Implementation notes: sparse_hash, dense_hash, sparsetable

# Implementation of sparse_hash_map, dense_hash_map, and sparsetable

This document contains a few notes on how the data structures in this package are implemented. This discussion refers at several points to the classic text in this area: Knuth, The Art of Computer Programming, Vol 3, Hashing.

## `sparsetable`

For specificity, consider the declaration

       sparsetable<Foo> t(100);        // a sparse array with 100 elements

A sparsetable is a random container that implements a sparse array, that is, an array that uses very little memory to store unassigned indices (in this case, between 1-2 bits per unassigned index). For instance, if you allocate an array of size 5 and assign a[2] = [big struct], then a[2] will take up a lot of memory but a[0], a[1], a[3], and a[4] will not. Array elements that have a value are called "assigned". Array elements that have no value yet, or have had their value cleared using erase() or clear(), are called "unassigned". For assigned elements, lookups return the assigned value; for unassigned elements, they return the default value, which for t is Foo().

sparsetable is implemented as an array of "groups". Each group is responsible for M array indices. The first group knows about t[0]..t[M-1], the second about t[M]..t[2M-1], and so forth. (M is 48 by default.) At construct time, t creates an array of (99/M + 1) groups. From this point on, all operations -- insert, delete, lookup -- are passed to the appropriate group. In particular, any operation on t[i] is actually performed on (t.group[i / M])[i % M].

Each group contains of a vector, which holds assigned values, and a bitmap of size M, which indicates which indices are assigned. A lookup works as follows: the group is asked to look up index i, where i < M. The group looks at bitmap[i]. If it's 0, the lookup fails. If it's 1, then the group has to find the appropriate value in the vector.

### `find()`

Finding the appropriate vector element is the most expensive part of the lookup. The code counts all bitmap entries <= i that are set to 1. (There's at least 1 of them, since bitmap[i] is 1.) Suppose there are 4 such entries. Then the right value to return is the 4th element of the vector: vector[3]. This takes time O(M), which is a constant since M is a constant.

### `insert()`

Insert starts with a lookup. If the lookup succeeds, the code merely replaces vector[3] with the new value. If the lookup fails, then the code must insert a new entry into the middle of the vector. Again, to insert at position i, the code must count all the bitmap entries <= i that are set to 1. This indicates the position to insert into the vector. All vector entries above that position must be moved to make room for the new entry. This takes time, but still constant time since the vector has size at most M.

(Inserts could be made faster by using a list instead of a vector to hold group values, but this would use much more memory, since each list element requires a full pointer of overhead.)

The only metadata that needs to be updated, after the actual value is inserted, is to set bitmap[i] to 1. No other counts must be maintained.

### `delete()`

Deletes are similar to inserts. They start with a lookup. If it fails, the delete is a noop. Otherwise, the appropriate entry is removed from the vector, all the vector elements above it are moved down one, and bitmap[i] is set to 0.

### iterators

Sparsetable iterators pose a special burden. They must iterate over unassigned array values, but the act of iterating should not cause an assignment to happen -- otherwise, iterating over a sparsetable would cause it to take up much more room. For const iterators, the matter is simple: the iterator is merely programmed to return the default value -- Foo() -- when dereferenced while pointing to an unassigned entry.

For non-const iterators, such simple techniques fail. Instead, dereferencing a sparsetable_iterator returns an opaque object that acts like a Foo in almost all situations, but isn't actually a Foo. (It does this by defining operator=(), operator value_type(), and, most sneakily, operator&().) This works in almost all cases. If it doesn't, an explicit cast to value_type will solve the problem:

       printf("%d", static_cast<Foo>(*t.find(0)));

To avoid such problems, consider using get() and set() instead of an iterator:

       for (int i = 0; i < t.size(); ++i)
          if (t.get(i) == ...)  t.set(i, ...);

Sparsetable also has a special class of iterator, besides normal and const: nonempty_iterator. This only iterates over array values that are assigned. This is particularly fast given the sparsetable implementation, since it can ignore the bitmaps entirely and just iterate over the various group vectors.

### Resource use

The space overhead for an sparsetable of size N is N + 48N/M bits. For the default value of M, this is exactly 2 bits per array entry. (That's for 32-bit pointers; for machines with 64-bit pointers, it's N + 80N/M bits, or 2.67 bits per entry.) A larger M would use less overhead -- approaching 1 bit per array entry -- but take longer for inserts, deletes, and lookups. A smaller M would use more overhead but make operations somewhat faster.

