如何实现检查 function 以检查 b 树属性的有效性？

Question

I have recently implemented a normal B-tree (without any variant) in C, but I would like to check if my implementation is valid ie if it does not violate the following properties:

1. Every node has at most m children.
2. Every non-leaf node (except root) has at least ⌈m/2⌉ child nodes.
3. The root has at least two children if it is not a leaf node.
4. A non-leaf node with k children contains k − 1 keys.
5. All leaves appear in the same level and carry no information.

If anyone could help me with the implementation of this procedure giving me an example with some code in C or with some suggestions. Many thanks in advance.

#include <stdio.h>
#include <stdlib.h>

#define TRUE 1
#define FALSE 0
#define EMPTY 0

#define NODE_ORDER      3 /*The degree of the tree.*/
#define NODE_POINTERS   (NODE_ORDER*2)
#define NODE_KEYS       NODE_POINTERS-1

typedef unsigned char bool;

typedef struct tree_node {
    int key_array[NODE_KEYS];
    struct tree_node *child_array[NODE_POINTERS];
    unsigned int key_index;
    bool leaf;
} node_t;

typedef struct {
    node_t *node_pointer;
    int key;
    bool found;
    unsigned int depth;
} result_t;

typedef struct {
    node_t *root;
    unsigned short order;
    bool lock;
} btree_t;

static int BTreeGetLeftMax(node_t *T);
static int BTreeGetRightMin(node_t *T);
/* The AllocateNode operation allocate a b-tree node.And then set the node's
** properties to the defualt value :
**   BTreeNode => K[i] = 0
**   BTreeNode => child_array[i] = NULL
**   BTreeNode => key_index = 0
**   BTreeNode => isLeaf = 1;
*/
static node_t *create_node()
{
    int i;
    node_t *new_node = (node_t *)malloc(sizeof(node_t));

    if(!new_node){
        printf("Out of memory");
        exit(0);
    }

    // Set Keys
    for(i = 0;i < NODE_KEYS; i++){
        new_node->key_array[i] = 0;
    }

    // Set ptr
    for(i = 0;i < NODE_POINTERS; i++){
        new_node->child_array[i] = NULL;
    }

    new_node->key_index = EMPTY;
    new_node->leaf = TRUE;

    return new_node;
}
/* The CreatBTree operation creates an empty b-tree by allocating a new root
** that has no keys and is a leaf node.Only the root node is permitted to
** have this properties.
*/
btree_t *create_btree()
{
    btree_t *new_root = (btree_t *)malloc(sizeof(btree_t));

    if(!new_root){
        return NULL;
    }

    node_t *head = create_node();

    if(!head){
        return NULL;
    }

    new_root->order = NODE_ORDER;
    new_root->root = head;
    new_root->lock = FALSE;

    return new_root;
}

static result_t *get_resultset()
{
    result_t *ret = (result_t *)malloc(sizeof(result_t));

    if(!ret){
        printf("ERROR! Out of memory.");
        exit(0);
    }

    ret->node_pointer = NULL;
    ret->key = 0;
    ret->found = FALSE;
    ret->depth = 0;

    return ret;
}

/* The BTreeSearch operation is to search X in T.Recursively traverse the tree
** from top to bottom.At each level, BTreeSearch choose the maximum key whose
** value is greater than or equal to the desired value X.If equal to the
** desired ,found.Otherwise continue to traverse.
*/
result_t *search(int key, node_t *node)
{
    print_node(node);
    int i = 0;

    while((i < node->key_index) && (key > node->key_array[i])){
        //printf("it %d is <= %d and key %d > than %d\n", i, node->key_index, key, node->key_array[i]);
        i++;
    }
//printf("end iterator: %d\n", i);

//printf("better: \n");
/*
    int c = 0;
    while((c < node->key_index) && (key > node->key_array[c])){
        printf("it %d is <= %d and key %d > than %d\n", c, node->key_index, key, node->key_array[c]);
        c++;
    }
*/
    // HACK /// may not be working 
    if(i == 6){
        i--;
    }

    // Check if we found it
    if((i <= node->key_index) && (key == node->key_array[i])){
        result_t *result = get_resultset();
        result->node_pointer = node;
        result->key = i;
        result->found = TRUE;
        return result;
    }

    // Not found check leaf or child
    if(node->leaf){
        result_t *result = get_resultset();
        result->node_pointer = node;
        result->found = FALSE;
        return result;
    }else{
        result_t *result = get_resultset();
        return search(key, node->child_array[i]);
    }
}
/* The split_child operation moves the median key of node child_array into
** its parent ptrParent where child_array is the ith child of ptrParent.
*/
static void split_child(node_t *parent_node, int i, node_t *child_array)
{
    int j;

