postgresql之integerset

一、简介

1.1 integerset是什么

正如其名，integer set, 整数集合。这里是指存储整数的数据结构。对于数学中集合的定义，集合中的元素是没有重复的。

integerset特点：

• 能存储64bit整数

• 内存数据结构，内存操作速度快

• 使用B-tree结构组织数据，插入、搜索 复杂度O(log2)

• 使用Simple-8b算法压缩数据，使存储空间更小，还能使用SIMD等指令来加速

1.2 能用来干什么

• 能存储最大64bit的任意正整数
• 快速检索某个整数是否在集合中

1.3 有什么限制

• 插入的值必须是有序的
• 在遍历集合的时候不能插入值
• 不支持删除值
• 不支持负数

1.4 对比redis的intset

redis之intset

数据结构	可存储最大值	组织形式	搜索方式	存储方式	支持负数	插入顺序
integerset	unsigned 64bit	B-tree	遍历树+折半搜索	simple-8b编码	No	必须有序
intset	signed 64bit	数组	折半搜索	直接存值	Yes	任意，自动排序

1.5 对比redis的listpack变长编码

redis之listpack

数据结构	数值编码方式	编码形式	编码长度影响因素
integerset	simple-8b编码	批量编码	多个连续数值之间的差值大小
listpack	变长字节编码	单数值编码	单个数值本身大小

二、 数据结构

2.1 数据结构定义

typedef struct intset_node intset_node;
typedef struct intset_leaf_node intset_leaf_node;
typedef struct intset_internal_node intset_internal_node;

/* Common structure of both leaf and internal nodes. */
struct intset_node
{
    
	uint16		level;			/* tree level of this node */
	uint16		num_items;		/* number of items in this node */
};

/* Internal node */
struct intset_internal_node
{
    
	/* common header, must match intset_node */
	uint16		level;			/* >= 1 on internal nodes */
	uint16		num_items;

	/* * 'values' is an array of key values, and 'downlinks' are pointers to * lower-level nodes, corresponding to the key values. */
	uint64		values[MAX_INTERNAL_ITEMS];
	intset_node *downlinks[MAX_INTERNAL_ITEMS];
};

/* Leaf node */
typedef struct
{
    
	uint64		first;			/* first integer in this item */
	uint64		codeword;		/* simple8b encoded differences from 'first' */
} leaf_item;

#define MAX_VALUES_PER_LEAF_ITEM (1 + SIMPLE8B_MAX_VALUES_PER_CODEWORD)

struct intset_leaf_node
{
    
	/* common header, must match intset_node */
	uint16		level;			/* 0 on leafs */
	uint16		num_items;

	intset_leaf_node *next;		/* right sibling, if any */

	leaf_item	items[MAX_LEAF_ITEMS];
};

/* * We buffer insertions in a simple array, before packing and inserting them * into the B-tree. MAX_BUFFERED_VALUES sets the size of the buffer. The * encoder assumes that it is large enough that we can always fill a leaf * item with buffered new items. In other words, MAX_BUFFERED_VALUES must be * larger than MAX_VALUES_PER_LEAF_ITEM. For efficiency, make it much larger. */
#define MAX_BUFFERED_VALUES (MAX_VALUES_PER_LEAF_ITEM * 2)

/* * IntegerSet is the top-level object representing the set. * * The integers are stored in an in-memory B-tree structure, plus an array * for newly-added integers. IntegerSet also tracks information about memory * usage, as well as the current position when iterating the set with * intset_begin_iterate / intset_iterate_next. */
struct IntegerSet
{
    
	/* * 'context' is the memory context holding this integer set and all its * tree nodes. * * 'mem_used' tracks the amount of memory used. We don't do anything with * it in integerset.c itself, but the callers can ask for it with * intset_memory_usage(). */
	MemoryContext context;
	uint64		mem_used;

	uint64		num_entries;	/* total # of values in the set */
	uint64		highest_value;	/* highest value stored in this set */

	/* * B-tree to hold the packed values. * * 'rightmost_nodes' hold pointers to the rightmost node on each level. * rightmost_parent[0] is rightmost leaf, rightmost_parent[1] is its * parent, and so forth, all the way up to the root. These are needed when * adding new values. (Currently, we require that new values are added at * the end.) */
	int			num_levels;		/* height of the tree */
	intset_node *root;			/* root node */
	intset_node *rightmost_nodes[MAX_TREE_LEVELS];
	intset_leaf_node *leftmost_leaf;	/* leftmost leaf node */

	/* * Holding area for new items that haven't been inserted to the tree yet. */
	uint64		buffered_values[MAX_BUFFERED_VALUES];
	int			num_buffered_values;

	/* * Iterator support. * * 'iter_values' is an array of integers ready to be returned to the * caller; 'iter_num_values' is the length of that array, and * 'iter_valueno' is the next index. 'iter_node' and 'iter_itemno' point * to the leaf node, and item within the leaf node, to get the next batch * of values from. * * Normally, 'iter_values' points to 'iter_values_buf', which holds items * decoded from a leaf item. But after we have scanned the whole B-tree, * we iterate through all the unbuffered values, too, by pointing * iter_values to 'buffered_values'. */
	bool		iter_active;	/* is iteration in progress? */

	const uint64 *iter_values;
	int			iter_num_values;	/* number of elements in 'iter_values' */
	int			iter_valueno;	/* next index into 'iter_values' */

	intset_leaf_node *iter_node;	/* current leaf node */
	int			iter_itemno;	/* next item in 'iter_node' to decode */

	uint64		iter_values_buf[MAX_VALUES_PER_LEAF_ITEM];
};

其中 intset_node可以看作是基类，其他继承它。

class intset_node {
    };
class intset_internal_node : intset_node {
    };
class intset_leaf_node : intset_node {
    };

2.2 结构图

三、 相关API

3.1 创建integerset

/* * Create a new, initially empty, integer set. * * The integer set is created in the current memory context. * We will do all subsequent allocations in the same context, too, regardless * of which memory context is current when new integers are added to the set. */
IntegerSet *
intset_create(void)
{
    
	IntegerSet *intset;

	intset = (IntegerSet *) palloc(sizeof(IntegerSet));
	intset->context = CurrentMemoryContext;
	intset->mem_used = GetMemoryChunkSpace(intset);

	intset->num_entries = 0;
	intset->highest_value = 0;

	intset->num_levels = 0;
	intset->root = NULL;
	memset(intset->rightmost_nodes, 0, sizeof(intset->rightmost_nodes));
	intset->leftmost_leaf = NULL;

	intset->num_buffered_values = 0;

	intset->iter_active = false;
	intset->iter_node = NULL;
	intset->iter_itemno = 0;
	intset->iter_valueno = 0;
	intset->iter_num_values = 0;
	intset->iter_values = NULL;

	return intset;
}

3.2 插入元素

/* * Add a value to the set. * * Values must be added in order. */
void
intset_add_member(IntegerSet *intset, uint64 x)
{
    
	if (intset->iter_active)
		elog(ERROR, "cannot add new values to integer set while iteration is in progress");

	if (x <= intset->highest_value && intset->num_entries > 0)
		elog(ERROR, "cannot add value to integer set out of order");

	if (intset->num_buffered_values >= MAX_BUFFERED_VALUES)
	{
    
		/* Time to flush our buffer */
		intset_flush_buffered_values(intset);
		Assert(intset->num_buffered_values < MAX_BUFFERED_VALUES);
	}

	/* Add it to the buffer of newly-added values */
	intset->buffered_values[intset->num_buffered_values] = x;
	intset->num_buffered_values++;
	intset->num_entries++;
	intset->highest_value = x;
}

