Searching Tables

Yao^[3] considered a static data-structure problem where one has to build a data structure ("table") to represent a set ${\displaystyle S}$ of ${\displaystyle n}$ elements out of ${\displaystyle 1,\dots ,m}$. The query parameter is a number ${\displaystyle x\leq m}$ and the query has to report whether ${\displaystyle x}$ is in the table. Yao showed that as long as the table size is bounded independently of ${\displaystyle m}$ and ${\displaystyle m}$ is large enough, a query must perform ${\displaystyle \lceil \log(n+1)\rceil }$ probes in the worst case. This shows that a sorted table together with binary search for queries is an optimal scheme.

Dynamic Partial Sums

The dynamic partial sum problem defines two operations Update(k,v) which sets the value in an array A at index k to be v, and Sum(k) which returns the sum of the values in A at indices 0 through k. A naïve implementation would take ${\displaystyle O(1)}$ time for Update and ${\displaystyle O(n)}$ time for Sum.^[5]

Instead, values can be stored as leaves in a tree whose inner nodes store the sum over the subtree rooted at that node. In this structure Update requires ${\displaystyle O(\log n)}$ time to update each node in the leaf to root path, and Sum similarly requires ${\displaystyle O(\log n)}$ time to traverse the tree from leaf to root summing the values of all subtrees left of the query index.

Improving on a result of Fredman and Saks, Mihai Pătraşcu used the cell-probe model and an information transfer argument to show that the partial sums problem requires ${\displaystyle \Omega \left(\log n\right)}$ time per operation in the worst case (i.e., the worst of query and update must consume such time), assuming ${\displaystyle b=\Omega (\log n)}$ bits per word.^[1][2] He further exhibited the trade-off curve between update time and query time and investigated the case that updates are restricted to small numbers (of ${\displaystyle \delta =o(b)}$ bits).

Disjoint Set Maintenance (Union-Find)

In the disjoint-set data structure, the structure represents a collection of disjoint sets; there is an update operation, called Union, which unites two sets, and a query operation, called Find, which identifies the set to which a given element belongs. Fredman and Saks proved that in the model CPROBE(log n), any solution for this problem requires ${\displaystyle \Omega (m\alpha (m,n))}$ probes in the worst case (even in expectation) to execute ${\displaystyle n-1}$ unions and ${\displaystyle m\geq n}$ finds.^[4] This shows that the classic data structure described in the article on disjoint-set data structure is optimal.

Approximate Nearest Neighbor Searching

The exact nearest neighbor search problem is to determine the closest in a set of input points to a given query point. An approximate version of this problem is often considered since many applications of this problem are in very high dimension spaces and solving the problem in high dimensions requires exponential time or space with respect to the dimension.^[6]

Chakrabarti and Regev proved that the approximate nearest neighbor search problem on the Hamming cube using polynomial storage and ${\displaystyle d^{O(1)}}$ word size requires a worst-case query time of ${\displaystyle \Omega \left({\frac {\log \log d}{\log \log \log d}}\right)}$. This proof used the cell-probe model and information theoretic techniques from communication complexity.