Algorithms, Spring 2014, CSE, OSU Lecture 2: Sorting
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1 Algorithms, Spring 2014, CSE, OSU Lecture 2: Sorting Instructor: Anastasios Sidiropoulos January 10, 2014
2 Sorting Given an array of integers A[1... n], rearrange its elements so that A[1] A[2]... A[n].
3 A simple sorting algorithm Bubble-Sort repeat swapped = false for i = 1 to n 1 do if A[i 1] > A[i] then swap(a[i 1], A[i]) swapped = true end if end for until not swapped
4 A simple sorting algorithm Bubble-Sort repeat swapped = false for i = 1 to n 1 do if A[i 1] > A[i] then swap(a[i 1], A[i]) swapped = true end if end for until not swapped What is the worst-case time complexity of this algorithm?
5 A simple sorting algorithm Bubble-Sort repeat swapped = false for i = 1 to n 1 do if A[i 1] > A[i] then swap(a[i 1], A[i]) swapped = true end if end for until not swapped What is the worst-case time complexity of this algorithm? We can do much better!
6 Heaps A Heap is a data structure representing a full binary tree.
7 Heaps A Heap is a data structure representing a full binary tree. A heap is stored in an array A[1... n]. The root is A[1]. parent(i) = i/2. left-child(i) = 2i. right-child(i) = 2i + 1.
8 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i]
9 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i]
10 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i] Where is the maximum element in the tree?
11 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i] Where is the maximum element in the tree? Where is the maximum element in the array?
12 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i] Where is the maximum element in the tree? Where is the maximum element in the array? Where is the minimum element in the tree?
13 Max-Heaps For all nodes other than the root, we have A[parent(i)] A[i] Where is the maximum element in the tree? Where is the maximum element in the array? Where is the minimum element in the tree? Where is are the leaves in the array?
14 Height The height of a node i is the maximum number of edges on a path from i to a leaf.
15 Height The height of a node i is the maximum number of edges on a path from i to a leaf. The height of a tree is the height of its root.
16 Height The height of a node i is the maximum number of edges on a path from i to a leaf. The height of a tree is the height of its root. What is the height of a heap?
17 Building and using heaps Procedure Max-Heapify (auxiliary procedure) Procedure Build-Max-Heap (building a max-heap) Procedure Heap-Sort (sorting using a heap)
18 Maintaining the max-heap property Suppose that the subtrees rooted at left-child(i) and right-child(i) are max-heaps. However, i might violate the max-heap property. E.g., A[i] < A[left-child(i)].
19 Maintaining the max-heap property Suppose that the subtrees rooted at left-child(i) and right-child(i) are max-heaps. However, i might violate the max-heap property. E.g., A[i] < A[left-child(i)]. How can we enforce the max-heap property?
20 Maintaining the max-heap property Procedure Max-Heapify(A, i) l = left-child(i) r = right-child(i) if l n and A[l] > A[i] largest = l else largest = i if r n and A[r] > largest largest = r if largest i exchange A[i] with A[largest] Max-Heapify(A, largest)
21 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)?
22 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest).
23 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion?
24 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion? What is the depth of the recursion?
25 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion? What is the depth of the recursion? Worst-case depth of recursion = height of i.
26 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion? What is the depth of the recursion? Worst-case depth of recursion = height of i. Worst-case running time is O(height(i)).
27 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion? What is the depth of the recursion? Worst-case depth of recursion = height of i. Worst-case running time is O(height(i)). Worst-case running time is O(log(n)).
28 Running time of Max-Heapify What is the running time of Max-Heapify(A, i)? Total time spent in Max-Heapify is at most O(1) + the time spent in the recursive call Max-Heapify(A, largest). Total running time of the recursion? What is the depth of the recursion? Worst-case depth of recursion = height of i. Worst-case running time is O(height(i)). Worst-case running time is O(log(n)). Is this tight?
29 Building a heap Procedure Build-Max-Heap(A) for i = n/2 downto 1 Max-Heapify(A, i)
30 Building a heap Procedure Build-Max-Heap(A) for i = n/2 downto 1 Max-Heapify(A, i) Loop invariant: At the start of each iteration, each node i + 1, i + 2,..., n is the root of a max-heap.
31 Building a heap Procedure Build-Max-Heap(A) for i = n/2 downto 1 Max-Heapify(A, i) Loop invariant: At the start of each iteration, each node i + 1, i + 2,..., n is the root of a max-heap. Initialization: i = n/2. The nodes n/2 + 1,..., n are leaves, and so they are max-heaps.
