Fuw-Yi Yang 1

演算法概論 Introduction to Algorithms 

Department of Computer Science and Information Engineering, Chaoyang University of Technology

朝陽科技大學資工系

Speaker: Fuw-Yi Yang 楊伏夷

伏夷非征番 , 道德經 察政章 (Chapter 58) 伏者潛藏也 道紀章 (Chapter 14) 道無形象 , 視之不可見者曰夷

Fuw-Yi Yang 2

Reference: T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 3rd ed.

Chapter Heap

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Chapter 6 Heap—Parent, Left, Right

Parent(i)

1. Return i/2

Left(i)

1. Return 2i

Right(i)

1. Return 2i + 1

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Chapter 6 Heap—Max-Heapify

Max-Heapify(A, i)

1. l = Left(i)

2. r = Right(i)

3. if l A.heap-size and A[l] > A[i] then largest = l

4. else largest = i

5. if r A.heap-size and A[r] > A[largest] then largest = r

6. if largest i then Exchange A[i] with A[largest]

7. Max-Heapify (A, largest)

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Chapter 6 Heap—Max-Heapify

Min-Heapify(A, i)

1. l = Left(i)

2. r = Right(i)

3. if l A.heap-size and A[l] < A[i] then smallest = l

4. else smallest = i

5. if r A.heap-size and A[r] < A[smallest] then smallest = r

6. if smallest i then Exchange A[i] with A[smallest]

7. Min-Heapify (A, smallest)

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Chapter 6 Heap—Build-Max-Heap

Build-Max-Heap(A)

1. A.heap-size = A.length

2. for i = A.length / 2 downto 1

3. Max-Heapify(A, i)

Build-Min-Heap(A)

1. A.heap-size = A.length

2. for i = A.length / 2 downto 1

3. Min-Heapify(A, i)

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Chapter 6 Heap — Complexity of Build-Max-Heap

1. Each call to Max-Heapify costs O(log n) time, there are O(n)

such calls. Thus the complexity is O(n log n).

2. Tighter analysis results in the complexity of O(n).

The time required by Max-Heapify when called on a node of height h is O(h), and so we can express the total cost of Build-Max-Heap as being bounded by

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Chapter 6 Heap— Complexity of Build-Max-Heap

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Chapter 6 Heap— Complexity of Build-Max-Heap

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Chapter 6 Heap—HeapSort

HeapSort(A) //O(n log n)

1. Build-Max-Heap(A) //O(n)

2. for i = A.length downto 2 //O(n)

3. Exchange A[1] with A[i] //O(1)

4. A.heap-size = A.heap-size – 1 //O(1)

5. Max-Heapify(A, 1) //O(log n)

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Chapter 6 Heap—Priority Queues

A priority queue is a data structure for maintaining a set S of elements, each with an associated value called a key.

A max-priority queue supports the following operations:

Insert(S, x) inserts the element x into the set S,

Maximum(S) returns the element of S with the largest key,

Extract-Max(S) removes and returns the element of S with the

largest key,

Increase-Key(S, x, k) increase the value of element x’s key to the

new value k, which is assumed to be at least as large as x’s

current key value.

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Chapter 6 Heap—Priority Queues

Maximum

Heap-Maximum (A)

1. Return A[1]

Heap-Minimum (A)

1. Return A[1]

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Chapter 6 Heap—Priority Queues

Extract-Max

Heap-Extract-Max (A)

1. If A.heap-size < 1

2. Error “heap empty”

3. Max = A[1]

4. A[1] = A[A.heap-size]

5. A.heap-size = A.heap-size - 1

6. Max-Heapify(A, 1)

7. Return Max

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Chapter 6 Heap—Priority Queues

Extract-Min

Heap-Extract-Min (A)

1. If A.heap-size < 1

2. Error “heap empty”

3. Min = A[1]

4. A[1] = A[A.heap-size]

5. A.heap-size = A.heap-size - 1

6. Min-Heapify(A, 1)

7. Return Max

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Chapter 6 Heap—Priority Queues

Increase key

Heap-Increase-Key (A, i, key)

1. If key < A[i]

2. Error “new key is smaller than current key”

3. A[i] = key

4. While i > 1 and A[Parent(i)] < A[i]

5. Exchange A[i] with A[Parent(i)]

6. i = Parent(i)

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Chapter 6 Heap—Priority Queues

Increase key

Heap-Decrease-Key (A, i, key)

1. If key > A[i]

2. Error “new key is larger than current key”

3. A[i] = key

4. While i > 1 and A[Parent(i)] > A[i]

5. Exchange A[i] with A[Parent(i)]

6. i = Parent(i)

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Chapter 6 Heap—Priority Queues

Insert

Max-Heap-Insert (A, key)

1. A.heap-size = A.heap-size + 1

2. A[A.heap-size] = -∞

4. Heap-Increase-Key(A, A.heap-size, key)

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Chapter 6 Heap—Priority Queues

Insert

Min-Heap-Insert (A, key)

1. A.heap-size = A.heap-size + 1

2. A[A.heap-size] = ∞

4. Heap-Decrease-Key(A, A.heap-size, key)