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Programming Assignment 2

In this assignment you will implement the page replacement procedure in the buffer and a B+ tree index for efficient lookups and range scans. We supply you with all the low-level code you will need to implement the tree structure. You will implement searching, splitting pages, redistributing tuples between pages, and merging pages (the last two are optional).

The remainder of this document gives some suggestions about how to start coding, describes a set of exercises to help you work through the assignment, and discusses how to hand in your code. This assignment requires you to write a fair amount of code, so we encourage you to start early!

Page eviction for BufferPool

In assignment 2, you will choose a page eviction policy and instrument any previous code that reads or creates pages to implement your policy.

When more than numPages pages are in the buffer pool, one page should be evicted from the pool before the next is loaded. The choice of eviction policy is up to you; it is not necessary to do something sophisticated. Describe your policy in the writeup. You need to implement a BufferPool::flushAllPages method. This is not something you would ever need in a real implementation of a buffer pool. However, we need this method for testing purposes. You should never call this method from any real code.

You will need to ensure that your flushPage and flushAllPages methods do not evict pages from the buffer pool to properly pass this test. flushAllPages should call flushPage on all pages in the BufferPool, and flushPage should write any dirty page to disk and mark it as not dirty, while leaving it in the BufferPool. The only method which should remove page from the buffer pool is evictPage, which should call flushPage on any dirty page it evicts.

Search in B+ Tree

Take a look at BTreeFile. This is the core file for the implementation of the B+Tree and where you will write all your code for this assignment. Unlike the HeapFile, the BTreeFile consists of four different kinds of pages. As you would expect, there are two different kinds of pages for the nodes of the tree: internal pages and leaf pages. Internal pages are implemented in BTreeInternalPage, and leaf pages are implemented in BTreeLeafPage. For convenience, we have created an abstract class in BTreePage which contains code that is common to both leaf and internal pages. In addition, header pages are implemented in BTreeHeaderPage and keep track of which pages in the file are in use. Lastly, there is one page at the beginning of every BTreeFile which points to the root page of the tree and the first header page. This singleton page is implemented in BTreeRootPtrPage. Familiarize yourself with the interfaces of these classes, especially BTreePage, BTreeInternalPage and BTreeLeafPage. You will need to use these classes in your implementation of the B+Tree.

Your first job is to implement the BTreeFile::findLeafPage function. This function is used to find the appropriate leaf page given a particular key value, and is used for both searches and inserts. For example, suppose we have a B+Tree with two leaf pages (See Figure 1). The root node is an internal page with one entry containing one key (6, in this case) and two child pointers. Given a value of 1, this function should return the first leaf page. Likewise, given a value of 8, this function should return the second page. The less obvious case is if we are given a key value of 6. There may be duplicate keys, so there could be 6's on both leaf pages. In this case, the function should return the first (left) leaf page.

Figure 1: A simple B+ Tree with duplicate keys

Figure 1: A simple B+ Tree with duplicate keys

Your findLeafPage function should search through internal nodes until it reaches the leaf page corresponding to the provided key value. In order to find the appropriate child page at each step, you should iterate through the entries in the internal page and compare the entry value to the provided key value. BTreeInternalPageIterator provides access to the entries in the internal page using the interface defined in BTreeEntry. This iterator allows you to iterate through the key values in the internal page and access the left and right child page ids for each key. The function returns when the passed-in BTreePageId has type equal to BTreePageId::LEAF, indicating that it is a leaf page. In this case, you should just fetch the page from the buffer pool and return it. You do not need to confirm that it actually contains the provided key value.

Your findLeafPage code must also handle the case a key value is not provided. In this case, go to the left-most child every time in order to find the left-most leaf page. Finding the left-most leaf page is useful for scanning the entire file. Once the correct leaf page is found, you should return it. As mentioned above, you can check the type of page using the type function in BTreePageId. You can assume that only leaf and internal pages will be passed to this function.

Instead of directly calling BufferPool::getPage to get each internal page and leaf page, we recommend calling the wrapper function we have provided, BTreeFile::getPage. It works exactly like BufferPool::getPage, but takes an extra argument to track the list of dirty pages. This function will be important for the next two exercises in which you will actually update the data and therefore need to keep track of dirty pages.

Every internal (non-leaf) page your findLeafPage implementation visits should be fetched with READ_ONLY permission, except the returned leaf page, which should be fetched with the permission provided as an argument to the function. These permission levels will not matter for this assignment, but they will be important for the code to function correctly in future works.

