLS_LINE_MAX
)  ,
BIN_NUM_ALL AS(--构造所有可能出现行的组合
SELECT 
REPLACE( SYS_CONNECT_BY_PATH( LINES.BIN,',') ,',') BIN_CONCAT,
SYS_CONNECT_BY_PATH( LINES.BIN,' 1#') BIN_EXP,
SYS_CONNECT_BY_PATH( RPAD(BIN, MATRIX_LINE_LIMIT + (LEVEL-1), '0') ,' 2#' ) BINL_EXP, --左移，升序补0--
SYS_CONNECT_BY_PATH( RPAD(BIN, MATRIX_LINE_LIMIT + (MATRIX_LINE_LIMIT - 1) - (LEVEL-1), '0') ,' 3#' ) BINR_EXP, --右移，降序补0--
'['|| (BOLLS_LIMIT + 1)||'-9]' REG_EXP 
FROM LINES
WHERE LEVEL = MATRIX_LINE_LIMIT
CONNECT BY LEVEL <= MATRIX_LINE_LIMIT
) ,
RESULT1 AS (
SELECT BXC.*
,
(SELECT LISTAGG(SUM(NUM)) WITHIN GROUP(ORDER BY 1)
  FROM XMLTABLE('DATA/ROW' PASSING
                XMLTYPE('' ||
                        REPLACE(REGEXP_REPLACE(BIN_EXP || ',' || BINL_EXP || ',' ||
                                               BINR_EXP,
                                               '([[:digit:]]+)#([[:digit:]]+)',
                                               '\1\2'),
                                ',') || '') COLUMNS ID NUMBER PATH 'ID',
                NUM NUMBER PATH 'NUM')
 GROUP BY ID ) SUM_NUM_LIST --利用正则表达式转成XML后解析求和，但速度不理想--
FROM BIN_NUM_ALL BXC
) ,
RESULT2 AS (
SELECT BIN_CONCAT ,REGEXP_COUNT(BIN_CONCAT,'1') NUM_COUNT ,SUM_NUM_LIST
FROM RESULT1
WHERE NOT REGEXP_LIKE(SUM_NUM_LIST,REG_EXP) --过滤掉不满足条件的
)
SELECT * FROM RESULT2 A;

解法三：参数化，灵活易扩展，速度理想，利用11G RSF特性，终级方案

WITH PARAMS AS --参数化
 (SELECT 2 BOLLS_LIMIT, 5 MATRIX_LINE_LIMIT FROM DUAL),
BIN_LIST AS --构造单行所有可能的摆位
 (SELECT REPLACE(SYS_CONNECT_BY_PATH(BIT_NUM, ','), ',') BIN
    FROM (SELECT ROWNUM - 1 BIT_NUM, MATRIX_LINE_LIMIT
            FROM PARAMS
          CONNECT BY ROWNUM <= 2)
   WHERE LEVEL = MATRIX_LINE_LIMIT
  CONNECT BY LEVEL <= MATRIX_LINE_LIMIT) ,
LINES AS --最多可以放两个球--
 (SELECT /*+MATERIALIZE*/
   BIN_LIST.BIN, ROWNUM NUM, BOLLS_LIMIT, MATRIX_LINE_LIMIT --BIN的索引号--
    FROM BIN_LIST
    JOIN PARAMS
      ON REGEXP_COUNT(BIN_LIST.BIN, '1') BETWEEN 1 AND BOLLS_LIMIT),
LINES_REV(BIN,
          NUM,
          BOLLS_LIMIT,
          MATRIX_LINE_LIMIT,
          LVL,
          BIN_LIST,
          BIN_SUM1,
          BIN_SUM2,
          BIN_SUM3) AS --关键处理，WITH递归构造行间排列组合
 (SELECT BIN,
         NUM,
         BOLLS_LIMIT,
         MATRIX_LINE_LIMIT,
         1 LVL,
         BIN BIN_LIST,
         TO_NUMBER(BIN) BIN_SUM1,
         TO_NUMBER(RPAD(BIN, MATRIX_LINE_LIMIT + 0, '0')) BIN_SUM2,
         TO_NUMBER(RPAD(BIN,
                        MATRIX_LINE_LIMIT + (MATRIX_LINE_LIMIT - 1) - 0,
                        '0')) BIN_SUM3
    FROM LINES
  UNION ALL
  SELECT LN.BIN,
         LN.NUM,
         LN.BOLLS_LIMIT,
         LN.MATRIX_LINE_LIMIT,
         LNR.LVL + 1 LVL,
         LNR.BIN_LIST || LN.BIN BIN_LIST,
         LNR.BIN_SUM1 + TO_NUMBER(LN.BIN) BIN_SUM1,
         LNR.BIN_SUM2 +
         TO_NUMBER(RPAD(LN.BIN, LNR.MATRIX_LINE_LIMIT + LNR.LVL, '0')) BIN_SUM2,
         LNR.BIN_SUM3 + TO_NUMBER(RPAD(LN.BIN,
                                       LNR.MATRIX_LINE_LIMIT +
                                       (LNR.MATRIX_LINE_LIMIT - 1) - LNR.LVL,
                                       '0')) BIN_SUM3
    FROM LINES_REV LNR
    JOIN LINES LN
      ON LNR.LVL < LNR.MATRIX_LINE_LIMIT
     AND TO_CHAR(LNR.BIN_SUM1 + TO_NUMBER(LN.BIN)) NOT LIKE
         '%' || TO_CHAR(LN.BOLLS_LIMIT + 1) || '%'
     AND LNR.BIN_SUM2 +
         TO_NUMBER(RPAD(LN.BIN, LNR.MATRIX_LINE_LIMIT + LNR.LVL, '0')) NOT LIKE
         '%' || TO_CHAR(LN.BOLLS_LIMIT + 1) || '%'
     AND LNR.BIN_SUM3 + TO_NUMBER(RPAD(LN.BIN,
                                       LNR.MATRIX_LINE_LIMIT +
                                       (LNR.MATRIX_LINE_LIMIT - 1) - LNR.LVL,
                                       '0')) NOT LIKE
         '%' || TO_CHAR(LN.BOLLS_LIMIT + 1) || '%') --及时进行冲突检查收敛集合达到有效提速的目的
,
RESULT AS
 (SELECT T.*, RANK() OVER(ORDER BY BIN_SUM1 DESC) RK
    FROM LINES_REV T
   WHERE T.LVL = MATRIX_LINE_LIMIT)
SELECT * FROM RESULT WHERE RK = 1;

至此，这个棋盘摆放问题的几种实现方法已经介绍完，通过循序渐进的总结修正，不断地深入使用与理解SQL，特别是第三种解法，ORACLE 11G的RSF特性使SQL具有了递归的能力。

总之，Thinking in SQL，各类复杂数据处理问题都会迎刃而解。