The Information Complexity of Hamming Distance

The Hamming distance function Hamn,d returns 1 on all pairs of inputs x and y that dier in at most d coordinates and returns 0 otherwise. We initiate the study of the information complexity of the Hamming distance function. We give a new optimal lower bound for the information complexity of the Hamn,d function in the small-error regime where the protocol is required to err with probability at most ‘ < d/n. We also give a new conditional lower bound for the information complexity of Hamn,d that is optimal in all regimes. These results imply the first new lower bounds on the communication complexity of the Hamming distance function for the shared randomness two-way communication model since Pang and El-Gamal (1986). These results also imply new lower bounds in the areas of property testing and parity decision tree complexity