Indsættelse i et B-træ

I denne vejledning lærer du, hvordan du indsætter en nøgle i et btree. Du finder også arbejdseksempler på indsættelse af nøgler i et B-træ i C, C ++, Java og Python.

Indsættelse af et element i et B-træ består af to begivenheder: at søge i den relevante knude for at indsætte elementet og opdele noden, hvis det er nødvendigt.

Lad os forstå disse begivenheder nedenfor.

Brug af indsættelse

  1. Hvis træet er tomt, skal du tildele en rodnode og indsætte nøglen.
  2. Opdater det tilladte antal nøgler i noden.
  3. Søg efter den relevante node for indsættelse.
  4. Hvis nedenstående knude er fuld, skal du følge nedenstående trin.
  5. Indsæt elementerne i stigende rækkefølge.
  6. Nu er der elementer, der er større end dens grænse. Så del på medianen.
  7. Skub median-tasten opad, og lav de venstre taster som et venstre barn og de højre taster som et højre barn.
  8. Hvis noden ikke er fuld, skal du følge nedenstående trin.
  9. Indsæt noden i stigende rækkefølge.

Eksempel på indsættelse

Lad os forstå indsættelsesfunktionen med nedenstående illustrationer.

Elementerne, der skal indsættes, er 8, 9, 10, 11, 15, 16, 17, 18, 20, 23.

Indsættelse af elementer i et B-træ

Algoritme til indsættelse af et element

 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

Python, Java og C / C ++ eksempler

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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