![]() ![]() Another solution (cf. again 12.2) to this problem is what is called arithmetic coding: we think of the concatenated elements as forming a ternary number between and, and we write down its binary representation. ![]() The space used is and we have linear redundancy. With such an encoding we can retrieve each by reading just bits (which is optimal). 12.2) to this problem is to use bits per. One can encode the input to as before using bits without loss of generality, but the current choice simplifies the exposition. Note that the input ranges over elements, and so the minimum space of the data structure is This will be our benchmark for space. To illustrate, consider the ECC problem where is an error-correcting code (with constant relative distance) and is linear in. Moreover, it was shown that improving the bounds would imply stronger circuit lower bounds. But, surprisingly, again these stronger bounds were shown to be tight. Unsurprisingly, stronger bounds can be probed in this setting. Specifically, we let for some called redundancy. Succinct data structures are those where the space is close to the minimum. We pick the graph at random and show that it has the latter property with non-zero probability. Note here we are using that the neighborhood has size, and so the memory is -wise uniform. And for that, finally, it suffices that has a neighborhood of size greater than (because if every element in the neighborhood of has two neighbors in then has a neighborhood of size less than ). For this in turn it suffices that has a unique neighbor. To show -wise uniformity it suffices to show that for any subset on the right-hand side of size, the sum of the corresponding memory cells is uniform in. We answer each query by summing the corresponding cells in. Then we pick a random bipartite graph with nodes on the left and nodes on the right. ![]() Proof. We fill the memory with evaluations of the input polynomial.
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