    //Allocate a new node to store child_array's node.
    node_t *new_node = create_node();
    new_node->leaf = child_array->leaf;
    new_node->key_index = NODE_ORDER-1;

    //Move child_array's right half nodes to the new node.
    for(j = 0;j < NODE_ORDER-1;j++){
        new_node->key_array[j] = child_array->key_array[NODE_ORDER+j];
    }

    //If child_array is not leaf node,then move child_array's [child_array]s to the new
    //node's [child_array]s.
    if(child_array->leaf == 0){
        for(j = 0;j < NODE_ORDER;j++){
            new_node->child_array[j] = child_array->child_array[NODE_ORDER+j];
        }
    }
    child_array->key_index = NODE_ORDER-1;

    //Right shift ptrParent's [child_array] from index i
    for(j = parent_node->key_index;j>=i;j--){
        parent_node->child_array[j+1] = parent_node->child_array[j];
    }

    //Set ptrParent's ith child_array to the newNode.
    parent_node->child_array[i] = new_node;

    //Right shift ptrParent's Keys from index i-1
    for(j = parent_node->key_index;j>=i;j--){
        parent_node->key_array[j] = parent_node->key_array[j-1];
    }

    //Set ptrParent's [i-1]th Key to child_array's median [child_array]
    parent_node->key_array[i-1] = child_array->key_array[NODE_ORDER-1];

    //Increase ptrParent's Key number.
    parent_node->key_index++;
}
/* The BTreeInsertNonFull operation insert X into a non-full node T.before
** execute this operation,guarantee T is not a full node.
*/
static void insert_nonfull(node_t *n, int key){
    int i = n->key_index;
    
    if(n->leaf){
        // Shift until we fit
        while(i>=1 && key<n->key_array[i-1]){
            n->key_array[i] = n->key_array[i-1];
            i--;
        }
        n->key_array[i] = key;
        n->key_index++;
    }else{
        // Find the position i to insert.
        while(i>=1 && key<n->key_array[i-1]){
            i--;
        }
        //If T's ith child_array is full,split first.
        if(n->child_array[i]->key_index == NODE_KEYS){
            split_child(n, i+1, n->child_array[i]);
            if(key > n->key_array[i]){
                i++;
            }
        }
        //Recursive insert.
        insert_nonfull(n->child_array[i], key);
    }
}
/* The BTreeInsert operation insert key into T.Before insert ,this operation
** check whether T's root node is full(root->key_index == 2*d -1) or not.If full,
** execute split_child to guarantee the parent never become full.And then
** execute BTreeInsertNonFull to insert X into a non-full node.
*/
node_t *insert(int key, btree_t *b)
{
    if(!b->lock){
        node_t *root = b->root;
        if(root->key_index == NODE_KEYS){ //If node root is full,split it.
            node_t *newNode = create_node();
            b->root = newNode; //Set the new node to T's Root.
            newNode->leaf = 0;
            newNode->key_index = 0;
            newNode->child_array[0] = root;
            split_child(newNode, 1, root);//Root is 1th child of newNode.
            insert_nonfull(newNode, key); //Insert X into non-full node.
        }else{ //If not full,just insert X in T.
            insert_nonfull(b->root, key);
        }
    }else{
        printf("Tree is locked\n");
    }

