barcode in ssrs 2008 Figure 13.1 A heap in Java

Printer European Article Number 13 in Java Figure 13.1 A heap

Figure 13.1 A heap
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Heaps could represent family descendant trees because the heap property means that every parent is older than its children. Heaps are used to implement priority queues (page 247) and to the heap sort algorithm (page 266). THE NATURAL MAPPING Every complete binary tree has a natural mapping into an array. (See Algorithm 11.1 on page 206.) The mapping is obtained from a level-order traversal of the tree. In the resulting array, the parent of the element at index i is at index i/2, and the children are at indexes 2i and 2i+1. EXAMPLE 13.2 Storing a Heap in an Array
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The heap shown in Figure 13.1 maps into the array shown Figure 13.2. For example, element 60 is at index i = 5, its parent is element 66 at index i/2 = 2, and its children are elements 58 and 25 at indexes 2i = 10 and 2i + 1=11.
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Figure 13.2 Array storage of a heap
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HEAPS AND PRIORITY QUEUES
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The natural mapping between a complete binary tree and an array is a two-way correspondence. To map the array elements back into a complete binary tree, simply number the tree nodes consecutively in a level-order traversal beginning with number 1 at the root. Then copy the array element at index i into the tree node numbered i. The locations for those indexes are shown in Figure 13.3. If the resulting tree has the heap property, then we also say that the array has the heap property.
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Figure 13.3 Array indexes in a heap
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EXAMPLE 13.3 Determining Whether an Array Has the Heap Property
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To determine whether this array has the heap property, we first map it into a binary tree, and then check each root-to-leaf path. The root-to-leaf path {88, 66, 44, 51} shown in Figure 13.4 is not descending because 44 < 51. Hence, the tree does not have the heap property. Therefore, the array does not have the heap property.
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An array with the heap property is partially ordered. That means that most of the larger keys come before most of the smaller keys. More precisely, it means that every heap-path subarray is sorted in descending order, where a heappath subarray is a subsequence of array elements in which each index number is half Figure 13.4 Checking the heap property that of its successor. For example, {a[1], a[2], a[5], a[11], a[22], a[45], a[90], a[180]} would be a heap-path subarray of an array a[] of 200 elements. The heap sort algorithm (Algorithm 14.8 on page 266) exploits this fact to obtain a fast and efficient method for sorting arrays. INSERTION INTO A HEAP Elements are inserted into a heap next to its right-most leaf at the bottom level. Then the heap property is restored by percolating the new element up the tree until it is no longer older (i.e., its key is greater) than its parent. On each iteration, the child is swapped with its parent. EXAMPLE 13.4 Inserting into a Heap
Figure 13.5 on page 247 shows how the key 75 would be inserted into the heap shown in Figure 13.4. The element 75 is added to the tree as a new last leaf. Then it is swapped with its parent element 44 because 75 > 44. Then it is swapped with its parent element 66 because 75 > 66. Now the heap property has been restored because the new element 75 is less than its parent and greater than its children.
Note that the insertion affects only the nodes along a single root-to-leaf path.
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HEAPS AND PRIORITY QUEUES
Figure 13.5 Inserting 75 into a heap
REMOVAL FROM A HEAP The heap removal algorithm always removes the root element from the tree. This is done by moving the last leaf element into the root element and then restoring the heap property by percolating the new root element down the tree until it is no longer younger (i.e., its key is less) than its children. On each iteration, the parent is swapped with the older of its two children. EXAMPLE 13.5 Removing from a Heap
Figure 13.6 shows how the root element (key 88) would be removed from a heap. The last leaf (key 44) is removed and copied into the root, replacing the previous root (key 88), which is removed. Then, to restore the heap property, the element 44 is swapped with the larger of its two children (77). That step is repeated until the element 44 is no longer smaller than any of its children. In this case, the result is that 44 ends up as a leaf again.
Note that the removal affects only the nodes along a single root-to-leaf path. That gives us this result from Corollary 11.2 on page 202: Theorem 13.1 Insertions into and removals from a heap run in O(lgn) time. PRIORITY QUEUES A stack is a LIFO container: The last one in comes out first. A queue is a FIFO container: The first one in comes out first. A priority queue is a BIFO container : The best one in comes
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