Hash Function with Order Preserving

In general case, such a function is impossible unless the size of the hash is at least the size of the object.

The argument is trivial: if there are N objects but M < N hash values, by pigeonhole principle, two different objects are mapped to one hash value, and so their order is not preserved.

If however we have additional properties of the objects guaranteed or the requirements relaxed, a custom or probabilistic solution may become possible.

CHM (Z.J. Czech, G. Havas, and B.S. Majewski) is an algorithm which generates a minimal perfect hash that preserves ordering (e.g. if A < B, then h(A) < h(B)). It uses approximately 8 bytes of storage per key.

See: http://cmph.sourceforge.net/chm.html