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This paper introduces a new data-structural object that we call the tiny pointer. In many applications, traditional log n-bit pointers can be replaced with o(log n)-bit tiny pointers at the cost of only a constant-factor time overhead and a small probability of failure. We develop a comprehensive theory of tiny pointers, and give optimal constructions for both fixed-size tiny pointers (i.e., settings in which all of the tiny pointers must be the same size) and variable-size tiny pointers (i.e., settings in which the average tiny-pointer size must be small, but some tiny pointers can be larger). If a tiny pointer references an element in an array filled to load factor 1 — δ, then the optimal tiny-pointer size is Θ(log log log n + log δ-1) bits in the fixed-size case, and Θ(log δ-1) expected bits in the variable-size case. Our tiny-pointer constructions also require us to revisit several classic problems having to do with balls and bins; these results may be of independent interest. Using tiny pointers, we revisit five classic data-structure problems. We show that: • A data structure storing n v-bit values for n keys with constant-time modifications/queries can be implemented to take space nv + O(n log(r) n) bits, for any constant r > 0, as long as the user stores a tiny pointer of expected size O(1) with each key—here, log(r) n is the r-th iterated logarithm. • Any binary search tree can be made succinct with constant-factor time overhead, and can even be made to be within O(n) bits of optimal if we allow for O(log* n)-time modifications—this holds even for rotation-based trees such as the splay tree and the red-black tree. • Any fixed-capacity key-value dictionary can be made stable (i.e., items do not move once inserted) with constant-time overhead and 1 + o(1) space overhead. • Any key-value dictionary that requires uniform-size values can be made to support arbitrary-size values with constant-time overhead and with an additional space consumption of log(r) n + O(log j) bits per j-bit value for an arbitrary constant r > 0 of our choice. • Given an external-memory array A of size (1 + ε)n containing a dynamic set of up to n key-value pairs, it is possible to maintain an internal-memory stash of size O(n log ε-1) bits so that the location of any key-value pair in A can be computed in constant time (and with no IOs). These are all well studied and classic problems, and in each case tiny pointers allow for us to take a natural space-inefficient solution that uses pointers and make it space-efficient for free.
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This content will become publicly available on January 1, 2027
Succinct Dynamic Rank/Select: Bypassing the Tree-Structure Bottleneck
We show how to construct a dynamic ordered dictionary, supporting insert/delete/rank/select on a set of $$n$$ elements from a universe of size $$U$$, that achieves the optimal amortized expected time complexity of $$O(1 + \log n / \log \log U)$$, while achieving a nearly optimal space consumption of $$\log \binom{U}{n} + n / 2^{(\log n)^{\Omega(1)}} + \polylog U$$ bits in the regime where $$U = \poly(n)$$. This resolves an open question by Pibiri and Venturini as to whether a redundancy (a.k.a.\ space overhead) of $o(n)$ bits is possible, and is the first dynamic solution to bypass the so-called tree-structure bottleneck, in which the bits needed to encode some dynamic tree structure are themselves enough to force a redundancy of $$\tilde{\Omega}(n)$$ bits. Our main technical building block is a dynamic balanced binary search tree, which we call the \emph{compressed tabulation-weighted treap}, that itself achieves a surprising time/space tradeoff. The tree supports $$\polylog n$$-time operations and requires a static lookup table of size $$\poly(n) + \polylog U$$---but, in exchange for these, the tree is able to achieve a remarkable space guarantee. Its total space redundancy is $$O(\log U)$$ bits. In fact, if the tree is given $$n$$ and $$U$$ for free, then the redundancy further drops to $O(1)$ bits.
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- Award ID(s):
- 2504471
- PAR ID:
- 10680934
- Publisher / Repository:
- Society for Industrial and Applied Mathematics
- Date Published:
- ISBN:
- 978-1-61197-897-1
- Page Range / eLocation ID:
- 3760 to 3804
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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