The numbers above assume that the allocator used doesn't require extra memory. The default allocator (using malloc/free) typically has some overhead for each allocation. If we assume 16 byte overhead per allocation, the overhead becomes 4.6 bit per array entry (32 bit pointers) or 5.3 bit per array entry (64 bit pointers)

Each sparsegroup has:

member | 32 bit | 64 bit
--- | --- | ---
pointer | 4 bytes | 8 bytes
num_buckets | 2 bytes | 2 bytes
bitmap | 6 bytes | 6 bytes
total | 12 bytes = 96 bits | 16 bytes = 128 bits
because this is the overhead for each sparsegroup (48 entries), we divide by 48 |  |
overhead / entry | 96 / 48 = 2 bits | 128 / 48 = 2.67 bits
additional overhead per allocation up to 16 bytes = 128 bits |  |
max overhead / entry | (96 + 128) / 48 = 4.67 bits | (128 + 128) / 48 = 5.33 bits

You can also look at some specific performance numbers.

## `sparse_hash_set`

For specificity, consider the declaration

       sparse_hash_set<Foo> t;

sparse_hash_set is a hashtable. For more information on hashtables, see Knuth. Hashtables are basically arrays with complicated logic on top of them. sparse_hash_set uses a sparsetable to implement the underlying array.

In particular, sparse_hash_set stores its data in a sparsetable using quadratic internal probing (see Knuth). Many hashtable implementations use external probing, so each table element is actually a pointer chain, holding many hashtable values. sparse_hash_set, on the other hand, always stores at most one value in each table location. If the hashtable wants to store a second value at a given table location, it can't; it's forced to look somewhere else.

### `insert()`

As a specific example, suppose t is a new sparse_hash_set. It then holds a sparsetable of size 32. The code for t.insert(foo) works as follows:

1) Call hash<Foo>(foo) to convert foo into an integer i. (hash<Foo> is the default hash function; you can specify a different one in the template arguments.)

2a) Look at t.sparsetable[i % 32]. If it's unassigned, assign it to foo. foo is now in the hashtable.

2b) If t.sparsetable[i % 32] is assigned, and its value is foo, then do nothing: foo was already in t and the insert is a noop.

2c) If t.sparsetable[i % 32] is assigned, but to a value other than foo, look at t.sparsetable[(i+1) % 32]. If that also fails, try t.sparsetable[(i+3) % 32], then t.sparsetable[(i+6) % 32]. In general, keep trying the next triangular number.

3) If the table is now "too full" -- say, 25 of the 32 table entries are now assigned -- grow the table by creating a new sparsetable that's twice as big, and rehashing every single element from the old table into the new one. This keeps the table from ever filling up.

4) If the table is now "too empty" -- say, only 3 of the 32 table entries are now assigned -- shrink the table by creating a new sparsetable that's half as big, and rehashing every element as in the growing case. This keeps the table overhead proportional to the number of elements in the table.

Instead of using triangular numbers as offsets, one could just use regular integers: try i, then i+1, then i+2, then i+3. This has bad 'clumping' behavior, as explored in Knuth. Quadratic probing, using the triangular numbers, avoids the clumping while keeping cache coherency in the common case. As long as the table size is a power of 2, the quadratic-probing method described above will explore every table element if necessary, to find a good place to insert.

(As a side note, using a table size that's a power of two has several advantages, including the speed of calculating (i % table_size). On the other hand, power-of-two tables are not very forgiving of a poor hash function. Make sure your hash function is a good one! There are plenty of dos and don'ts on the web (and in Knuth), for writing hash functions.)

The "too full" value, also called the "maximum occupancy", determines a time-space tradeoff: in general, the higher it is, the less space is wasted but the more probes must be performed for each insert. sparse_hash_set uses a high maximum occupancy, since space is more important than speed for this data structure.

The "too empty" value is not necessary for performance but helps with space use. It's rare for hashtable implementations to check this value at insert() time -- after all, how will inserting cause a hashtable to get too small? However, the sparse_hash_set implementation never resizes on erase(); it's nice to have an erase() that does not invalidate iterators. Thus, the first insert() after a long string of erase()s could well trigger a hashtable shrink.

### `find()`

find() works similarly to insert. The only difference is in step (2a): if the value is unassigned, then the lookup fails immediately.

