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Heap and Others

黃兆武

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Heap• A max tree is a tree in which the key value in

each node is no smaller than the key values in its children. A max heap is a complete binary tree that is also a max tree.

• A min tree is a tree in which the key value in each node is no larger than the key values in its children. A min heap is a complete binary tree that is also a min tree.

• Operations on heaps– creation of an empty heap– insertion of a new element into the heap; – deletion of the largest element from the heap

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*Figure 5.25: Sample max heaps (p.219)

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Property:The root of max heap (min heap) contains the largest (smallest).

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*Figure 5.26:Sample min heaps (p.220)

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Application: priority queue

• machine service– amount of time (min heap)– amount of payment (max heap)

• factory– time tag

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Example of Insertion to Max Heap

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15 2

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initial location of new node

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15 20

14 10 2

insert 21 into heap

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15 5

14 10 2

insert 5 into heap

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Insertion into a Max Heapvoid insert_max_heap(element item, int *n){ int i; if (HEAP_FULL(*n)) { fprintf(stderr, “the heap is full.\n”); exit(1); } i = ++(*n); while ((i!=1)&&(item.key>heap[i/2].key)) { heap[i] = heap[i/2]; i /= 2; } heap[i]= item;} 2k-1=n ==> k=log2(n+1)

O(log2n)

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Example of Deletion from Max Heap

20remove

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Deletion from a Max Heapelement delete_max_heap(int *n){ int parent, child; element item, temp; if (HEAP_EMPTY(*n)) { fprintf(stderr, “The heap is empty\n”); exit(1); } /* save value of the element with the highest key */

item = heap[1]; /* use last element in heap to adjust heap */ temp = heap[(*n)--]; parent = 1; child = 2;

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while (child <= *n) { /* find the larger child of the current parent */ if ((child < *n)&& (heap[child].key<heap[child+1].key)) child++; if (temp.key >= heap[child].key) break; /* move to the next lower level */ heap[parent] = heap[child]; child *= 2; } heap[parent] = temp; return item;}

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26[1]

5[2] 77[3]

1[4] 61[5] 11[6] 59[7]

15[8] 48[9] 19[10]

Heap Sort

1 2 3 4 5 6 7 8 9 1026 5 77 1 61 11 59 15 48 19

input file

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77[1]

61[2] 59[3]

48[4] 19[5] 11[6] 26[7]

15[8] 1[9] 5[10]

initial heap

exchange

Heap Sort

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61[1]

48[2] 59[3]

15[4] 19[5] 11[6]26[7]

5[8] 1[9] 77[10]

59[1]

48[2] 26[3]

15[4] 19[5] 11[6]1[7]

5[8] 61[9] 77[10]

(a)

(b)

Heap Sort

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48[1]

19[2] 26[3]

15[4] 5[5] 11[6]1[7]

59[8] 61[9] 77[10]

26[1]

19[2] 11[3]

15[4] 5[5] 1[6]48[7]

59[8] 61[9] 77[10]

(c)

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6159

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Heap Sort

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Heap Sortvoid adjust(element list[], int root, int n){ int child, rootkey; element temp; temp=list[root]; rootkey=list[root].key; child=2*root; while (child <= n) { if ((child < n) && (list[child].key < list[child+1].key)) child++; if (rootkey > list[child].key) break; else { list[child/2] = list[child]; child *= 2; } } list[child/2] = temp;}

i2i 2i+1

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Heap Sort

void heapsort(element list[], int n){ int i, j; element temp; for (i=n/2; i>0; i--) adjust(list, i, n); for (i=n-1; i>0; i--) { SWAP(list[1], list[i+1], temp); adjust(list, 1, i); }}

ascending order (max heap)

n-1 cylces

top-down

bottom-up

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AVL樹 (Balanced Binary Tree)

1.T是一個非空的二元樹， Tl及 Tr分別是它的左右子樹，若符合下列兩條件，則稱Ｔ為高度平衡樹。（ Height Balancing Binary Tree)

　　 (1) TL及 TR也是高度平衡樹　　 (2) | HL-HR | <= 1 ， HL及 HR分別為 TL及 TR的高度。

X ○

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1-2. Route balanced treeA binary tree with n nodes, where all nodes are at depth └ lg(n) ┘ or less, and where there are exactly 2d nodes at depth d for each depth 0 <= d < └ lg(n) ┘ , will be called route balanced.

Perfectly balanced tree

All six are route balanced

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Threaded Binary Trees• Two many null pointers incurrent

representation of binary trees n: number of nodes number of non-null links: n-1 total links: 2n null links: 2n-(n-1)=n+1

• Replace these null pointers with some useful “threads”.