    return b->root;
}
/* The merge_children operation merge the root->K[index] and its two child
** and then set chlid1 to the new root.
*/
static void merge_children(node_t *root, int index, node_t *child1, node_t *child2){
    child1->key_index = NODE_KEYS;
    int i;
    //Move child2's key to child1's right half.
    for(i=NODE_ORDER;i<NODE_KEYS;i++)
        child1->key_array[i] = child2->key_array[i-NODE_ORDER];
    child1->key_array[NODE_ORDER-1] = root->key_array[index]; //Shift root->K[index] down.
    //If child2 is not a leaf node,must copy child2's [ptrchlid] to the new
    //root(child1)'s [child_array].
    if(0 == child2->leaf){
        for(i=NODE_ORDER;i<NODE_POINTERS;i++)
            child1->child_array[i] = child2->child_array[i-NODE_ORDER];
    }
    //Now update the root.
    for(i=index+1;i<root->key_index;i++){
        root->key_array[i-1] = root->key_array[i];
        root->child_array[i] = root->child_array[i+1];
    }
    root->key_index--;
    free(child2);
}
/* The BTreeBorrowFromLeft operation borrows a key from leftPtr.curPtr borrow
** a node from leftPtr.root->K[index] shift down to curPtr,shift leftPtr's
** right-max key up to root->K[index].
*/
static void BTreeBorrowFromLeft(node_t *root, int index, node_t *leftPtr, node_t *curPtr){
    curPtr->key_index++;
    int i;
    for(i=curPtr->key_index-1;i>0;i--)
        curPtr->key_array[i] = curPtr->key_array[i-1];
    curPtr->key_array[0] = root->key_array[index];
    root->key_array[index] = leftPtr->key_array[leftPtr->key_index-1];
    if(0 == leftPtr->leaf)
        for(i=curPtr->key_index;i>0;i--)
            curPtr->child_array[i] = curPtr->child_array[i-1];
    curPtr->child_array[0] = leftPtr->child_array[leftPtr->key_index];
    leftPtr->key_index--;
}
/* The BTreeBorrowFromLeft operation borrows a key from rightPtr.curPtr borrow
** a node from rightPtr.root->K[index] shift down to curPtr,shift RightPtr's
** left-min key up to root->K[index].
*/
static void BTreeBorrowFromRight(node_t *root, int index, node_t *rightPtr, node_t *curPtr){
    curPtr->key_index++;
    curPtr->key_array[curPtr->key_index-1] = root->key_array[index];
    root->key_array[index] = rightPtr->key_array[0];
    int i;
    for(i=0;i<rightPtr->key_index-1;i++)
        rightPtr->key_array[i] = rightPtr->key_array[i+1];
    if(0 == rightPtr->leaf){
        curPtr->child_array[curPtr->key_index] = rightPtr->child_array[0];
        for(i=0;i<rightPtr->key_index;i++)
            rightPtr->child_array[i] = rightPtr->child_array[i+1];
    }
    rightPtr->key_index--;
}
/* The BTreeDeleteNoNone operation recursively delete X in root,handle both leaf
** and internal node:
**   1. If X in a leaf node,just delete it.
**   2. If X in a internal node P:
**      a): If P's left neighbor -> prePtr has at least d keys,replace X with
**          prePtr's right-max key and then recursively delete it.
**      b): If P's right neighbor -> nexPtr has at least d keys,replace X with
**          nexPtr's left-min key and then recursively delete it.
**      c): If both of prePtr and nexPtr have d-1 keys,merge X and nexPtr into
**          prePtr.Now prePtr have 2*d-1 keys,and then recursively delete X in
**          prePtr.
**   3. If X not in a internal node P,X must in P->child_array[i] zone.If child_array[i]
**      only has d-1 keys:
**      a): If child_array[i]'s neighbor have at least d keys,borrow a key from
**          child_array[i]'s neighbor.
**      b): If both of child_array[i]'s left and right neighbor have d-1 keys,merge
**          child_array[i] with one of its neighbor.
**      finally,recursively delete X.
*/
static void BTreeDeleteNoNone(int X, node_t *root){
    int i;
    //Is root is a leaf node ,just delete it.
    if(1 == root->leaf){
        i=0;
        while( (i<root->key_index) && (X>root->key_array[i])) //Find the index of X.
            i++;
        //If exists or not.
        if(X == root->key_array[i]){
            for(;i<root->key_index-1;i++)
                root->key_array[i] = root->key_array[i+1];
            root->key_index--;
        }
        else{
            printf("Node not found.\n");
            return ;
        }
    }
    else{  //X is in a internal node.
        i = 0;
        node_t *prePtr = NULL, *nexPtr = NULL;
        //Find the index;
        while( (i<root->key_index) && (X>root->key_array[i]) )
            i++;
        if( (i<root->key_index) && (X == root->key_array[i]) ){ //Find it in this level.
            prePtr = root->child_array[i];
            nexPtr = root->child_array[i+1];
            /*If prePtr at least have d keys,replace X by X's precursor in
             *prePtr*/
            if(prePtr->key_index > NODE_ORDER-1){
                int aPrecursor = BTreeGetLeftMax(prePtr);
                root->key_array[i] = aPrecursor;
                //Recursively delete aPrecursor in prePtr.
                BTreeDeleteNoNone(aPrecursor,prePtr);
            }
            else
            if(nexPtr->key_index > NODE_ORDER-1){
            /*If nexPtr at least have d keys,replace X by X's successor in
             * nexPtr*/
                int aSuccessor = BTreeGetRightMin(nexPtr);
                root->key_array[i] = aSuccessor;
                BTreeDeleteNoNone(aSuccessor,nexPtr);
            }
            else{
            /*If both of root's two child have d-1 keys,then merge root->K[i]
             * and prePtr nexPtr. Recursively delete X in the prePtr.*/
                merge_children(root,i,prePtr,nexPtr);
                BTreeDeleteNoNone(X,prePtr);
            }
        }
        else{ //Not find in this level,delete it in the next level.
            prePtr = root->child_array[i];
            node_t *leftBro = NULL;
            if(i<root->key_index)
                nexPtr = root->child_array[i+1];
            if(i>0)
                leftBro = root->child_array[i-1];
            /*root->child_array[i] need to borrow from or merge with his neighbor
             * and then recursively delete. */
            if(NODE_ORDER-1 == prePtr->key_index){
            //If left-neighbor have at least d-1 keys,borrow.
                if( (leftBro != NULL) && (leftBro->key_index > NODE_ORDER-1))
                    BTreeBorrowFromLeft(root,i-1,leftBro,prePtr);
                else //Borrow from right-neighbor
                if( (nexPtr != NULL) && (nexPtr->key_index > NODE_ORDER-1))
                    BTreeBorrowFromRight(root,i,nexPtr,prePtr);
                //OR,merge with its neighbor.
                else if(leftBro != NULL){
                    //Merge with left-neighbor
                    merge_children(root,i-1,leftBro,prePtr);
                    prePtr = leftBro;
                }
                else //Merge with right-neighbor
                    merge_children(root,i,prePtr,nexPtr);
            }
            /*Now prePtr at least have d keys,just recursively delete X in
             * prePtr*/
            BTreeDeleteNoNone(X,prePtr);
        }
    }
}
/*Get T's left-max key*/
static int BTreeGetLeftMax(node_t *T){
    if(0 == T->leaf){
        return BTreeGetLeftMax(T->child_array[T->key_index]);
    }else{
        return T->key_array[T->key_index-1];
    }
}
/*Get T's right-min key*/
static int BTreeGetRightMin(node_t *T){
    if(0 == T->leaf){
        return BTreeGetRightMin(T->child_array[0]);
    }else{
        return T->key_array[0];
    }
}
/* The BTreeDelete operation delete X from T up-to-down and no-backtrack.
** Before delete,check if it's necessary to merge the root and its children
** to reduce the tree's height.Execute BTreeDeleteNoNone to recursively delete
*/
node_t *delete(int key, btree_t *b)
{
    if(!b->lock){
        //if the root of T only have 1 key and both of T's two child have d-1
        //key,then merge the children and the root. Guarantee not need to backtrack.
        if(b->root->key_index == 1){
            node_t *child1 = b->root->child_array[0];
            node_t *child2 = b->root->child_array[1];
            if((!child1) && (!child2)){
                if((NODE_ORDER-1 == child1->key_index) && (NODE_ORDER-1 == child2->key_index)){
                    //Merge the children and set child1 to the new root.
                    merge_children(b->root, 0, child1, child2);
                    free(b->root);
                    BTreeDeleteNoNone(key, child1);
                    return child1;
                }
            }
        }
        BTreeDeleteNoNone(key, b->root);
    }else{
        printf("Tree is locked\n");
    }
    return b->root;
}

