Data Structures with C++ using STL Chapter 15.ppt

1、1,Building a Ruler: drawRuler() Merge Algorithm Example (4 slides) Partitioning and Merging of Sublists in mergeSort() Function calls in mergeSort() Quicksort Example (8 slides) Finding kth Largest Element powerSet() Example,Chapter 15 Advanced Recursive Algorithms,Recursive calls for fib(5) Effect

2、of fib(5) Using Dynamic Programming The 8-Queens Example (3 slides) Summary Slides (6 slides),2,Building a Ruler: drawRuler(),Problem: create a program that draws marks at regular intervals on a line. The sizes of the marks differ, depending on the specific interval. - The recursive function drawRul

3、er() assumes the existence of the function drawMark(), which takes a point x and an integer value h as arguments and draws a mark at point x with size proportional to h.,3,The Merge Algorithm Example,The merge algorithm takes a sequence of elements in a vector v having index range first, last). The

4、sequence consists of two ordered sublists separated by an intermediate index, called mid.,4,The Merge Algorithm (Cont),5,The Merge Algorithm (Cont),6,The Merge Algorithm (Cont),7,Partitioning and Merging of Sublists in mergeSort(),8,Function calls in mergeSort(),9,Quicksort Example,The quicksort alg

5、orithm uses a series of recursive calls to partition a list into smaller and smaller sublists about a value called the pivot. Example: Let v be a vector containing 10 integer values: v = 800, 150, 300, 650, 550, 500, 400, 350, 450, 900,10,Quicksort Example (Cont),11,Quicksort Example (Cont),12,Quick

6、sort Example (Cont),13,Quicksort Example (Cont),14,Quicksort Example (Cont),15,Quicksort Example (Cont),16,Quicksort Example (Cont),17,Finding Kth Largest Element,To locate the position of the kth-largest value (kLargest) in the list, partition the elements into two disjoint sublists. The lower subl

7、ist must contain k elements that are less than or equal to kLargest and the upper sublist must contain elements that are greater than or equal to kLargest. The elements in the lower sublist do not need to be ordered but only to have values that are less than or equal to kLargest. The opposite condit

8、ion applies to the upper sublist.,18,powerSet() Example,19,Recursive calls for fib(5),20,Affect of fib(5) Using Dynamic Programming,21,The 8-Queens Example,22,The 8-Queens Example (Cont),23,The 8-Queens Example (Cont),24,Summary Slide 1,- Divide-and-Conquer Algorithms - splits a problem into subprob

9、lems and works on each part separately - Two type of divide-and-conquer strategy: 1) mergesort algorithm - Split the range of elements to be sorted in half, sort each half, and then merge the sorted sublists together. - running time: O(n log2n), requires the use of an auxiliary vector to perform the

10、 merge steps.,25,Summary Slide 2,2) quicksort algorithm - uses a partitioning strategy that finds the final location of a pivot element within an interval first,last). - The pivot splits the interval into two parts, first, pivotIndex), pivotIndex,last). All elements in the lower interval have values

11、 pivot and all elements in the upper interval have values pivot. - running time: O(n log2n) - worst case: of O(n2), highly unlikely to occur,26,Summary Slide 3,- Recursion in Combinatorics - A set of n elements has 2n subsets, and the set of those subsets is called the power set. By using a divide a

12、nd conquer strategy that finds the power set after removing an element from the set and then adds the element back into each subset, we implement a function that computes the power set. The section also uses recursion to list all the n! permutations of the integers from 1 through n. The success of t

13、his algorithm depends on the passing of a vector by value to the recursive function.,27,Summary Slide 4,- Dynamic Programming - Two type of dynamic programming: 1) top-down dynamic programming - uses a vector to store Fibonacci numbers as a recursive function computes them - avoids costly redundant

14、recursive calls and leads to an O(n) algorithm that computes the nth Fibonacci number. - recursive function that does not apply dynamic programming has exponential running time. - improve the recursive computation for C(n,k), the combinations of n things taken k at a time.,28,Summary Slide 5,2) bott

15、om-up dynamic programming - evaluates a function by computing all the function values in order, starting at the lowest level and using previously computed values at each step to compute the current value. - 0/1 knapsack problem,29,Summary Slide 6,-Backtracking Algorithm - finds a consistent partial solution to a problem and then tries to extend the partial solution to a complete solution by executing a recursive step. If the recursive step fails to find a solution, it returns to the previous state and the algorithm tries a