[Data Structure] XOR LinkedList

An XOR linked list is a type of double-linked list that uses bitwise XOR operation to store address of the previous and next address in one space (thus, reducing storage space).

In the traditional double-linked list, each node has to store a prev and next address field. 
A  B  C  D
..  n->n->n ..
..  p<-p<-p ..

An XOR linked list combines the same information into 1 address field via bitwise XOR (denoted by ⊕) of the prev and next address.
A    B      C      D      E
..  p⊕n  p⊕n  p⊕n    ..

Getting the next address space

When traversing from left to right, suppose the cursor is at C, the address of previous item B can be XORed with the value of linked field of C (B⊕D) to get the address for D. 

ie. addr(D) = link(C) ⊕ addr(B)
where 
    link(C) = addr(B) ⊕ addr(D)
so 
    addr(D) = addr(B) ⊕ addr(D) ⊕ addr(B)
since
    X⊕X = 0
    => addr(D) = 0 ⊕ addr(D)
since 
    X⊕0 = X
    => addr(D) = addr(D)
The XOR operation cancels addr(B) appearing twice in the equation and all we are left is the addr(D)

Properties of XOR:
  • X⊕X = 0
  • X⊕0 = X
  • X⊕Y = Y⊕X
  • (X⊕Y)⊕Z = X⊕(Y⊕Z)

Drawbacks

  • General-purpose debugging tools cannot follow the XOR chain, making debugging more difficult.
  • Price for decrease in memory usage is an increase in code complexity; making maintenance expensive
  • Most garbage collection schemes do not work with data structures that do not contain literal pointers.
  • Computer systems have increasingly cheap and plentiful memory, therefore storage overhead is not generally a problem.

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