Quadratic probing

Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking the original hash index and adding successive values of an arbitrary quadratic polynomial until an open slot is found.

An example sequence using quadratic probing is:

$$H + 1^2, H + 2^2 , H + 3^2 , H + 4^2 , ... , H + k^2$$

Quadratic probing is often recommended as an alternative to linear probing because it incurs less clustering. Quadratic probing exhibits better locality of reference than many other hash table such as chaining; however, for queries, quadratic probing does not have as good locality as linear probing, causing the latter to be faster in some settings.

Quadratic probing was first introduced by Ward Douglas Maurer in 1968.

Quadratic function
Let h(k) be a hash function that maps an element k to an integer in [0, m−1], where m is the size of the table. Let the ith probe position for a value k be given by the function
 * $$h(k,i) = h(k) + c_1 i + c_2 i^2 \pmod{m}$$

where c2 ≠ 0 (If c2 = 0, then h(k,i) degrades to a linear probe). For a given hash table, the values of c1 and c2 remain constant.

Examples:
 * If $$h(k,i) = (h(k) + i + i^2) \pmod{m}$$, then the probe sequence will be $$h(k), h(k)+2, h(k)+6, ...$$
 * For m = 2n, a good choice for the constants are c1 = c2 = 1/2, as the values of h(k,i) for i in [0, m−1] are all distinct (in fact, it is a permutation on [0, m−1] ). This leads to a probe sequence of $$h(k), h(k)+1, h(k)+3, h(k)+6, ...$$ (the triangular numbers) where the values increase by 1, 2, 3, ...
 * For prime m > 2, most choices of c1 and c2 will make h(k,i) distinct for i in [0, (m−1)/2]. Such choices include c1 = c2 = 1/2, c1 = c2 = 1, and c1 = 0, c2 = 1.  However, there are only m/2 distinct probes for a given element, requiring other techniques to guarantee that insertions will succeed when the load factor exceeds 1/2.
 * For $$m = n^p$$, where m, n, and p are integer greater or equal 2 (degrades to linear probe when p = 1), then $$h(k,i) = (h(k) + i + ni^2)\pmod{m}$$ gives cycle of all distinct probes. It can be computed in loop as: $$h(k, 0) = h(k)$$, and $$h(k, i+1) = (h(k,i) + 2in+n+1)\pmod{m}$$
 * For any m, full cycle with quadratic probing can be achieved by rounding up m to closest power of 2, compute probe index: $$h(k,i) = h(k) + ((i^2+i)/2)\pmod{roundUp2(m)}$$, and skip iteration when $$h(k,i)>=m$$. There is maximum $$roundUp2(m)-m < m/2$$ skipped iterations, and these iterations do not refer to memory, so it is fast operation on most modern processors. Rounding up m can be computed by:

Alternating signs
If the sign of the offset is alternated (e.g. +1, −4, +9, −16, etc.), and if the number of buckets is a prime number $$p$$ congruent to 3 modulo 4 (e.g. 3, 7, 11, 19, 23, 31, etc.), then the first $$p$$ offsets will be unique (modulo $$p$$). In other words, a permutation of 0 through $$p-1$$ is obtained, and, consequently, a free bucket will always be found as long as at least one exists.