### `delete()`

delete() is tricky in an internal-probing scheme. The obvious implementation of just "unassigning" the relevant table entry doesn't work. Consider the following scenario:

        t.insert(foo1);         // foo1 hashes to 4, is put in table[4]
        t.insert(foo2);         // foo2 hashes to 4, is put in table[5]
        t.erase(foo1);          // table[4] is now 'unassigned'
        t.lookup(foo2);         // fails since table[hash(foo2)] is unassigned

To avoid these failure situations, delete(foo1) is actually implemented by replacing foo1 by a special 'delete' value in the hashtable. This 'delete' value causes the table entry to be considered unassigned for the purposes of insertion -- if foo3 hashes to 4 as well, it can go into table[4] no problem -- but assigned for the purposes of lookup.

What is this special 'delete' value? The delete value has to be an element of type Foo, since the table can't hold anything else. It obviously must be an element the client would never want to insert on its own, or else the code couldn't distinguish deleted entries from 'real' entries with the same value. There's no way to determine a good value automatically. The client has to specify it explicitly. This is what the set_deleted_key() method does.

Note that set_deleted_key() is only necessary if the client actually wants to call t.erase(). For insert-only hash-sets, set_deleted_key() is unnecessary.

When copying the hashtable, either to grow it or shrink it, the special 'delete' values are not copied into the new table. The copy-time rehash makes them unnecessary.

### Resource use

The data is stored in a sparsetable, so space use is the same as for sparsetable. However, by default the sparse_hash_set implementation tries to keep about half the table buckets empty, to keep lookup-chains short. Since sparsehashmap has about 2 bits overhead per bucket (or 2.5 bits on 64-bit systems), sparse_hash_map has about 4-5 bits overhead per hashtable item.

Time use is also determined in large part by the sparsetable implementation. However, there is also an extra probing cost in hashtables, which depends in large part on the "too full" value. It should be rare to need more than 4-5 probes per lookup, and usually significantly less will suffice.

A note on growing and shrinking the hashtable: all hashtable implementations use the most memory when growing a hashtable, since they must have room for both the old table and the new table at the same time. sparse_hash_set is careful to delete entries from the old hashtable as soon as they're copied into the new one, to minimize this space overhead. (It does this efficiently by using its knowledge of the sparsetable class and copying one sparsetable group at a time.)

You can also look at some specific performance numbers.

## `sparse_hash_map`

sparse_hash_map is implemented identically to sparse_hash_set. The only difference is instead of storing just Foo in each table entry, the data structure stores pair<Foo, Value>.

## `dense_hash_set`

The hashtable aspects of dense_hash_set are identical to sparse_hash_set: it uses quadratic internal probing, and resizes hashtables in exactly the same way. The difference is in the underlying array: instead of using a sparsetable, dense_hash_set uses a C array. This means much more space is used, especially if Foo is big. However, it makes all operations faster, since sparsetable has memory management overhead that C arrays do not.

The use of C arrays instead of sparsetables points to one immediate complication dense_hash_set has that sparse_hash_set does not: the need to distinguish assigned from unassigned entries. In a sparsetable, this is accomplished by a bitmap. dense_hash_set, on the other hand, uses a dedicated value to specify unassigned entries. Thus, dense_hash_set has two special values: one to indicate deleted table entries, and one to indicated unassigned table entries. At construct time, all table entries are initialized to 'unassigned'.

dense_hash_set provides the method set_empty_key() to indicate the value that should be used for unassigned entries. Like set_deleted_key(), set_empty_key() requires a value that will not be used by the client for any legitimate purpose. Unlike set_deleted_key(), set_empty_key() is always required, no matter what hashtable operations the client wishes to perform.

### Resource use

This implementation is fast because even though dense_hash_set may not be space efficient, most lookups are localized: a single lookup may need to access table[i], and maybe table[i+1] and table[i+3], but nothing other than that. For all but the biggest data structures, these will frequently be in a single cache line.

This implementation takes, for every unused bucket, space as big as the key-type. Usually between half and two-thirds of the buckets are empty.

The doubling method used by dense_hash_set tends to work poorly with most memory allocators. This is because memory allocators tend to have memory 'buckets' which are a power of two. Since each doubling of a dense_hash_set doubles the memory use, a single hashtable doubling will require a new memory 'bucket' from the memory allocator, leaving the old bucket stranded as fragmented memory. Hence, it's not recommended this data structure be used with many inserts in memory-constrained situations.

You can also look at some specific performance numbers.

## `dense_hash_map`

dense_hash_map is identical to dense_hash_set except for what values are stored in each table entry.

Craig Silverstein
Thu Jan 6 20:15:42 PST 2005