A **Sparse Table** is a data structure that allows for efficient range queries on an array. It is particularly useful when dealing with static arrays or arrays that undergo only a few updates. By precomputing some information, sparse tables provide a quick way to answer queries such as finding the minimum, maximum, sum, or any other associative operation over a given range of elements.

To understand how a sparse table works, let’s consider an example. Suppose we have an array of numbers: [4, 1, 5, 2, 7, 3, 6].

We want to find the minimum element between indices 2 and 5 (inclusive). Instead of iterating over the range and checking each element individually, we can use a sparse table to speed up this process.

To construct a sparse table, we start by initializing a two-dimensional array with the same dimensions as the original array. Each cell (i,j) in the sparse table represents the result of applying the desired operation over the range starting at index i and having length 2^j.

Let’s see how this works in practice. For our example array [4, 1, 5, 2, 7, 3, 6], we can construct the following sparse table:

Index | j = 0 | j = 1 | j = 2 |
---|---|---|---|

i = 0 | 4 | 1 | 1 |

i = 1 | 1 | 1 | 1 |

i = 2 | 5 | 2 | 2 |

i = 3 | 2 | 2 | 2 |

i = 4 | 7 | 3 | – |

i = 5 | 3 | – | – |

i = 6 | 6 | – | – |

In this table, the cell (i,j) represents the minimum element in the range [i, i + 2^j – 1]. For example, cell (0,1) represents the minimum element in the range [0, 1], which is 1.

To fill in the sparse table, we use a dynamic programming approach. For each j value (column), we iterate over all i values (rows) and compute the result using the previously filled cells. The formula to compute the value of cell (i,j) is:

sparse_table[i][j] = min(sparse_table[i][j-1], sparse_table[i + 2^(j-1)][j-1])

With this precomputed information, we can now answer queries efficiently. To find the minimum element between indices l and r, we use a logarithmic number of table lookups and a constant number of operations:

min_element = min(sparse_table[l][k], sparse_table[r – 2^k + 1][k])

Here, k is the largest power of 2 that fits within the range [l, r]. By taking the minimum of two ranges that cover the entire query range, we can obtain the desired result.

In conclusion, a sparse table is a powerful data structure for efficient range queries on static or rarely updated arrays. It allows us to precompute information and answer queries with a logarithmic number of table lookups. By incorporating this technique into our algorithms, we can optimize performance and improve overall efficiency in various scenarios.