huge sparse array

(data structure)

Definition: Let N be the number of items to store and R be the size of the range of key values; R >> N. Allocate, but don't initialize, two arrays: an item array I, where |I|≥N, and a location array L, where |L|=R. Initialize a variable, next, the number of items, to zero (with 0-based indexing). To insert an item, put it in the next place in the item array and save where to find it in the location array.

   I[next] = item;
L[item.key] = next;
To look up an item by key, get the index from the location array. If the index is invalid or refers to the wrong item, the item is not found.
   index = L[key];
if (index < 0 OR index >= next) return NOTFOUND;
if (I[index].key != key) return NOTFOUND;
return I[index];
Inserting N items takes Θ(N) total time, assuming allocation takes constant time. Retrieving an item by key (or responding "not found") takes constant time. Listing all items takes Θ(N) time using I.

Generalization (I am a kind of ...)

See also sparse matrix, hash table.

Note: Keys must be unique. Duplicate keys could be handled with collision resolution schemes. This data structure is usually impractical since the range is enormous.

This effectively allocates a huge array without explicitly initializing it, at the cost of Θ(N) space and a little time for each access.

This is what hash tables want to be: constant time for insertion and look up.

Author: PEB

More information

Historical Note
Motivating problem first posed as exercise 2.12 in Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman, The Design and Analysis of Computer Algorithms, page 71, Addison-Wesley, 1974.

Go to the Dictionary of Algorithms and Data Structures home page.

If you have suggestions, corrections, or comments, please get in touch with Paul Black.

Entry modified 14 August 2008.
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Cite this as:
Paul E. Black, "huge sparse array", in Dictionary of Algorithms and Data Structures [online], Vreda Pieterse and Paul E. Black, eds. 14 August 2008. (accessed TODAY) Available from: