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Types of Algorithms Types of Algorithms Material adapted – courtesy of Prof. Dave Matuszek at UPENN 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 2 1 A short list of categories Algorithm types we will consider include: Simple recursive algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy algorithms Brute force algorithms Randomized algorithms Note: we may even classify algorithms based on the type of problem they are trying to solve – example sorting algorithms, searching algorithms etc. 3 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 We call these “simple” because several of the other algorithm types are inherently recursive 4 2 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 5 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 *We will cover depth-first search very soon (once we get to tree-traversal and trees/graphs). 6 3 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 at least two recursive calls 7 Examples Quicksort: Partition the array into two parts (smaller numbers in one part, larger numbers in the other part) Quicksort each of the parts No additional work is required to combine the two sorted parts Another example is Mergesort Not yet reviewed in class (we can do so if time permits) Cut the array in half, and mergesort each half Combine the two sorted arrays into a single sorted array by merging them 8 4 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 subtree If the key is greater, search the right subtree 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 9 Fibonacci numbers th To find the n 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 10 5 Dynamic programming algorithms A dynamic programming algorithm remembers past results (“memoization”) 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 11 Fibonacci numbers again th To find the n 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 th Since finding the n Fibonacci number involves finding all smaller Fibonacci numbers, the second recursive call has little work to do (this is easiest observed by drawing the tree out) The table may be preserved and used again later 12 6 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 13 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 traveling salesman Satisficing: Stop as soon as a solution is found that is good enough Example: Finding a traveling salesman path that is within 10% of optimal 14 7 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 15 Randomized algorithms A randomized algorithm uses a random number at least once during the computation to make a decision Example: In Quicksort, using a random number to choose a pivot Example: Trying to factor a large number by choosing random numbers as possible divisors 16 8 More on Dynamic Programming Recall Recursion… The function that does the work calls itself to solve a smaller version of its input successively. Need a base case to terminate! Solutions look very terse and elegant Common examples (especially for programming interviews) include factorial/fibonacci But think about it…every call falls on a Stack…this approach is memory intensive Furthermore, it doesn’t take into account the ability to avoid repeated operations. 18 9 Counting coins To find the minimum number of US coins to make any amount, the greedy method always works At each step, just choose the largest coin that does not overshoot the desired amount: 31¢=25 The greedy method would not work if we did not have 5¢ coins For 31 cents, the greedy method gives seven coins (25+1+1+1+1+1+1), but we can do it with four (10+10+10+1) The greedy method also would not work if we had a 21¢ coin For 63 cents, the greedy method gives six coins (25+25+10+1+1+1), but we can do it with three (21+21+21) How can we find the minimum number of coins for any given coin set? 19 A dynamic programming solution Idea: Solve first for one cent, then two cents, then three cents, etc., up to the desired amount Save each answer in an array ! For each new amount N, compute all the possible pairs of previous answers which sum to N For example, to find the solution for 13¢, First, solve for all of 1¢, 2¢, 3¢, ..., 12¢ Next, choose the best solution among: Solution for 1¢ + solution for 12¢ Solution for 2¢ + solution for 11¢ Solution for 3¢ + solution for 10¢ Solution for 4¢ + solution for 9¢ Solution for 5¢ + solution for 8¢ Solution for 6¢ + solution for 7¢ 20 10 Example Suppose coins are 1¢, 3¢, and 4¢ There’s only one way to make 1¢ (one coin) To make 2¢, try 1¢+1¢ (one coin + one coin = 2 coins) To make 3¢, just use the 3¢ coin (one coin) To make 4¢, just use the 4¢ coin (one coin) To make 5¢, try 1¢ + 4¢ (1 coin + 1 coin = 2 coins) 2¢ + 3¢ (2 coins + 1 coin = 3 coins) The first solution is better, so best solution is 2 coins To make 6¢, try 1¢ + 5¢ (1 coin + 2 coins = 3 coins) 2¢ + 4¢ (2 coins + 1 coin = 3 coins) 3¢ + 3¢ (1 coin + 1 coin = 2 coins) – best solution Etc. 21 Comparison with divide-and-conquer Divide-and-conquer algorithms split a problem into separate subproblems, solve the subproblems, and combine the results for a solution to the original problem Example: Quicksort, Mergesort, Binary Search Divide-and-conquer algorithms can be thought of as top-down algorithms In contrast, a dynamic programming algorithm proceeds by solving small problems, remembering the results, then combining them to find the solution to larger problems Dynamic programming can be thought of as bottom-up 22 11 Example 2: Binomial Coefficients 2 2 2 (x + y) = x + 2xy + y , coefficients are 1,2,1 3 3 2 2 3 (x + y) = x + 3x y + 3xy + y , coefficients are 1,3,3,1 4 4 3 2 2 3 4 (x + y) = x + 4x y + 6x y + 4xy + y , coefficients are 1,4,6,4,1 5 5 4 3 2 2 3 4 5 (x + y) = x + 5x y + 10x y + 10x y + 5xy + y , coefficients are 1,5,10,10,5,1 n The n+1 coefficients can be computed for (x + y) according to the formula c(n, i) = n! / (i! * (n – i)!) for each of i = 0..n The repeated computation of all the factorials gets to be expensive We can use dynamic programming to save the factorials as we go 23 Solution by dynamic programming n c(n,0) c(n,1) c(n,2) c(n,3) c(n,4) c(n,5) c(n,6) 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 Each row depends only on
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