Abstract

We describe a novel randomized method, the method of color-coding for finding simple paths and cycles of a specified length k, and other small subgraphs, within a given graph G = (V; E). The randomized algorithms obtained using this method can be derandomized using families of perfect hash functions. Using the color-coding method we obtain, in particular, the following new results: ffl For every fixed k, if a graph G = (V; E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ! ) expected time or O(V ! log V ) worst-case time, where ! ! 2:376 is the exponent of matrix multiplication. (Here and in what follows we use V and E instead of jV j and jEj whenever no confusion may arise.) ffl For every fixed k, if a planar graph G = (V; E) contains a simple cycle of size exactly k, then such a cycle can be found in either O(V ) expected time or O(V log V ) worst-case time. The same algorithm applies, in fact, not only to planar graphs, but to any min...

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