• 首先会加入到缓存区中
• 等缓冲区满后，再插入B树中

(1) 为啥先插入缓冲区，而不是直接插入B-tree

• simple-8b编码方式决定的

• B-tree中叶子节点中存储的是通过simple-8b编码后的值, 而不是原始值。

(2) 什么是simple-8b编码

simple-8b, 其中b是byte, 使用8字节编码，每次编码后的结果都是8字节。

8byte, 64bit, 其中开始4bit用于选择使用编码模式，不同的编码模式，决定了后面的60bit的如何分配来存储数据。

static const struct simple8b_mode
{
    
	uint8		bits_per_int;
	uint8		num_ints;
}			simple8b_modes[17] =

{
    
	{
    0, 240},					/* mode 0: 240 zeroes */
	{
    0, 120},					/* mode 1: 120 zeroes */
	{
    1, 60},					/* mode 2: sixty 1-bit integers */
	{
    2, 30},					/* mode 3: thirty 2-bit integers */
	{
    3, 20},					/* mode 4: twenty 3-bit integers */
	{
    4, 15},					/* mode 5: fifteen 4-bit integers */
	{
    5, 12},					/* mode 6: twelve 5-bit integers */
	{
    6, 10},					/* mode 7: ten 6-bit integers */
	{
    7, 8},						/* mode 8: eight 7-bit integers (four bits * are wasted) */
	{
    8, 7},						/* mode 9: seven 8-bit integers (four bits * are wasted) */
	{
    10, 6},					/* mode 10: six 10-bit integers */
	{
    12, 5},					/* mode 11: five 12-bit integers */
	{
    15, 4},					/* mode 12: four 15-bit integers */
	{
    20, 3},					/* mode 13: three 20-bit integers */
	{
    30, 2},					/* mode 14: two 30-bit integers */
	{
    60, 1},					/* mode 15: one 60-bit integer */

	{
    0, 0}						/* sentinel value */
};

根据编码模式可以看出，数值越小，能编码的个数就越多，越节省空间。

那如何将需要编码的数值变小呢

对相邻两数差值进行编码，因此数据集越集中，差值越小，能编码的个数越多

只存储差值，如何恢复原始值呢?

为了能恢复原始值，因此编码分为两部分

• 基数base
• 后续数据差值编码

typedef struct
{
    
	uint64		first;		// base
	uint64		codeword;	 //编码结果
} leaf_item;

比如使用mode10, 则每个数值使用10bit，总共可以编码6个数值

1010 xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx xxxxxxxxxx

只有60 bit用来编码数据，那大于60bit的数值如何编码的呢？

对于大于60bit的数值，将数据存入base, 编码8byte就写UINT64CONST(0x0FFFFFFFFFFFFFFF)，编码数据长度0，相当于浪费了8byte。

a. simple-8b编码实现

/* * EMPTY_CODEWORD is a special value, used to indicate "no values". * It is used if the next value is too large to be encoded with Simple-8b. * * This value looks like a mode-0 codeword, but we can distinguish it * because a regular mode-0 codeword would have zeroes in the unused bits. */
#define EMPTY_CODEWORD UINT64CONST(0x0FFFFFFFFFFFFFFF)

/* * Encode a number of integers into a Simple-8b codeword. * * (What we actually encode are deltas between successive integers. * "base" is the value before ints[0].) * * The input array must contain at least SIMPLE8B_MAX_VALUES_PER_CODEWORD * elements, ensuring that we can produce a full codeword. * * Returns the encoded codeword, and sets *num_encoded to the number of * input integers that were encoded. That can be zero, if the first delta * is too large to be encoded. */
static uint64
simple8b_encode(const uint64 *ints, int *num_encoded, uint64 base)
{
    
	int			selector;
	int			nints;
	int			bits;
	uint64		diff;
	uint64		last_val;
	uint64		codeword;
	int			i;

	Assert(ints[0] > base);

	/* * Select the "mode" to use for this codeword. * * In each iteration, check if the next value can be represented in the * current mode we're considering. If it's too large, then step up the * mode to a wider one, and repeat. If it fits, move on to the next * integer. Repeat until the codeword is full, given the current mode. * * Note that we don't have any way to represent unused slots in the * codeword, so we require each codeword to be "full". It is always * possible to produce a full codeword unless the very first delta is too * large to be encoded. For example, if the first delta is small but the * second is too large to be encoded, we'll end up using the last "mode", * which has nints == 1. */
	selector = 0;
	nints = simple8b_modes[0].num_ints;
	bits = simple8b_modes[0].bits_per_int;
	diff = ints[0] - base - 1;
	last_val = ints[0];
	i = 0;						/* number of deltas we have accepted */
	for (;;)
	{
    
		if (diff >= (UINT64CONST(1) << bits))
		{
    
			/* too large, step up to next mode */
			selector++;
			nints = simple8b_modes[selector].num_ints;
			bits = simple8b_modes[selector].bits_per_int;
			/* we might already have accepted enough deltas for this mode */
			if (i >= nints)
				break;
		}
		else
		{
    