32 Building a heap Procedure Build-Max-Heap(A) for i = n/2 downto 1 Max-Heapify(A, i) Loop invariant: At the start of each iteration, each node i + 1, i + 2,..., n is the root of a max-heap. Initialization: i = n/2. The nodes n/2 + 1,..., n are leaves, and so they are max-heaps. Maintenance: By the loop invariant, the children of i are roots of max-heaps. Therefore, running Max-Heapify makes i the root of a max-heap.
33 Building a heap Procedure Build-Max-Heap(A) for i = n/2 downto 1 Max-Heapify(A, i) Loop invariant: At the start of each iteration, each node i + 1, i + 2,..., n is the root of a max-heap. Initialization: i = n/2. The nodes n/2 + 1,..., n are leaves, and so they are max-heaps. Maintenance: By the loop invariant, the children of i are roots of max-heaps. Therefore, running Max-Heapify makes i the root of a max-heap. Termination: i = 0. By the loop invariant, 1 is the root of a heap.
34 Running time of Max-Heapify Each call to Max-Heapify takes time O(log n).
35 Running time of Max-Heapify Each call to Max-Heapify takes time O(log n). There are O(n) calls to Max-Heapify.
36 Running time of Max-Heapify Each call to Max-Heapify takes time O(log n). There are O(n) calls to Max-Heapify. Total running time O(n log(n)).
37 Running time of Max-Heapify Each call to Max-Heapify takes time O(log n). There are O(n) calls to Max-Heapify. Total running time O(n log(n)). This is not asymptotically tight!
38 Running time of Max-Heapify: Better analysis Each call Max-Heapify(A, i) takes time O(height(i)).
39 Running time of Max-Heapify: Better analysis Each call Max-Heapify(A, i) takes time O(height(i)). A heap has height log(n).
40 Running time of Max-Heapify: Better analysis Each call Max-Heapify(A, i) takes time O(height(i)). A heap has height log(n). There are at most n/2 h+1 nodes of height h.
41 Running time of Max-Heapify: Better analysis Each call Max-Heapify(A, i) takes time O(height(i)). A heap has height log(n). There are at most n/2 h+1 nodes of height h. Total running time: log n h=0 n 2 h+1 O(h) = O ( n = O(n). h=0 ) h 2 h
42 Sorting using a heap Procedure Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size 1 Max-Heapify(A, 1)
43 Sorting using a heap Procedure Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size 1 Max-Heapify(A, 1) Build-Max-Heap takes time O(n).
44 Sorting using a heap Procedure Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size 1 Max-Heapify(A, 1) Build-Max-Heap takes time O(n). There are n 1 calls to Max-Heapify.
45 Sorting using a heap Procedure Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size 1 Max-Heapify(A, 1) Build-Max-Heap takes time O(n). There are n 1 calls to Max-Heapify. Each call to Max-Heapify takes time O(log n).
46 Sorting using a heap Procedure Heapsort(A) Build-Max-Heap(A) for i = A.length downto 2 exchange A[1] with A[i] A.heap-size = A.heap-size 1 Max-Heapify(A, 1) Build-Max-Heap takes time O(n). There are n 1 calls to Max-Heapify. Each call to Max-Heapify takes time O(log n). Total running time O(n log(n)).
47 Priority queues A priority queue is a data structure for maintaining a set S of elements, each having a key.
48 Priority queues A priority queue is a data structure for maintaining a set S of elements, each having a key. There are max-priority queues, and min-priority queues.
49 Priority queues A priority queue is a data structure for maintaining a set S of elements, each having a key. There are max-priority queues, and min-priority queues. Operations of a max-priority queue: Insert(S, x): S = S {x}. Maximum(S): Return the element in S with the maximum key. Extract-Max(S): Removes and returns the element in S with the maximum key. Increase-Key(S, x, k): Increases the value of the key of x to k, assuming that k is larger than the current value.
50 Implementing a max-priority queue using a max-heap Procedure Heap-Maximum(A)
51 Implementing a max-priority queue using a max-heap Procedure Heap-Maximum(A) return A[1]
52 Implementing a max-priority queue using a max-heap Procedure Heap-Extract-Max(A)
53 Implementing a max-priority queue using a max-heap Procedure Heap-Extract-Max(A) if n < 1 error empty heap max = A[1] A[1] = A[n] n = n 1 Max-Heapify(A, 1) return max
54 Implementing a max-priority queue using a max-heap Procedure Heap-Increase-Key(A, i, key)
55 Implementing a max-priority queue using a max-heap Procedure Heap-Increase-Key(A, i, key) if key < A[i] error A[i] = key while i > 1 and A[parent(i)] < A[i] exchange A[i] with A[parent(i)] i = parent(i)
56 Implementing a max-priority queue using a max-heap Procedure Max-Heap-Insert(A, key)
57 Implementing a max-priority queue using a max-heap Procedure Max-Heap-Insert(A, key) n = n + 1 A[n] = Heap-Increase-Key(A, n, key)
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