Fractional Cascading - The General Problem

The General Problem

In general, fractional cascading begins with a catalog graph, a directed graph in which each vertex is labeled with an ordered list. A query in this data structure consists of a path in the graph and a query value q; the data structure must determine the position of q in each of the ordered lists associated with the vertices of the path. For the simple example above, the catalog graph is itself a path, with just four nodes. It is possible for later vertices in the path to be determined dynamically as part of a query, in response to the results found by the searches in earlier parts of the path.

To handle queries of this type, for a graph in which each vertex has at most d incoming and at most d outgoing edges for some constant d, the lists associated with each vertex are augmented by a fraction of the items from each outgoing neighbor of the vertex; the fraction must be chosen to be smaller than 1/d, so that the total amount by which all lists are augmented remains linear in the input size. Each item in each augmented list stores with it the position of that item in the unaugmented list stored at the same vertex, and in each of the outgoing neighboring lists. In the simple example above, d = 1, and we augmented each list with a 1/2 fraction of the neighboring items.

A query in this data structure consists of a standard binary search in the augmented list associated with the first vertex of the query path, together with simpler searches at each successive vertex of the path. If a 1/r fraction of items are used to augment the lists from each neighboring item, then each successive query result may be found within at most r steps of the position stored at the query result from the previous path vertex, and therefore may be found in constant time without having to perform a full binary search.