Inserimento in un albero B.

In questo tutorial imparerai come inserire una chiave in un btree. Inoltre, troverai esempi funzionanti di inserimento di chiavi in ​​un albero B in C, C ++, Java e Python.

L'inserimento di un elemento su un albero B consiste in due eventi: la ricerca del nodo appropriato per inserire l'elemento e la suddivisione del nodo se necessario. L'operazione di inserimento avviene sempre con approccio bottom-up.

Cerchiamo di capire questi eventi di seguito.

Operazione di inserimento

  1. Se l'albero è vuoto, alloca un nodo radice e inserisci la chiave.
  2. Aggiorna il numero di chiavi consentito nel nodo.
  3. Cerca il nodo appropriato per l'inserimento.
  4. Se il nodo è pieno, segui i passaggi seguenti.
  5. Inserisci gli elementi in ordine crescente.
  6. Ora, ci sono elementi maggiori del suo limite. Quindi, diviso in mediana.
  7. Spingi la chiave mediana verso l'alto e crea i tasti sinistro come bambino sinistro e i tasti destro come bambino destro.
  8. Se il nodo non è pieno, segui i passaggi seguenti.
  9. Inserisci il nodo in ordine crescente.

Esempio di inserimento

Comprendiamo l'operazione di inserimento con le illustrazioni sottostanti.

Gli elementi da inserire sono 8, 9, 10, 11, 15, 16, 17, 18, 20, 23.

Inserimento di elementi in un albero B.

Algoritmo per l'inserimento di un elemento

 BreeInsertion(T, k) r root(T) if n(r) = 2t - 1 s = AllocateNode() root(T) = s leaf(s) = FALSE n(s) <- 0 c1(s) <- r BtreeSplitChild(s, 1, r) BtreeInsertNonFull(s, k) else BtreeInsertNonFull(r, k) BtreeInsertNonFull(x, k) i = n(x) if leaf(x) while i ≧ 1 and k < keyi(x) keyi+1 (x) = keyi(x) i = i - 1 keyi+1(x) = k n(x) = n(x) + 1 else while i ≧ 1 and k < keyi(x) i = i - 1 i = i + 1 if n(ci(x)) == 2t - 1 BtreeSplitChild(x, i, ci(x)) if k &rt; keyi(x) i = i + 1 BtreeInsertNonFull(ci(x), k) BtreeSplitChild(x, i) BtreeSplitChild(x, i, y) z = AllocateNode() leaf(z) = leaf(y) n(z) = t - 1 for j = 1 to t - 1 keyj(z) = keyj+t(y) if not leaf (y) for j = 1 to t cj(z) = cj + t(y) n(y) = t - 1 for j = n(x) + 1 to i + 1 cj+1(x) = cj(x) ci+1(x) = z for j = n(x) to i keyj+1(x) = keyj(x) keyi(x) = keyt(y) n(x) = n(x) + 1

