✨ Solutions of 6.5

Author: MurphysL

src/C6/CLRS 6.4.md

@@ -16,7 +16,9 @@ https://github.com/MurphysL/Introduction-to-Algorithms/blob/master/src/C6/code/s
 **Answer：**
 
 **初始化：**第一次循环之前，此时为最大堆，子数组为空。
+
 **保持：**每次迭代，最大堆堆顶元素被交换，交换后调用 MAX-HEAPIFY( A,  1)保持最大堆性质。经 i 此迭代后，子数组为 A[i+1..n]，为从小到大排序的数组。
+
 **终止：**i = 1 时过程终止。
 
 

src/C6/CLRS 6.5.md

@@ -0,0 +1,71 @@
+# CLRS 6.5
+
+## 6.5-3
+
+**Write pseudocode for the procedures HEAP-MINIMUM, HEAP-EXTRACT-MIN, HEAP- DECREASE-KEY, and MIN-HEAP-INSERT that implement a min-priority queue with a min-heap.**
+
+**Answer：**
+
+https://github.com/MurphysL/Introduction-to-Algorithms/blob/master/src/C6/code/s6.5-3.kt
+
+## 6.5-4
+
+**Why do we bother setting the key of the inserted node to -∞ in line 2 of MAX-HEAP- INSERT when the next thing we do is increase its key to the desired value?**
+
+**Answer：**
+
+确保 key 的值小于 A[heap.size]
+
+## 6.5-5
+
+**Argue the correctness of HEAP-INCREASE-KEY using the following loop invariant:**
+> At the start of each iteration of the while loop of lines 4-6, the array A[1...heap- size[A]] satisfies the max-heap property, except that there may be one violation: A[i] may be larger than A[PARENT(i)].
+
+**Answer：**
+
+**初始化：**第一次循环迭代开始之前，除 A[heap.size] 外，其他元素均满足最大堆的性质。
+**保持：**循环过程中，不断交换 key 与 其父结点的值使其满足最大堆性质。
+**终止：**当 i = 1 时，所有元素已按最大堆的性质排好。
+
+## 6.5-6
+
+**Each exchange operation on line 5 of HEAP-INCREASE-KEY typically requires three asignments. Show how to use the idea of the inner loop of INSERTION-SORT to reduce the three assignments down to just one assignment.**
+
+**Answer：**
+```
+HEAP-INCREASE-KEY(A, I, key):
+	if key < A[i]
+		error "new key is smaller than current key"
+	A[i] = key
+	while i > 1 and A[PARENT(i)] < key:
+		A[i] = A[PARENT]
+		i = PARENT(i)
+	A[i] = key 
+```
+
+## 6.5-7
+
+**Show how to implement a first-in, first-out queue with a priority queue. Show how to implement a stack with a priority queue. (Queues and stacks are defined in Section 10.1.)**
+
+**Answer：**
+* **队列：**将新插入的值赋予更低的优先级
+* **栈：**将新插入的值赋予更高的优先级
+
+## 6.5-8
+
+**The operation HEAP-DELETE(A, i) deletes the item in node i from heap A. Give an implementation of HEAP-DELETE that runs in O(lg n) time for an n-element max-heap.**
+
+**Answer：**
+
+* 当 **A[A.heap-size] == A[i]** 时，时间复杂度为 O(1)
+* 当 **A[A.heap-size] > A[i]** 时，时间复杂度为 O(lgn)
+* 当 **A[A.heap-size] < A[i]** 时，时间复杂度为 O(lgn)，例：
+
+>      10
+>    /    \
+>   5      9
+>  / \    / \
+> 2   3  7   8
+
+https://github.com/MurphysL/Introduction-to-Algorithms/blob/master/src/C6/code/s6.5-8.kt
+