Algorithm Design

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Chapter 2 Basics of Algorithm Analysis
~a
EaCh node’s key is at least~
s large as its parent’s.
1 2 5 10 3 7 11 15 17 20 9 15 8 16 X
Figure 2.3 Values in a heap shown as a binaD, tree on the left, and represented as an
array on the right. The arrows show the children for the top three nodes in the tree.
at position i, the children are the nodes at positions leftChild(i) = 2i and
rightChild(f) = 2i + 1. So the two children of the root are at positions 2 and
3, and the parent of a node at position i is at position parent(f) =/i/2J. If
the heap has n < N elements at some time, we will use the first rt positions
of the array to store the n heap elements, and use lenggh(H) to denote the
number of elements in H. This representation keeps the heap balanced at all
times. See the right-hand side of Figure 2.3 for the array representation of the
heap on the left-hand side.
Implementing the Heap Operations
The heap element with smallest key is at the root, so it takes O(1) time to
identify the minimal element. How do we add or delete heap elements? First
conside~ adding a new heap element v, and assume that our heap H has n < N
elements so far. Now it will have n + 1 elements. To start with, we can add the
new element v to the final position i = n + 1, by setting H[i] = v. Unfortunately,
this does not maintain the heap property, as the key of element v may be
smaller than the key of its parent. So we now have something that is almost-a
heap, except for a small "damaged" part where v was pasted on at the end.
We will use the procedure Heap±f y-up to fix our heap. Letj = parent(i) =
L//2] be the parent of the node i, and assume H[j] = w. If key[v] < key[w],
then we will simply swap the positions of v and w. This wil! fix the heap
property at position i, but the resulting structure will possibly fail to satisfy
the heap property at position j--in other words, the site of the "damage" has
moved upward from i to j. We thus call the process recursively from position
2.5 A More Complex Data Structure: Priority Queues
The H e a p i fy - u p process is moving I
element v toward the root.
Figure 2.4 The Heapify-up process. Key 3 (at position 16) is too small (on the left).
After swapping keys 3 and 11, the heap xdolation moves one step closer to the root of
the tree (on the right).
] = parent(i) to continue fixing the heap by pushing the damaged part upward.
Figure 2.4 shows the first two steps of the process after an insertion.
Heapify-up (H, i) :
If i> 1 then
let ] = parent(i) = Lil2J
If key[H[i]] key(v) such that raising the value of key(v) to c~ would make
the resulting array satisfy the heap property. (In other words, element v in H[i]
is too small, but raising it to cz would fix the problem.) One important point
to note is that if H is almost a heap with the key of the root (i.e., H[1]) too
small, then in fact it is a~heap. To see why this is true, consider that if raising
the value of H[1] to c~ would make H a heap, then the value of H[!] must
also be smaller than both its children, and hence it already has the heap-order
property.
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