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Heaps Chapter 10 has several programming projects, including a project that uses heaps. This presentation shows you what a heap is, and demonstrates two of the important heap algorithms. This chapter also takes a look at B-trees Data Structures and Other Objects Using Java Heaps A heap is a certain kind of complete binary tree. Heaps Root A heap is a certain kind of complete binary tree. When a complete binary tree is built, its first node must be the root. Heaps Left child Complete of the binary tree. root The second node is always the left child of the root. Heaps Right child Complete of the binary tree. root The third node is always the right child of the root. Heaps Complete binary tree. The next nodes always fill the next level from left-to-right. Heaps Complete binary tree. The next nodes always fill the next level from left-to-right. Heaps Complete binary tree. The next nodes always fill the next level from left-to-right. Heaps Complete binary tree. The next nodes always fill the next level from left-to-right. Heaps Complete binary tree. Heaps 45 A heap is a certain kind 35 23 of complete binary tree. 27 21 22 4 19 Each node in a heap contains a key that can be compared to other nodes' keys. Heaps 45 A heap is a certain kind 35 23 of complete binary tree. 27 21 22 4 19 The "heap property" requires that each node's key is >= the keys of its children Adding a Node to a Heap 45 ¶ Put the new node in the next available spot. 35 23 Push the new node upward, swapping with its parent until the new 27 21 22 4 node reaches an acceptable location. 19 42 Adding a Node to a Heap 45 ¶ Put the new node in the next available spot. 35 23 Push the new node upward, swapping with its parent until the new 42 21 22 4 node reaches an acceptable location. 19 27 Adding a Node to a Heap 45 ¶ Put the new node in the next available spot. 42 23 Push the new node upward, swapping with its parent until the new 35 21 22 4 node reaches an acceptable location. 19 27 Adding a Node to a Heap 45 The parent has a key that is >= new node, or 42 23 The node reaches the root. The process of pushing 35 21 22 4 the new node upward is called 19 27 reheapification upward. Removing the Top of a Heap 45 ¶ Move the last node onto the root. 42 23 35 21 22 4 19 27 Removing the Top of a Heap 27 ¶ Move the last node onto the root. 42 23 35 21 22 4 19 Removing the Top of a Heap 27 ¶ Move the last node onto the root. 42 23 Push the out-of-place node downward, swapping with its larger 35 21 22 4 child until the new node reaches an acceptable 19 location. Removing the Top of a Heap 42 ¶ Move the last node onto the root. 27 23 Push the out-of-place node downward, swapping with its larger 35 21 22 4 child until the new node reaches an acceptable 19 location. Removing the Top of a Heap 42 ¶ Move the last node onto the root. 35 23 Push the out-of-place node downward, swapping with its larger 27 21 22 4 child until the new node reaches an acceptable 19 location. Removing the Top of a Heap 42 The children all have keys <= the out-of-place 35 23 node, or The node reaches the leaf. 27 21 22 4 The process of pushing the new node 19 downward is called reheapification downward. Implementing a Heap 42 We will store the data from the 35 23 nodes in a partially-filled 27 21 array. An array of data Implementing a Heap 42 Data from the root goes in the 35 23 first location 27 21 of the array. 42 An array of data Implementing a Heap 42 Data from the next row goes in the 35 23 next two array locations. 27 21 42 35 23 An array of data Implementing a Heap 42 Data from the next row goes in the 35 23 next two array locations. 27 21 42 35 23 27 21 An array of data Implementing a Heap 42 Data from the next row goes in the 35 23 next two array locations. 27 21 42 35 23 27 21 An array of data We don't care what's in this part of the array. Important Points about the Implementation 42 The links between the tree's nodes are not actually stored as 35 23 pointers, or in any other way. The only way we "know" that "the array is a tree" is from the 27 21 way we manipulate the data. 42 35 23 27 21 An array of data Important Points about the Implementation 42 If you know the index of a node, then it is easy to figure 35 23 out the indexes of that node's parent and children. Formulas are given in the book. 27 21 42 35 23 27 21 [0] [1] [2] [3] [4] Summary A heap is a complete binary tree, where the entry at each node is greater than or equal to the entries in its children. To add an entry to a heap, place the new entry at the next available spot, and perform a reheapification upward. To remove the biggest entry, move the last node onto the root, and perform a reheapification downward. Depth of a binary search tree The first tree has a large depth that would not have to be if it was like the more balanced second tree Count 1 6 5 Count 1 4 100 3 100 1 6 150 1 4 6 150 2 5 2 4 6 4 Count 2 3 Count 2 B-trees Problem of Unbalanced Trees Solutions Allinvolve trees whose depth remains small Could balance trees periodically AVL trees Red-Black Trees We’ll look at B-trees B-Tree Rules Depends on a positive constant integer called MINIMUM Rule 1: The root may have as few as one element (or none if no children) ; every other node has at least MINIMUM elements Rule 2: The maximum number of elements in a node is twice the value of MINIMUM More rules about B-tree Rule 3: The elements of each B-tree node are stored in a partially filled array, sorted from the smallest element (at index 0) to the largest element (at the final position of the array) Rule 4: The number of subtrees below node depends on how many elements are in a node: always one more Subtrees below a B-tree node Rule 5: For any non-leaf node An element at index i is greater than all the elements in subtree number i of the node An element at index i is less than all the elements in subtree number i+1 of the node A B-tree is balanced Rule 6: Every leaf in a B-tree has the same depth Sample B-tree 6 2 and 4 9 1 3 5 7 and 8 10 Every child of the root node is also the root node of a smaller B-tree Non-leaf node with two elements 93 and 107 Subtree 0 Subtree 1 Subtree 2 Searching for a number Subset [0] dataCount =1 childCount=2 6 Subset[1] 2 and 4 9 1 3 5 7 and 8 10

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posted: | 11/25/2011 |

language: | English |

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