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Data Structures with Generic Types in Java

Searching and Addition in B-Tree

Learn about searching and addition in the B-trees.

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Searching

The implementation of the find(x) operation, which is illustrated below generalizes the find(x) operation in a binary search tree. The search for x starts at the root and uses the keys stored at a node, u, to determine in which of u’s children the search should continue.

A successful search (for the value 44) and an unsuccessful search (for the value 16.516.5) in a BB-tree is shown below. Shaded nodes show where the value of z is updated during the searches.

Searching in B-tree
Searching in B-tree

More specifically, at a node u, the search checks if x is stored in u.keys. If so, x has been found and the search is complete. Otherwise, the search finds the smallest integer, i, such that u.key[i] > x and continues the search in the subtree rooted at u.children[i]. If no key in u.keys is greater than x, then the search continues in u’s rightmost child. Just like binary search trees, the algorithm keeps track of the most recently seen key, z that is larger than x. In case x is not found, z is returned as the smallest value that is greater than or equal to x.

Java
T find(T x) {

  T z = null;

  int ui = ri;

  while (ui >= 0) {

    BTree

    Node u = bs.readBlock(ui);

    int i = findIt(u.keys, x);

    if (i < 0) return u.keys[-(i + 1)]; // found it

    if (u.keys[i] != null)

      z = u.keys[i];

    ui = u.children[i];

  }

  return z;

}

Central to the find(x) method is the findIt(a,x) method that searches in a null-padded sorted array, a, for the value x.

The execution of findIt(a,27)
The execution of findIt(a,27)

This method, illustrated above, works for any array, a, where a[0],...,a[k−1]\text{a}[0],...,\text{a}[k −1] is a sequence of keys in sorted order and a[k],...,a[a.length−1]\text{a}[k],...,\text{a}[\text{a.length}−1] are all set to null. If x is in the array at position i, then findIt(a, x) returns −i − 1. Otherwise, it returns the smallest index, i, such that a[i] > x or a[i] = null.

Java
int findIt(T[] a, T x) {

  int lo = 0, hi = a.length;

  while (hi != lo) {

    int m = (hi + lo) / 2;

    int cmp = a[m] == null ? -1 : compare(x, a[m]);

    if (cmp < 0)

      hi = m; // look in first half

    else if (cmp > 0)

      lo = m + 1; // look in second half

    else

      return -m - 1; // found it

  }

  return lo;

}

The findIt(a, x) method uses a binary search that halves the search space at each step, so it runs in O(log⁡(a.length))O(\log(\text{a.length})) time. In our setting, a.length=2B\text{a.length}=2B, so findIt(a,x) runs in O(log⁡B)O(\log B) time.

We can analyze the running time of a BB-tree find(x) operation both in ...