Esempi di Python, Java e C / C ++

Python Java C C ++ 
# Inserting a key on a B-tree in Python # Create a node class BTreeNode: def __init__(self, leaf=False): self.leaf = leaf self.keys = () self.child = () # Tree class BTree: def __init__(self, t): self.root = BTreeNode(True) self.t = t # Insert node def insert(self, k): root = self.root if len(root.keys) == (2 * self.t) - 1: temp = BTreeNode() self.root = temp temp.child.insert(0, root) self.split_child(temp, 0) self.insert_non_full(temp, k) else: self.insert_non_full(root, k) # Insert nonfull def insert_non_full(self, x, k): i = len(x.keys) - 1 if x.leaf: x.keys.append((None, None)) while i>= 0 and k(0)  = 0 and k(0)  x.keys(i)(0): i += 1 self.insert_non_full(x.child(i), k) # Split the child def split_child(self, x, i): t = self.t y = x.child(i) z = BTreeNode(y.leaf) x.child.insert(i + 1, z) x.keys.insert(i, y.keys(t - 1)) z.keys = y.keys(t: (2 * t) - 1) y.keys = y.keys(0: t - 1) if not y.leaf: z.child = y.child(t: 2 * t) y.child = y.child(0: t - 1) # Print the tree def print_tree(self, x, l=0): print("Level ", l, " ", len(x.keys), end=":") for i in x.keys: print(i, end=" ") print() l += 1 if len(x.child)> 0: for i in x.child: self.print_tree(i, l) def main(): B = BTree(3) for i in range(10): B.insert((i, 2 * i)) B.print_tree(B.root) if __name__ == '__main__': main()  
// Inserting a key on a B-tree in Java public class BTree ( private int T; // Node Creation public class Node ( int n; int key() = new int(2 * T - 1); Node child() = new Node(2 * T); boolean leaf = true; public int Find(int k) ( for (int i = 0; i < this.n; i++) ( if (this.key(i) == k) ( return i; ) ) return -1; ); ) public BTree(int t) ( T = t; root = new Node(); root.n = 0; root.leaf = true; ) private Node root; // split private void split(Node x, int pos, Node y) ( Node z = new Node(); z.leaf = y.leaf; z.n = T - 1; for (int j = 0; j < T - 1; j++) ( z.key(j) = y.key(j + T); ) if (!y.leaf) ( for (int j = 0; j = pos + 1; j--) ( x.child(j + 1) = x.child(j); ) x.child(pos + 1) = z; for (int j = x.n - 1; j>= pos; j--) ( x.key(j + 1) = x.key(j); ) x.key(pos) = y.key(T - 1); x.n = x.n + 1; ) // insert key public void insert(final int key) ( Node r = root; if (r.n == 2 * T - 1) ( Node s = new Node(); root = s; s.leaf = false; s.n = 0; s.child(0) = r; split(s, 0, r); _insert(s, key); ) else ( _insert(r, key); ) ) // insert node final private void _insert(Node x, int k) ( if (x.leaf) ( int i = 0; for (i = x.n - 1; i>= 0 && k  = 0 && k x.key(i)) ( i++; ) ) _insert(x.child(i), k); ) ) public void display() ( display(root); ) // Display the tree private void display(Node x) ( assert (x == null); for (int i = 0; i < x.n; i++) ( System.out.print(x.key(i) + " "); ) if (!x.leaf) ( for (int i = 0; i < x.n + 1; i++) ( display(x.child(i)); ) ) ) public static void main(String() args) ( BTree b = new BTree(3); b.insert(8); b.insert(9); b.insert(10); b.insert(11); b.insert(15); b.insert(20); b.insert(17); b.display(); ) ) 
// insertioning a key on a B-tree in C #include #include #define MAX 3 #define MIN 2 struct btreeNode ( int item(MAX + 1), count; struct btreeNode *link(MAX + 1); ); struct btreeNode *root; // Node creation struct btreeNode *createNode(int item, struct btreeNode *child) ( struct btreeNode *newNode; newNode = (struct btreeNode *)malloc(sizeof(struct btreeNode)); newNode->item(1) = item; newNode->count = 1; newNode->link(0) = root; newNode->link(1) = child; return newNode; ) // Insert void insertValue(int item, int pos, struct btreeNode *node, struct btreeNode *child) ( int j = node->count; while (j> pos) ( node->item(j + 1) = node->item(j); node->link(j + 1) = node->link(j); j--; ) node->item(j + 1) = item; node->link(j + 1) = child; node->count++; ) // Split node void splitNode(int item, int *pval, int pos, struct btreeNode *node, struct btreeNode *child, struct btreeNode **newNode) ( int median, j; if (pos> MIN) median = MIN + 1; else median = MIN; *newNode = (struct btreeNode *)malloc(sizeof(struct btreeNode)); j = median + 1; while (j item(j - median) = node->item(j); (*newNode)->link(j - median) = node->link(j); j++; ) node->count = median; (*newNode)->count = MAX - median; if (pos item(node->count); (*newNode)->link(0) = node->link(node->count); node->count--; ) // Set the value of node int setNodeValue(int item, int *pval, struct btreeNode *node, struct btreeNode **child) ( int pos; if (!node) ( *pval = item; *child = NULL; return 1; ) if (item item(1)) ( pos = 0; ) else ( for (pos = node->count; (item item(pos) && pos> 1); pos--) ; if (item == node->item(pos)) ( printf("Duplicates not allowed"); return 0; ) ) if (setNodeValue(item, pval, node->link(pos), child)) ( if (node->count link(pos); for (; dummy->link(0) != NULL;) dummy = dummy->link(0); myNode->item(pos) = dummy->item(1); ) // Do rightshift void rightShift(struct btreeNode *myNode, int pos) ( struct btreeNode *x = myNode->link(pos); int j = x->count; while (j> 0) ( x->item(j + 1) = x->item(j); x->link(j + 1) = x->link(j); ) x->item(1) = myNode->item(pos); x->link(1) = x->link(0); x->count++; x = myNode->link(pos - 1); myNode->item(pos) = x->item(x->count); myNode->link(pos) = x->link(x->count); x->count--; return; ) // Do leftshift void leftShift(struct btreeNode *myNode, int pos) ( int j = 1; struct btreeNode *x = myNode->link(pos - 1); x->count++; x->item(x->count) = myNode->item(pos); x->link(x->count) = myNode->link(pos)->link(0); x = myNode->link(pos); myNode->item(pos) = x->item(1); x->link(0) = x->link(1); x->count--; while (j count) ( x->item(j) = x->item(j + 1); x->link(j) = x->link(j + 1); j++; ) return; ) // Merge the nodes void mergeNodes(struct btreeNode *myNode, int pos) ( int j = 1; struct btreeNode *x1 = myNode->link(pos), *x2 = myNode->link(pos - 1); x2->count++; x2->item(x2->count) = myNode->item(pos); x2->link(x2->count) = myNode->link(0); while (j count) ( x2->count++; x2->item(x2->count) = x1->item(j); x2->link(x2->count) = x1->link(j); j++; ) j = pos; while (j count) ( myNode->item(j) = myNode->item(j + 1); myNode->link(j) = myNode->link(j + 1); j++; ) myNode->count--; free(x1); ) // Adjust the node void adjustNode(struct btreeNode *myNode, int pos) ( if (!pos) ( if (myNode->link(1)->count> MIN) ( leftShift(myNode, 1); ) else ( mergeNodes(myNode, 1); ) ) else ( if (myNode->count != pos) ( if (myNode->link(pos - 1)->count> MIN) ( rightShift(myNode, pos); ) else ( if (myNode->link(pos + 1)->count> MIN) ( leftShift(myNode, pos + 1); ) else ( mergeNodes(myNode, pos); ) ) ) else ( if (myNode->link(pos - 1)->count> MIN) rightShift(myNode, pos); else mergeNodes(myNode, pos); ) ) ) // Traverse the tree void traversal(struct btreeNode *myNode) ( int i; if (myNode) ( for (i = 0; i count; i++) ( traversal(myNode->link(i)); printf("%d ", myNode->item(i + 1)); ) traversal(myNode->link(i)); ) ) int main() ( int item, ch; insertion(8); insertion(9); insertion(10); insertion(11); insertion(15); insertion(16); insertion(17); insertion(18); insertion(20); insertion(23); traversal(root); )
// Inserting a key on a B-tree in C++ #include using namespace std; class Node ( int *keys; int t; Node **C; int n; bool leaf; public: Node(int _t, bool _leaf); void insertNonFull(int k); void splitChild(int i, Node *y); void traverse(); friend class BTree; ); class BTree ( Node *root; int t; public: BTree(int _t) ( root = NULL; t = _t; ) void traverse() ( if (root != NULL) root->traverse(); ) void insert(int k); ); Node::Node(int t1, bool leaf1) ( t = t1; leaf = leaf1; keys = new int(2 * t - 1); C = new Node *(2 * t); n = 0; ) // Traverse the nodes void Node::traverse() ( int i; for (i = 0; i traverse(); cout << " " 
 keys(0) = k; root->n = 1; ) else ( if (root->n == 2 * t - 1) ( Node *s = new Node(t, false); s->C(0) = root; s->splitChild(0, root); int i = 0; if (s->keys(0) C(i)->insertNonFull(k); root = s; ) else root->insertNonFull(k); ) ) // Insert non full condition void Node::insertNonFull(int k) ( int i = n - 1; if (leaf == true) ( while (i>= 0 && keys(i)> k) ( keys(i + 1) = keys(i); i--; ) keys(i + 1) = k; n = n + 1; ) else ( while (i>= 0 && keys(i)> k) i--; if (C(i + 1)->n == 2 * t - 1) ( splitChild(i + 1, C(i + 1)); if (keys(i + 1) insertNonFull(k); ) ) // split the child void Node::splitChild(int i, Node *y) ( Node *z = new Node(y->t, y->leaf); z->n = t - 1; for (int j = 0; j keys(j) = y->keys(j + t); if (y->leaf == false) ( for (int j = 0; j C(j) = y->C(j + t); ) y->n = t - 1; for (int j = n; j>= i + 1; j--) C(j + 1) = C(j); C(i + 1) = z; for (int j = n - 1; j>= i; j--) keys(j + 1) = keys(j); keys(i) = y->keys(t - 1); n = n + 1; ) int main() ( BTree t(3); t.insert(8); t.insert(9); t.insert(10); t.insert(11); t.insert(15); t.insert(16); t.insert(17); t.insert(18); t.insert(20); t.insert(23); cout << "The B-tree is: "; t.traverse(); )

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