Algorithm types
- 2. Algorithm classification • Algorithms that use a similar problem-solving approach can be grouped together • This classification scheme is neither exhaustive nor disjoint • The purpose is not to be able to classify an algorithm as one type or another, but to highlight the various ways in which a problem can be attacked
- 3. A short list of categories • Algorithm types we will consider include: – Simple recursive algorithms – Backtracking algorithms – Divide and conquer algorithms – Dynamic programming algorithms – Greedy algorithms – Branch and bound algorithms – Brute force algorithms – Randomized algorithms
- 4. Simple recursive algorithms I • A simple recursive algorithm: – Solves the base cases directly – Recurs with a simpler subproblem – Does some extra work to convert the solution to the simpler subproblem into a solution to the given problem • I call these “simple” because several of the other algorithm types are inherently recursive
- 5. Example recursive algorithms • To count the number of elements in a list: – If the list is empty, return zero; otherwise, – Step past the first element, and count the remaining elements in the list – Add one to the result • To test if a value occurs in a list: – If the list is empty, return false; otherwise, – If the first thing in the list is the given value, return true; otherwise – Step past the first element, and test whether the value occurs in the remainder of the list
- 6. Backtracking algorithms • Backtracking algorithms are based on a depth-first recursive search • A backtracking algorithm: – Tests to see if a solution has been found, and if so, returns it; otherwise – For each choice that can be made at this point, • Make that choice • Recur • If the recursion returns a solution, return it – If no choices remain, return failure
- 7. Example backtracking algorithm • To color a map with no more than four colors: – color(Country n) • If all countries have been colored (n > number of countries) return success; otherwise, • For each color c of four colors, – 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 (if loop exits)
- 8. Divide and Conquer • A divide and conquer algorithm consists of two parts: – Divide the problem into smaller subproblems of the same type, and solve these subproblems recursively – Combine the solutions to the subproblems into a solution to the original problem • Traditionally, an algorithm is only called divide and conquer if it contains two or more recursive calls
- 9. Examples • Quick sort: – Partition the array into two parts, and quick sort each of the parts – No additional work is required to combine the two sorted parts • Merge sort: – Cut the array in half, and merge sort each half – Combine the two sorted arrays into a single sorted array by merging them
- 10. Binary tree lookup • Here’s how to look up something in a sorted binary tree: – Compare the key to the value in the root • If the two values are equal, report success • If the key is less, search the left sub tree • If the key is greater, search the right sub tree • This is not a divide and conquer algorithm because, although there are two recursive calls, only one is used at each level of the recursion
- 11. Fibonacci numbers • To find the nth Fibonacci number: – If n is zero or one, return one; otherwise, – Compute fibonacci(n-1) and fibonacci(n-2) – Return the sum of these two numbers • This is an expensive algorithm – It requires O(fibonacci(n)) time – This is equivalent to exponential time, that is, O(2n)
- 12. Dynamic programming algorithms • A dynamic programming algorithm remembers past results and uses them to find new results • Dynamic programming is generally used for optimization problems – Multiple solutions exist, need to find the “best” one – Requires “optimal substructure” and “overlapping subproblems” • Optimal substructure: Optimal solution contains optimal solutions to subproblems • Overlapping subproblems: Solutions to subproblems can be stored and reused in a bottom-up fashion • This differs from Divide and Conquer, where subproblems generally need not overlap
- 13. Fibonacci numbers again • To find the nth Fibonacci number: – If n is zero or one, return one; otherwise, – Compute, or look up in a table, fibonacci(n-1) and fibonacci(n-2) – Find the sum of these two numbers – Store the result in a table and return it • Since finding the nth Fibonacci number involves finding all smaller Fibonacci numbers, the second recursive call has little work to do • The table may be preserved and used again later
- 14. Greedy algorithms • An optimization problem is one in which you want to find, not just a solution, but the best solution • A “greedy algorithm” sometimes works well for optimization problems • A greedy algorithm works in phases: At each phase: – You take the best you can get right now, without regard for future consequences – You hope that by choosing a local optimum at each step, you will end up at a global optimum
- 15. Example: Counting money • Suppose you want to count out a certain amount of money, using the fewest possible bills and coins • A greedy algorithm would do this would be: At each step, take the largest possible bill or coin that does not overshoot – Example: To make $6.39, you can choose: • a $5 bill • a $1 bill, to make $6 • a 25¢ coin, to make $6.25 • A 10¢ coin, to make $6.35 • four 1¢ coins, to make $6.39 • For US money, the greedy algorithm always gives the optimum solution
- 16. A failure of the greedy algorithm • In some (fictional) monetary system, “krons” come in 1 kron, 7 kron, and 10 kron coins • Using a greedy algorithm to count out 15 krons, you would get – A 10 kron piece – Five 1 kron pieces, for a total of 15 krons – This requires six coins • A better solution would be to use two 7 kron pieces and one 1 kron piece – This only requires three coins • The greedy algorithm results in a solution, but not in an optimal solution
- 17. Branch and bound algorithms • Branch and bound algorithms are generally used for optimization problems – As the algorithm progresses, a tree of subproblems is formed – The original problem is considered the “root problem” – A method is used to construct an upper and lower bound for a given problem – At each node, apply the bounding methods • If the bounds match, it is deemed a feasible solution to that particular subproblem • If bounds do not match, partition the problem represented by that node, and make the two subproblems into children nodes – Continue, using the best known feasible solution to trim sections of the tree, until all nodes have been solved or trimmed
- 18. Example branch and bound algorithm • Travelling salesman problem: A salesman has to visit each of n cities (at least) once each, and wants to minimize total distance travelled – Consider the root problem to be the problem of finding the shortest route through a set of cities visiting each city once – Split the node into two child problems: • Shortest route visiting city A first • Shortest route not visiting city A first – Continue subdividing similarly as the tree grows
- 19. Brute force algorithm • A brute force algorithm simply tries all possibilities until a satisfactory solution is found – Such an algorithm can be: • Optimizing: Find the best solution. This may require finding all solutions, or if a value for the best solution is known, it may stop when any best solution is found – Example: Finding the best path for a travelling salesman • Satisfying: Stop as soon as a solution is found that is good enough – Example: Finding a travelling salesman path that is within 10% of optimal
- 20. Improving brute force algorithms • Often, brute force algorithms require exponential time • Various heuristics and optimizations can be used – Heuristic: A “rule of thumb” that helps you decide which possibilities to look at first – Optimization: In this case, a way to eliminate certain possibilities without fully exploring them
- 21. Randomized algorithms • A randomized algorithm uses a random number at least once during the computation to make a decision – Example: In Quick sort, using a random number to choose a pivot – Example: Trying to factor a large prime by choosing random numbers as possible divisors
- 22. The End