Also note that as part of this implementation, you will need to implement IndexPredicate and Predicate in order to support comparison operations.

Insert in B+ Tree

In order to keep the tuples of the B+Tree in sorted order and maintain the integrity of the tree, we must insert tuples into the leaf page with the enclosing key range. As was mentioned above, findLeafPage can be used to find the correct leaf page into which we should insert the tuple. However, each page has a limited number of slots, and we need to be able to insert tuples even if the corresponding leaf page is full.

Attempting to insert a tuple into a full leaf page should cause that page to split so that the tuples are evenly distributed between the two new pages. Each time a leaf page splits, a new entry corresponding to the first tuple in the second page will need to be added to the parent node. Occasionally, the internal node may also be full and unable to accept new entries. In that case, the parent should split and add a new entry to its parent. This may cause splits and ultimately the creation of a new root node.

In this exercise you will implement splitLeafPage and splitInternalPage in BTreeFile. If the page being split is the root page, you will need to create a new internal node to become the new root page, and update the BTreeRootPtrPage. Otherwise, you will need to fetch the parent page with READ_WRITE permissions, split it if necessary, and add a new entry. You will find the function getParentWithEmptySlots extremely useful for handling these different cases. In splitLeafPage you should "copy" the key up to the parent page, while in splitInternalPage you should "push" the key up to the parent page. See Figure 2 and review lecture slides and section 10.5 in the text book if this is confusing. Remember to update the parent pointers of the new pages as needed (for simplicity, we do not show parent pointers in the figures). When an internal node is split, you will need to update the parent pointers of all the children that were moved. You may find the function updateParentPointers() useful for this task. Additionally, remember to update the sibling pointers of any leaf pages that were split. Finally, return the page into which the new tuple or entry should be inserted, as indicated by the provided key field. (Hint: You do not need to worry about the fact that the provided key may actually fall in the exact center of the tuples/entries to be split. You should ignore the key during the split, and only use it to determine which of the two pages to return.)

Figure 2: Splitting pages

Figure 2: Splitting pages

Figure 2: Splitting pages

Whenever you create a new page, either because of splitting a page or creating a new root page, call getEmptyPage to get the new page. This function is an abstraction which will allow us to reuse pages that have been deleted due to merging (covered in the next section).

We expect that you will interact with leaf and internal pages using BTreeLeafPageIterator and BTreeInternalPageIterator to iterate through the tuples/entries in each page. For convenience, we have also provided reverse iterators for both types of pages: BTreeLeafPageReverseIterator and BTreeInternalPageReverseIterator. These reverse iterators will be especially useful for moving a subset of tuples/entries from a page to its right sibling.

As mentioned above, the internal page iterators use the interface defined in BTreeEntry, which has one key and two child pointers. It also has a recordId, which identifies the location of the key and child pointers on the underlying page. We think working with one entry at a time is a natural way to interact with internal pages, but it is important to keep in mind that the underlying page does not actually store a list of entries, but stores ordered lists of m keys and m+1 child pointers. Since the BTreeEntry is just an interface and not an object actually stored on the page, updating the fields of BTreeEntry will not modify the underlying page. In order to change the data on the page, you need to call BTreeInternalPage::updateEntry. Furthermore, deleting an entry actually deletes only a key and a single child pointer, so we provide the functions BTreeInternalPage::deleteKeyAndLeftChild and BTreeInternalPage::deleteKeyAndRightChild to make this explicit. The entry's recordId is used to find the key and child pointer to be deleted. Inserting an entry also only inserts a key and single child pointer (unless it's the first entry), so BTreeInternalPage::insertEntry checks that one of the child pointers in the provided entry overlaps an existing child pointer on the page, and that inserting the entry at that location will keep the keys in sorted order.

In both splitLeafPage and splitInternalPage, you will need to update the set of dirtypages with any newly created pages as well as any pages modified due to new pointers or new data. This is where BTreeFile::getPage will come in handy. Each time you fetch a page, BTreeFile::getPage will check to see if the page is already stored in the local cache (dirtypages), and if it can't find the requested page there, it fetches the page from the buffer pool. BTreeFile::getPage also adds pages to the dirtypages cache if they are fetched with read-write permission, since presumably they will soon be dirtied.