			/* accept this delta; then done if codeword is full */
			i++;
			if (i >= nints)
				break;
			/* examine next delta */
			Assert(ints[i] > last_val);
			diff = ints[i] - last_val - 1;
			last_val = ints[i];
		}
	}

	if (nints == 0)
	{
    
		/* * The first delta is too large to be encoded with Simple-8b. * * If there is at least one not-too-large integer in the input, we * will encode it using mode 15 (or a more compact mode). Hence, we * can only get here if the *first* delta is >= 2^60. */
		Assert(i == 0);
		*num_encoded = 0;
		return EMPTY_CODEWORD;
	}

	/* * Encode the integers using the selected mode. Note that we shift them * into the codeword in reverse order, so that they will come out in the * correct order in the decoder. */
	codeword = 0;
	if (bits > 0)
	{
    
		for (i = nints - 1; i > 0; i--)
		{
    
			diff = ints[i] - ints[i - 1] - 1;
			codeword |= diff;
			codeword <<= bits;
		}
		diff = ints[0] - base - 1;
		codeword |= diff;
	}

	/* add selector to the codeword, and return */
	codeword |= (uint64) selector << 60;

	*num_encoded = nints;
	return codeword;
}

b. simple-8b解码实现

将8byte的编码数据，恢复到一个数组中。

/* * Decode a codeword into an array of integers. * Returns the number of integers decoded. */
static int
simple8b_decode(uint64 codeword, uint64 *decoded, uint64 base)
{
    
	int			selector = (codeword >> 60);
	int			nints = simple8b_modes[selector].num_ints;
	int			bits = simple8b_modes[selector].bits_per_int;
	uint64		mask = (UINT64CONST(1) << bits) - 1;
	uint64		curr_value;

	if (codeword == EMPTY_CODEWORD)
		return 0;

	curr_value = base;
	for (int i = 0; i < nints; i++)
	{
    
		uint64		diff = codeword & mask;

		curr_value += 1 + diff;
		decoded[i] = curr_value;
		codeword >>= bits;
	}
	return nints;
}

(3) 比如插入1-600

• 首先是插入缓冲区

• 当缓冲区写满后，将进行编码以及插入B-tree

• 剩余数据继续插入缓冲区

每次编码结束后，如果缓冲区中还有剩余的数据，将会拷贝到缓冲区开头

我尝试将这个缓冲区改造成一个循环队列，这样就不用进行数据的拷贝了。但是对于一些使用缓冲区 的地方都要加求余操作，看起来就没有原来的代码更清晰简单，而且随着队列的使用，将会将数组分成两个部分，两个部分依然有序，还是使用折半搜索，一次尝试还是不错的，调试，当能正常工作后，还是挺高兴的

3.2 判断某个值是否在integerset中

• 首先判断是否在缓冲区中，缓冲区是有序递增的，所以使用折半搜索
• 判断值在哪个节点范围内，因为B-tree特性，也使用折半搜索
• 最后在simple-8b编码数据中搜索， simple8b_contains函数和simple8b_decode类似

/* * Does the set contain the given value? */
bool
intset_is_member(IntegerSet *intset, uint64 x)
{
    
	intset_node *node;
	intset_leaf_node *leaf;
	int			level;
	int			itemno;
	leaf_item  *item;

	/* * The value might be in the buffer of newly-added values. */
	if (intset->num_buffered_values > 0 && x >= intset->buffered_values[0])
	{
    
		int			itemno;

		itemno = intset_binsrch_uint64(x,
									   intset->buffered_values,
									   intset->num_buffered_values,
									   false);
		if (itemno >= intset->num_buffered_values)
			return false;
		else
			return (intset->buffered_values[itemno] == x);
	}

