  We reformulate the Kerov-Kirillov-Reshetikhin (KKR) map in the combinatorial
Bethe ansatz from paths to rigged configurations by introducing local energy
distribution in crystal base theory. Combined with an earlier result on the
inverse map, it completes the crystal interpretation of the KKR bijection for
U_q(\hat{sl}_2). As an application, we solve an integrable cellular automaton,
a higher spin generalization of the periodic box-ball system, by an inverse
scattering method and obtain the solution of the initial value problem in terms
of the ultradiscrete Riemann theta function.
