Given a Boolean function $f$, the $f$-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to $f$ up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions $f$ we can test $f$-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions---functions invariant to the reordering of all but a constant number of their variables---are efficiently isomorphism testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism testable.

Given a Boolean function f, the f-isomorphism testing problem requires a randomized algorithm to distinguish functions that are identical to f up to relabeling of the input variables from functions that are far from being so. An important open question in property testing is to determine for which functions f we can test f-isomorphism with a constant number of queries. Despite much recent attention to this question, essentially only two classes of functions were known to be efficiently isomorphism testable: symmetric functions and juntas. We unify and extend these results by showing that all partially symmetric functions -- functions invariant to the reordering of all but a constant number of their variables -- are efficiently isomorphism-testable. This class of functions, first introduced by Shannon, includes symmetric functions, juntas, and many other functions as well. We conjecture that these functions are essentially the only functions efficiently isomorphism-testable. To prove our main result, we also show that partial symmetry is efficiently testable. In turn, to prove this result we had to revisit the junta testing problem. We provide a new proof of correctness of the nearly-optimal junta tester. Our new proof replaces the Fourier machinery of the original proof with a purely combinatorial argument that exploits the connection between sets of variables with low influence and intersecting families. Another important ingredient in our proofs is a new notion of symmetric influence. We use this measure of influence to prove that partial symmetry is efficiently testable and also to construct an efficient sample extractor for partially symmetric functions. We then combine the sample extractor with the testing-by-implicit-learning approach to complete the proof that partially symmetric functions are efficiently isomorphism-testable.

A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,...,Rm. Each receiver Riis interested in one block, and has prior side information consisting of some subset of the other blocks. Let betatbe the minimum number of bits that has to be transmitted when each block is of length t, and let beta be the limit beta=limtrarrinfinbetat/t. Informally, beta is the average communication cost per bit in each block (for long blocks). Finding the coding rate beta, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on betat, and prove that for any constant C there is an explicit broadcast setting in which beta = 2 but beta1> C. One of these examples answers a question of . In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of. The proofs are based on a relation between this problem and results in the study of Witsenhausen's rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.