Zig-zag product

In graph theory, the zig-zag product of regular graphs , denoted by , takes a large graph () and a small graph (), and produces a graph that approximately inherits the size of the large one but the degree of the small one. An important property of the zig-zag product is that if is a good expander, then the expansion of the resulting graph is only slightly worse than the expansion of .

Roughly speaking, the zig-zag product replaces each vertex of with a copy (cloud) of , and connects the vertices by moving a small step (zig) inside a cloud, followed by a big step (zag) between two clouds, and finally performs another small step inside the destination cloud.

The zigzag product was introduced by Reingold, Vadhan & Wigderson (2002). When the zig-zag product was first introduced, it was used for the explicit construction of constant degree expanders and extractors. Later on the zig-zag product was used in computational complexity theory to prove that symmetric logspace and logspace are equal (Reingold 2008).