Bit-probe lower bounds for succinct data structures

We prove lower bounds on the redundancy necessary to represent a set S of objects using a number of bits close to the information-theoretic minimum log2 |S|, while answering various queries by probing few bits. Our main results are: ? To represent n ternary values t {0, 1, 2}n in terms of u bits b {0, 1}u while accessing a single value ti {0, 1, 2} by probing q bits of b, one needs u (log2 3)n + n/2O(q) . This matches an exciting representation by Patra?scu (FOCS 2008), later refined with Thorup, where u (log2 3)n + n/2(q). We also note that results on logarithmic forms imply the lower bound u (log2 3)n + n/ logO(1) n if we access ti by probing one cell of log n bits. ? To represent sets of size n/3 from a universe of n elements in terms of u bits b {0, 1}u while answering membership queries by probing q bits of b, one needs u log2 n n/3 + n/2O(q) - log n. Both results above hold even if the probe locations are determined adaptively. Ours are the first lower bounds for these f...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Emanuele Viola
• Keywords: Algorithms, Information-theoretic Minimum Log2, Lower Bounds, Single Value Ti, STOC 2009