ADT(Algorithm Design Technique Backtracking algorithm).ppt
- 2. General Concepts Algorithm strategy Approach to solving a problem May combine several approaches Algorithm structure Iterative execute action in loop Recursive reapply action to subproblem(s) Problem type Satisfying find any satisfactory solution Optimization find best solutions (vs. cost metric)
- 3. Some Algorithm Strategies Recursive algorithms Backtracking algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy algorithms
- 4. Recursive Algorithm Based on reapplying algorithm to subproblem Approach 1. Solves base case(s) directly 2. Recurs with a simpler subproblem 3. May need to convert solution(s) to subproblems
- 5. Recursive Algorithm – Examples To count elements in list If list is empty, return 0 Else skip 1st element and recur on remainder of list Add 1 to result To find element in list If list is empty, return false Else if first element in list is given value, return true Else skip 1st element and recur on remainder of list
- 6. Backtracking Algorithm Based on depth-first recursive search Approach 1. Tests whether solution has been found 2. If found solution, return it 3. Else for each choice that can be made a) Make that choice b) Recur c) If recursion returns a solution, return it 4. If no choices remain, return failure Some times called “search tree” Basically it is exhaustive search using divide and conquer. Sometimes the best algorithm for a problem is to try all possibilities. This is always slow. Backtracking speeds the exhaustive search by pruning.
- 7. Backtracking Algorithm Application Application to: The knapsack problem The Hamiltonian cycle problem The travelling salesperson problem The eight queen problem Eight Queen Problem
- 8. Backtracking Algorithm – Example Find path through maze Start at beginning of maze If at exit, return true Else for each step from current location Recursively find path Return with first successful step Return false if all steps fail
- 9. Backtracking Algorithm – Example Color a map with no more than four colors If all countries have been colored return success Else for each color c of four colors and country n If country n is not adjacent to a country that has been colored c Color country n with color c Recursively color country n+1 If successful, return success Return failure
- 10. Divide and Conquer Based on dividing problem into subproblems Approach 1. Divide problem into smaller subproblems Subproblems must be of same type Subproblems do not need to overlap 2. Solve each subproblem recursively 3. Combine solutions to solve original problem Usually contains two or more recursive calls
- 11. Divide and Conquer – Examples Quicksort Partition array into two parts around pivot Recursively quicksort each part of array Concatenate solutions Average Case Analysis of Quick Sort
- 12. Divide and Conquer – Examples Average Case Analysis of Quick Sort
- 13. Divide and Conquer – Examples
- 14. Divide and Conquer – Examples
- 15. Divide and Conquer – Examples Mergesort Partition array into two parts Recursively mergesort each half Merge two sorted arrays into single sorted array
- 16. Dynamic Programming Algorithm Based on remembering past results Approach 1.Divide problem into smaller subproblems Subproblems must be of same type Subproblems must overlap 2.Solve each subproblem recursively May simply look up solution 3.Combine solutions into to solve original problem 4.Store solution to problem Generally applied to optimization problems
- 17. Fibonacci Algorithm Fibonacci numbers fibonacci(0) = 1 fibonacci(1) = 1 fibonacci(n) = fibonacci(n-1) + fibonacci(n-2) Recursive algorithm to calculate fibonacci(n) If n is 0 or 1, return 1 Else compute fibonacci(n-1) and fibonacci(n-2) Return their sum Simple algorithm exponential time O(2n ) BY USING DP Dynamic programming version of fibonacci(n) If n is 0 or 1, return 1 Else solve fibonacci(n-1) and fibonacci(n-2) Look up value if previously computed Else recursively compute Find their sum and store Return result Dynamic programming algorithm O(n) time Since solving fibonacci(n-2) is just looking up value
- 18. Dynamic Programming – Example Combinations Knapsack problem Matrix product Dijkstra Algorithm Floyds Algorithm
- 19. Greedy Algorithm Based on trying best current (local) choice Approach At each step of algorithm Choose best local solution Avoid backtracking, exponential time O(2n ) Hope local optimum lead to global optimum
- 20. Greedy Algorithm – Example Kruskal’s Minimal Spanning Tree Algorithm sort edges by weight (from least to most) tree = for each edge (X,Y) in order if it does not create a cycle add (X,Y) to tree stop when tree has N–1 edgesPicks best local solution at each step