Note that in a major departure from HeapFile::insertTuple, BTreeFile::insertTuple could return a large set of dirty pages, especially if any internal pages are split. As you may remember from previous assignment, the set of dirty pages is returned to prevent the buffer pool from evicting dirty pages before they have been flushed.

Warning: as the B+Tree is a complex data structure, it is helpful to understand the properties necessary of every legal B+Tree before modifying it. Here is an informal list:

  1. If a parent node points to a child node, the child nodes must point back to those same parents.

  2. If a leaf node points to a right sibling, then the right sibling points back to that leaf node as a left sibling.

  3. The first and last leaves must point to null left and right siblings respectively.

  4. RecordId's must match the page they are actually in.

  5. A key in a node with non-leaf children must be larger than any key in the left child, and smaller than any key in the right child.

  6. A key in a node with leaf children must be larger or equal than any key in the left child, and smaller or equal than any key in the right child.

  7. A node has either all non-leaf children, or all leaf children.

  8. A non-root node cannot be less than half full.

Bonus: Delete in B+ Tree

This part is not required but if you can pass the tests you will earn some extra credits!

In order to keep the tree balanced and not waste unnecessary space, deletions in a B+Tree may cause pages to redistribute tuples (Figure 3) or, eventually, to merge (see Figure 4).

Figure 3: Redistributing internal pages

Figure 3: Redistributing leaf pages

Figure 3: Redistributing pages

Figure 4: Merging internal pages

Figure 4: Merging leaf pages

Figure 4: Merging pages

Attempting to delete a tuple from a leaf page that is less than half full should cause that page to either steal tuples from one of its siblings or merge with one of its siblings. If one of the page's siblings has tuples to spare, the tuples should be evenly distributed between the two pages, and the parent's entry should be updated accordingly (see Figure 3). However, if the sibling is also at minimum occupancy, then the two pages should merge and the entry deleted from the parent (Figure 4). In turn, deleting an entry from the parent may cause the parent to become less than half full. In this case, the parent should steal entries from its siblings or merge with a sibling. This may cause merges or even deletion of the root node if the last entry is deleted from the root node.

In this exercise you will implement stealFromLeafPage(), stealFromLeftInternalPage(), stealFromRightInternalPage(), mergeLeafPages() and mergeInternalPages() in BTreeFile.java. In the first three functions you will implement code to evenly redistribute tuples/entries if the siblings have tuples/entries to spare. Remember to update the corresponding key field in the parent (look carefully at how this is done in Figure 3 - keys are effectively "rotated" through the parent). In stealFromLeftInternalPage()/stealFromRightInternalPage(), you will also need to update the parent pointers of the children that were moved. You should be able to reuse the function updateParentPointers() for this purpose.

In mergeLeafPages() and mergeInternalPages() you will implement code to merge pages, effectively performing the inverse of splitLeafPage() and splitInternalPage(). You will find the function deleteParentEntry() extremely useful for handling all the different cases. Be sure to call setEmptyPage() on deleted pages to make them available for reuse. As with the previous exercises, we recommend using BTreeFile.getPage() to encapsulate the process of fetching pages and keeping the list of dirty pages up to date.

Exercises

Implement the methods as indicated by the // TODO pa2.x comments in the following files:

Exercise 1: Buffer Pool flush

Implement:

  • BufferPool::evictPage
  • BufferPool::flushAllPages
  • BufferPool::discardPage
  • BufferPool::flushPage
  • BufferPool::flushPages

Exercise 2: Find Leaf

Add private members:

  • Predicate
  • IndexPredicate

Implement:

  • Predicate::Predicate
  • Predicate::getField
  • Predicate::getOp
  • Predicate::getOperand
  • Predicate::filter
  • IndexPredicate::IndexPredicate
  • IndexPredicate::getField
  • IndexPredicate::getOp
  • IndexPredicate::operator==
  • std::hash<IndexPredicate>::operator()
  • BTreeFile::findLeafPage

Exercise 3: Splitting Pages

Implement:

  • BTreeFile::splitLeafPage
  • BTreeFile::splitInternalPage
  • BufferPool::insertTuple
  • BufferPool::deleteTuple

Exercise 4: Redistributing Pages (Bonus)

Implement:

  • BTreeFile::stealFromLeafPage
  • BTreeFile::stealFromLeftInternalPage
  • BTreeFile::stealFromRightInternalPage

Exercise 5: Merging Pages (Bonus)

Implement:

  • BTreeFile::mergeLeafPages
  • BTreeFile::mergeInternalPages