void tree_unlock(btree_t *r)
{
    r->lock = FALSE;
}

bool tree_lock(btree_t *r)
{
    if(r->lock){
        return FALSE;
    }   
    r->lock = TRUE;

    return TRUE;
}

Answer 1

You have not shown any code, which makes it difficult to come up with code examples that could fit to your implementation. However, in principle you could take the following approaches:

• Write unit-tests for your code. With the B-Tree that would mean to start with small trees (even with an empty tree), and use checks in your tests to verify the properties. You would then add more and more tests, specifically checking for bugs also in the "tricky" scenarios. There is a lot of general information about unit-testing available, you should be able to adapt it to your specific problem.
• Add assertions to your code (read about the assert macro in C). Many of the properties you have mentioned could be checked directly within the code at appropriate places.

Certainly, there is more you could do, like, having the code reviewed by some colleague, or using some formal verification tools, but the abovementioned two approaches are good starting points.

UPDATE (after code was added):

Some more hints about how you could approach unit-testing. In principle, you should write your tests with the help of a so called test framework, which is a helper library to make writing tests easier. To explain the concept, however, I just use plain C or even pseudo-code.

Moreover, you would also put some declarations and/or definitions into a header file, like "btree.h". For the sake of example, however, I will just #include "btree.c" in the code examples below.

• Create a file "btree-test.c" (the name is a proposal, you can name it as you like).
• A first test would look a bit like:

#include "btree.c"
#include <assert.h>

void test_create_empty_btree() {
    btree_t *actual_btree = create_btree();
    // now, check that the created btree has all desired properties
    // for example:
    assert(actual_btree != NULL);
    assert(actual_btree->order == NODE_ORDER);
    assert(actual_btree->lock == FALSE);
    assert(actual_btree->root->key_index == EMPTY);
    assert(actual_btree->root->leaf == TRUE);
    printf("PASSED: test_create_empty_btree");
}

The code above is just an example, I have not even tried compiling it. Note also that the test is not quite clean yet: there will be memory leaks, because the btree is not properly deleted at the end of the test, which would be better practice. It should, however, give you an idea how to start writing unit-tests.

A second test could then again create a btree, but in addition insert some data. In your tests you would then check that the btree has the expected form. And so on, adding more and more tests. It is good practice to have one function per test case...