1	CS 332: Algorithms Solving Recurrences Continued The Master Theorem Introduction to heapsort
2	Review: Merge Sort MergeSort(A, left, right) { if (left < right) { mid = floor((left + right) / 2); MergeSort(A, left, mid); MergeSort(A, mid+1, right); Merge(A, left, mid, right); } } // Merge() takes two sorted subarrays of A and // merges them into a single sorted subarray of A. // Code for this is in the book. It requires O(n) // time, and does require allocating O(n) space
3	Review: Analysis of Merge Sort Statement Effort So T(n) = Q(1) when n = 1, and 2T(n/2) + Q(n) when n > 1 This expression is a recurrence
4	Review: Solving Recurrences Substitution method Iteration method Master method
5	Review: Solving Recurrences The substitution method A.k.a. the “making a good guess method” Guess the form of the answer, then use induction to find the constants and show that solution works Run an example: merge sort T(n) = 2T(n/2) + cn We guess that the answer is O(n lg n) Prove it by induction Can similarly show T(n) = Ω(n lg n), thus Θ(n lg n)
6	Review: Solving Recurrences The “iteration method” Expand the recurrence Work some algebra to express as a summation Evaluate the summation We showed several examples, were in the middle of:
7	"T(n)" T(n) = aT(n/b) + cn a(aT(n/b/b) + cn/b) + cn a²T(n/b²) + cna/b + cn a²T(n/b²) + cn(a/b + 1) a²(aT(n/b²/b) + cn/b²) + cn(a/b + 1) a³T(n/b³) + cn(a²/b²) + cn(a/b + 1) a³T(n/b³) + cn(a²/b² + a/b + 1) … a^kT(n/b^k) + cn(a^k-1/b^k-1 + a^k-2/b^k-2 + … + a²/b² + a/b + 1)
8	"So we have" So we have T(n) = a^kT(n/b^k) + cn(a^k-1/b^k-1+ ... + a²/b²+ a/b + 1) For k = log_b n n = b^k T(n) = a^kT(1) + cn(a^k-1/b^k-1+ ... + a²/b²+ a/b + 1) = a^kc + cn(a^k-1/b^k-1+ ... + a²/b²+ a/b + 1) = ca^k + cn(a^k-1/b^k-1+ ... + a²/b²+ a/b + 1) = cna^k/b^k + cn(a^k-1/b^k-1+ ... + a²/b²+ a/b + 1) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1)
9	"So with k = log_b..." So with k = log_bn T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a = b? T(n) = cn(k + 1) = cn(log_b n + 1) = Q(n log n)
10	"So with k = log_b..." So with k = log_bn T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a < b?
11	"So with k = log_b..." So with k = log_bn T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a < b? Recall that S(x^k + x^k-1 + … + x + 1) = (x^k+1 -1)/(x-1)
12	"So with k = log_b..." So with k = log_bn T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a < b? Recall that (x^k + x^k-1 + … + x + 1) = (x^k+1 -1)/(x-1) So:
13	"So with k = log_b..." So with k = log_bn T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a < b? Recall that S(x^k + x^k-1 + … + x + 1) = (x^k+1 -1)/(x-1) So: T(n) = cn ·Q(1) = Q(n)
14	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b?
15	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b?
16	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b? T(n) = cn · Q(a^k / b^k)
17	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b? T(n) = cn · Q(a^k / b^k) = cn · Q(a^{log n} / b^{log n}) = cn · Q(a^{log n} / n)
18	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b? T(n) = cn · Q(a^k / b^k) = cn · Q(a^{log n} / b^{log n}) = cn · Q(a^{log n} / n) recall logarithm fact: a^{log n} = n^{log a}
19	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b? T(n) = cn · Q(a^k / b^k) = cn · Q(a^{log n} / b^{log n}) = cn · Q(a^{log n} / n) recall logarithm fact: a^{log n} = n^{log a} = cn · Q(n^{log a} / n) = Q(cn · n^{log a} / n)
20	"So with k = log_b..." So with k = log_b n T(n) = cn(a^k/b^k+ ... + a²/b²+ a/b + 1) What if a > b? T(n) = cn · Q(a^k / b^k) = cn · Q(a^{log n} / b^{log n}) = cn · Q(a^{log n} / n) recall logarithm fact: a^{log n} = n^{log a} = cn · Q(n^{log a} / n) = Q(cn · n^{log a} / n) = Q(n^{log a} )
21	"So…" So…
22	The Master Theorem Given: a divide and conquer algorithm An algorithm that divides the problem of size n into a subproblems, each of size n/b Let the cost of each stage (i.e., the work to divide the problem + combine solved subproblems) be described by the function f(n) Then, the Master Theorem gives us a cookbook for the algorithm’s running time:
23	The Master Theorem if T(n) = aT(n/b) + f(n) then
24	Using The Master Method T(n) = 9T(n/3) + n a=9, b=3, f(n) = n n^log^b^a = n^log³⁹ = Q(n²) Since f(n) = O(n^log³^{9 -}^e), where e=1, case 1 applies: Thus the solution is T(n) = Q(n²)
25	Sorting Revisited So far we’ve talked about two algorithms to sort an array of numbers What is the advantage of merge sort? What is the advantage of insertion sort? Next on the agenda: Heapsort Combines advantages of both previous algorithms
26	Heaps A heap can be seen as a complete binary tree: What makes a binary tree complete? Is the example above complete?
27	Heaps A heap can be seen as a complete binary tree: The book calls them “nearly complete” binary trees; can think of unfilled slots as null pointers
28	Heaps In practice, heaps are usually implemented as arrays:
29	Heaps To represent a complete binary tree as an array: The root node is A[1] Node i is A[i] The parent of node i is A[i/2] (note: integer divide) The left child of node i is A[2i] The right child of node i is A[2i + 1]
30	Referencing Heap Elements So… Parent(i) { return ëi/2û; } Left(i) { return 2i; } right(i) { return 2i + 1; } An aside: How would you implement this most efficiently? Another aside: Really?