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Threaded Binary Trees (Continued)

If ptr->left_child is null, replace it with a pointer to the node that would be visited before ptr in an inorder traversal

If ptr->right_child is null, replace it with a pointer to the node that would be visited after ptr in an inorder traversal

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A Threaded Binary Tree

A

B C

GE

I

D

H

F

root

dangling

dangling

inorder traversal:H, D, I, B, E, A, F, C, G

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TRUE FALSE

Data Structures for Threaded BT

typedef struct threaded_tree *threaded_pointer;

typedef struct threaded_tree { short int left_thread; threaded_pointer left_child; char data; threaded_pointer right_child; short int right_thread; };

left_thread left_child data right_child right_thread

FALSE: childTRUE: thread

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Memory Representation of A Threaded BT

f f--

f fA

f fCf fB

t tE t tF t tGf fD

t tIt tH

root

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Next Node in Threaded BTthreaded_pointer insucc(threaded_pointer tree){ threaded_pointer temp; temp = tree->right_child; if (!tree->right_thread) while (!temp->left_thread) temp = temp->left_child; return temp;}

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Inorder Traversal of Threaded BTvoid tinorder(threaded_pointer tree){/* traverse the threaded binary tree inorder */ threaded_pointer temp = tree; for (;;) { temp = insucc(temp); if (temp==tree) break; printf(“%3c”, temp->data); }}

O(n)

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Inserting Nodes into Threaded BTs

• Insert child as the right child of node parent– change parent->right_thread to FALSE– set child->left_thread and child->right_thread to

TRUE– set child->left_child to point to parent– set child->right_child to parent->right_child– change parent->right_child to point to child

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Examples

root

parent

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B

C Dchild

root

parent

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C D child

empty

Insert a node D as a right child of B.

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*Figure 5.24: Insertion of child as a right child of parent in a threaded binary tree (p.217)

nonempty

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Selection Trees

(1) winner tree(2) loser tree

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winner tree6

6 8

9 6 8 17

8 9 90 1720 610 91516

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run 1 run 2 run 3 run 4 run 5 run 6 run 7 run 8

ordered sequence

sequentialallocationscheme(completebinarytree)

Each node representsthe smaller of its twochildren.1

2 3

4 5 6 7

8 9 10 11 12 13 14 15

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*Figure 5.35: Selection tree of Figure 5.34 after one record has been output and the tree restructured(nodes that were changed are ticked)

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Analysis

• K: # of runs• n: # of records• setup time: O(K) (K-1)• restructure time: O(log2K) log2(K+1)

• merge time: O(nlog2K) • slight modification: tree of loser

– consider the parent node only (vs. sibling nodes)

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overallwinner

*Figure 5.36: Tree of losers corresponding to Figure 5.34 (p.235)

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Activity on Vertex (AOV) Network definitionA directed graph in which the vertices represent tasks or activities and the edges represent precedence relations between tasks.predecessor (successor)vertex i is a predecessor of vertex j iff there is a directed path from i to j. j is a successor of i.partial ordera precedence relation which is both transitive (i, j, k, ij & jk => ik ) and irreflexive (x xx).acylic grapha directed graph with no directed cycles

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*Figure 6.38: An AOV network (p.305)

Topological order:linear ordering of verticesof a graphi, j if i is a predecessor ofj, then i precedes j in thelinear ordering

C1, C2, C4, C5, C3, C6, C8,C7, C10, C13, C12, C14, C15, C11, C9

C4, C5, C2, C1, C6, C3, C8,C15, C7, C9, C10, C11, C13,C12, C14

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*Program 6.13: Topological sort (p.306)

for (i = 0; i <n; i++) { if every vertex has a predecessor { fprintf(stderr, “Network has a cycle. \n “ ); exit(1); } pick a vertex v that has no predecessors; output v; delete v and all edges leading out of v from the network;}

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*Figure 6.39:Simulation of Program 6.13 on an AOV network (p.306)

v0 no predecessordelete v0->v1, v0->v2, v0->v3

v1, v2, v3 no predecessorselect v3delete v3->v4, v3->v5

select v2delete v2->v4, v2->v5

select v5 select v1delete v1->v4

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*Figure 6.40:Adjacency list representation of Figure 6.30(a)(p.309)

0 1 2 3 NULL

1 4 NULL

1 4 5 NULL

1 5 4 NULL

3 NULL

2 NULL

V0

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count linkheadnodes

vertex linknode

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typedef struct node *node_pointer;typedef struct node { int vertex; node_pointer link; };typedef struct { int count; node_pointer link; } hdnodes;hdnodes graph[MAX_VERTICES];

Topological sort

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*Program 6.14: Topological sort (p.308)

O(n)

void topsort (hdnodes graph [] , int n){ int i, j, k, top; node_pointer ptr; /* create a stack of vertices with no predecessors */ top = -1; for (i = 0; i < n; i++) if (!graph[i].count) {no predecessors, stack is linked through graph[i].count = top; count field top = i; }for (i = 0; i < n; i++) if (top == -1) { fprintf(stderr, “\n Network has a cycle. Sort terminated. \n”); exit(1);}

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O(e)

O(e+n)

Continued} else { j = top; /* unstack a vertex */ top = graph[top].count; printf(“v%d, “, j); for (ptr = graph [j]. link; ptr ;ptr = ptr ->link ){ /* decrease the count of the successor vertices of j */ k = ptr ->vertex; graph[k].count --; if (!graph[k].count) { /* add vertex k to the stack*/ graph[k].count = top; top = k; } } } }