	/* * Start from the root, and walk down the B-tree to find the right leaf * node. */
	if (!intset->root)
		return false;
	node = intset->root;
	for (level = intset->num_levels - 1; level > 0; level--)
	{
    
		intset_internal_node *n = (intset_internal_node *) node;

		Assert(node->level == level);

		itemno = intset_binsrch_uint64(x, n->values, n->num_items, true);
		if (itemno == 0)
			return false;
		node = n->downlinks[itemno - 1];
	}
	Assert(node->level == 0);
	leaf = (intset_leaf_node *) node;

	/* * Binary search to find the right item on the leaf page */
	itemno = intset_binsrch_leaf(x, leaf->items, leaf->num_items, true);
	if (itemno == 0)
		return false;
	item = &leaf->items[itemno - 1];

	/* Is this a match to the first value on the item? */
	if (item->first == x)
		return true;
	Assert(x > item->first);

	/* Is it in the packed codeword? */
	if (simple8b_contains(item->codeword, x, item->first))
		return true;

	return false;
}

3.3 遍历整个integerset

/* * Begin in-order scan through all the values. * * While the iteration is in-progress, you cannot add new values to the set. */
void
intset_begin_iterate(IntegerSet *intset)
{
    
	/* Note that we allow an iteration to be abandoned midway */
	intset->iter_active = true;
	intset->iter_node = intset->leftmost_leaf;
	intset->iter_itemno = 0;
	intset->iter_valueno = 0;
	intset->iter_num_values = 0;
	intset->iter_values = intset->iter_values_buf;
}

/* * Returns the next integer, when iterating. * * intset_begin_iterate() must be called first. intset_iterate_next() returns * the next value in the set. Returns true, if there was another value, and * stores the value in *next. Otherwise, returns false. */
bool
intset_iterate_next(IntegerSet *intset, uint64 *next)
{
    
	Assert(intset->iter_active);
	for (;;)
	{
    
		/* Return next iter_values[] entry if any */
		if (intset->iter_valueno < intset->iter_num_values)
		{
    
			*next = intset->iter_values[intset->iter_valueno++];
			return true;
		}

		/* Decode next item in current leaf node, if any */
		if (intset->iter_node &&
			intset->iter_itemno < intset->iter_node->num_items)
		{
    
			leaf_item  *item;
			int			num_decoded;

			item = &intset->iter_node->items[intset->iter_itemno++];

			intset->iter_values_buf[0] = item->first;
			num_decoded = simple8b_decode(item->codeword,
										  &intset->iter_values_buf[1],
										  item->first);
			intset->iter_num_values = num_decoded + 1;
			intset->iter_valueno = 0;
			continue;
		}

		/* No more items on this leaf, step to next node */
		if (intset->iter_node)
		{
    
			intset->iter_node = intset->iter_node->next;
			intset->iter_itemno = 0;
			continue;
		}

		/* * We have reached the end of the B-tree. But we might still have * some integers in the buffer of newly-added values. */
		if (intset->iter_values == (const uint64 *) intset->iter_values_buf)
		{
    
			intset->iter_values = intset->buffered_values;
			intset->iter_num_values = intset->num_buffered_values;
			intset->iter_valueno = 0;
			continue;
		}

		break;
	}

	/* No more results. */
	intset->iter_active = false;
	*next = 0;					/* prevent uninitialized-variable warnings */
	return false;
}

• 从最左边的叶子节点开始遍历，叶子节点通过next指针连接成了一个链表，逐一遍历

• 每个叶子节点中有多个item, 逐一遍历item

• 每个item又是经过simple-8b编码的数据，通过解码，将数据写入临时缓冲区中

• 每次都从临时缓冲区中返回一个数据

• 当遍历完所有节点后，进行遍历缓冲区

四、总结

• integerset是一个正整数集合，最大值为64bit

• 使用B-tree结构进行管理数据

• 使用simple-8b编码数据，进行压缩

• 插入顺序必须有序

• 不支持删除数据

• 数据压缩率依赖数据之间的差距

• 迭代遍历过程中不能插入数据，未实现

• 搜索过程都使用了折半搜索

• 批量编码数据

待把B-tree学习清楚后，可以尝试实现非有序方式插入