31	The Heap Property Heaps also satisfy the heap property: A[Parent(i)] ³ A[i] for all nodes i > 1 In other words, the value of a node is at most the value of its parent Where is the largest element in a heap stored? Definitions: The height of a node in the tree = the number of edges on the longest downward path to a leaf The height of a tree = the height of its root
32	Heap Height What is the height of an n-element heap? Why? This is nice: basic heap operations take at most time proportional to the height of the heap
33	Heap Operations: Heapify() Heapify(): maintain the heap property Given: a node i in the heap with children l and r Given: two subtrees rooted at l and r, assumed to be heaps Problem: The subtree rooted at i may violate the heap property (How?) Action: let the value of the parent node “float down” so subtree at i satisfies the heap property What do you suppose will be the basic operation between i, l, and r?
34	Heap Operations: Heapify() Heapify(A, i) { l = Left(i); r = Right(i); if (l <= heap_size(A) && A[l] > A[i]) largest = l; else largest = i; if (r <= heap_size(A) && A[r] > A[largest]) largest = r; if (largest != i) Swap(A, i, largest); Heapify(A, largest); }
35	Heapify() Example
36	Heapify() Example
37	Heapify() Example
38	Heapify() Example
39	Heapify() Example
40	Heapify() Example
41	Heapify() Example
42	Heapify() Example
43	Heapify() Example
44	Analyzing Heapify(): Informal Aside from the recursive call, what is the running time of Heapify()? How many times can Heapify() recursively call itself? What is the worst-case running time of Heapify() on a heap of size n?
45	Analyzing Heapify(): Formal Fixing up relationships between i, l, and r takes Q(1) time If the heap at i has n elements, how many elements can the subtrees at l or r have? Draw it Answer: 2n/3 (worst case: bottom row 1/2 full) So time taken by Heapify() is given by T(n) £ T(2n/3) + Q(1)
46	Analyzing Heapify(): Formal So we have T(n) £ T(2n/3) + Q(1) By case 2 of the Master Theorem, T(n) = O(lg n) Thus, Heapify() takes linear time
47	Heap Operations: BuildHeap() We can build a heap in a bottom-up manner by running Heapify() on successive subarrays Fact: for array of length n, all elements in range A[ën/2û + 1 .. n] are heaps (Why?) So: Walk backwards through the array from n/2 to 1, calling Heapify() on each node. Order of processing guarantees that the children of node i are heaps when i is processed
48	BuildHeap() // given an unsorted array A, make A a heap BuildHeap(A) { heap_size(A) = length(A); for (i = ëlength[A]/2û downto 1) Heapify(A, i); }
49	BuildHeap() Example Work through example A = {4, 1, 3, 2, 16, 9, 10, 14, 8, 7}
50	Analyzing BuildHeap() Each call to Heapify() takes O(lg n) time There are O(n) such calls (specifically, ën/2û) Thus the running time is O(n lg n) Is this a correct asymptotic upper bound? Is this an asymptotically tight bound? A tighter bound is O(n) How can this be? Is there a flaw in the above reasoning?
51	Analyzing BuildHeap(): Tight To Heapify() a subtree takes O(h) time where h is the height of the subtree h = O(lg m), m = # nodes in subtree The height of most subtrees is small Fact: an n-element heap has at most én/2^h⁺¹ù nodes of height h CLR 7.3 uses this fact to prove that BuildHeap() takes O(n) time
52	Heapsort Given BuildHeap(), an in-place sorting algorithm is easily constructed: Maximum element is at A[1] Discard by swapping with element at A[n] Decrement heap_size[A] A[n] now contains correct value Restore heap property at A[1] by calling Heapify() Repeat, always swapping A[1] for A[heap_size(A)]
53	Heapsort Heapsort(A) { BuildHeap(A); for (i = length(A) downto 2) { Swap(A[1], A[i]); heap_size(A) -= 1; Heapify(A, 1); } }
54	Analyzing Heapsort The call to BuildHeap() takes O(n) time Each of the n - 1 calls to Heapify() takes O(lg n) time Thus the total time taken by HeapSort() = O(n) + (n - 1) O(lg n) = O(n) + O(n lg n) = O(n lg n)
55	Priority Queues Heapsort is a nice algorithm, but in practice Quicksort (coming up) usually wins But the heap data structure is incredibly useful for implementing priority queues A data structure for maintaining a set S of elements, each with an associated value or key Supports the operations Insert(), Maximum(), and ExtractMax() What might a priority queue be useful for?
56	Priority Queue Operations Insert(S, x) inserts the element x into set S Maximum(S) returns the element of S with the maximum key ExtractMax(S) removes and returns the element of S with the maximum key How could we implement these operations using a heap?