2. Instance simplification Solve a problem s instance by transforming it into another simpler/easier instance of the same problem

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1 CmSc 20 Intro to Algorithms Chapter 6. Transform and Conquer Algorithms. Introduction This group of techniques solves a problem by a transformation to a simpler/more convenient instance of the same problem (instance simplification) to a different representation of the same instance (representation change) to a different problem for which an algorithm is already available (problem reduction) 2. Instance simplification Solve a problem s instance by transforming it into another simpler/easier instance of the same problem 2.. Presorting Many problems involving lists are easier when list is sorted. searching computing the median (selection problem) checking if all elements are distinct (element uniqueness) Also: Topological sorting helps solving some problems for directed acyclic graphs (dags). Presorting is used in many geometric algorithms. Efficiency of algorithms involving sorting depends on efficiency of sorting. Theorem (see Sec..2): log 2 n! n log 2 n comparisons are necessary in the worst case to sort a list of size n by any comparison-based algorithm. Note: About nlog 2 n comparisons are also sufficient to sort array of size n (by mergesort) Searching with presorting Problem: Search for a given K in A[0..n-] Stage Sort the array by an efficient sorting algorithm Stage 2 Apply binary search Efficiency: Θ(nlog n) + O(log n) = Θ(nlog n)

2 Good or bad? Why do we have our dictionaries, telephone directories, etc. sorted? Element uniqueness with presorting Presorting-based algorithm Stage : sort by efficient sorting algorithm (e.g. mergesort) Stage 2: scan array to check pairs of adjacent elements Efficiency: Θ(nlog n) + O(n) = Θ(nlog n) Brute force algorithm : Compare all pairs of elements Efficiency: O(n 2 ) 2.2. Instance simplification Gaussian Elimination Given: A system of n linear equations in n unknowns with an arbitrary coefficient matrix. Transform to: An equivalent system of n linear equations in n unknowns with an upper triangular coefficient matrix. Solve the latter by substitutions starting with the last equation and moving up to the first one. a x + a 2 x a n x n = b a x+ a 2 x a n x n = b a 2 x + a 22 x a 2n x n = b 2 a 22 x a 2n x n = b a n x + a n2 x a nn x n = b n a nn x n = b n Efficiency: Θ(n 3 ) 3. Representation change 3.. Binary Search Tree Instead of linear representation use tree representation Arrange keys in a binary tree with the binary search tree property: 2

3 K <K >K Efficiency depends of the tree s height: log 2 n h n-, with height average (random files) about 3log 2 n Search, insert and delete: worst case efficiency: (n) average case efficiency: (log n) Advantage: Inorder traversal produces sorted list Disadvantage: worst case efficiency is (n) Several representations have been developed to overcome the disadvantages of BSTs AVL trees, multiway search trees trees, trees, red-black trees Heaps Heaps are important data structure used for sorting and priority queue implementation Definition A heap is a binary tree with keys at its nodes (one key per node) such that: It is complete, i.e., all its levels are full except possibly the last level, where only some rightmost keys may be missing The key at each node is of higher priority than the keys at its children a heap not a heap not a heap 3

4 Note : Heap s elements are ordered top down (along any path down from its root), but they are not ordered left to right. Note 2: The heap can be built with the key at each node less than or equal to the keys of its children. In this case the smallest key will be at the root Heap representation with an array: The children of the node at position k are stored at positions 2*k and 2*k See Binary Heaps and Heapsort 3.3. Horner s rule for polynomial evaluation Given a polynomial of degree n p(x) = a n x n + a n- x n- + + a x + a 0 and a specific value of x, find the value of p at that point. Two brute-force algorithms: Algorithm : Algorithm 2: p 0 p a0; power for i n down to 0 do power for j to i do power power * x p p + a i * power return p Efficiency of algorithm: (n 2 ) Efficiency of algorithm2: (n) for i to n do power power * x p p + a i * power return p 4

5 Horner s Rule p(x) = a n x n + a n- x n- + + a x + a 0 = x(x(x( (a n x + a n- ) + a n-2 ) + ) + a 0 Example: p(x) = 2x 4 - x 3 + 3x 2 + x - = = x(2x 3 - x 2 + 3x + ) - = = x(x(2x 2 - x + 3) + ) - = = x(x(x(2x - ) + 3) + ) - Horner s Rule pseudocode: Efficiency of Horner s Rule: # multiplications = # additions = n 4. Problem Reduction This variation of transform-and-conquer solves a problem by a transforming it into different problem for which an algorithm is already available. To be of practical value, the combined time of the transformation and solving the other problem should be smaller than solving the problem as given by another method. Examples of Solving Problems by Reduction computing lcm(m, n) via computing gcd(m, n) counting number of paths of length n in a graph by raising the graph s adjacency matrix to the n-th power transforming a maximization problem to a minimization problem and vice versa (also, minheap construction) linear programming reduction to graph problems (e.g., solving puzzles via state-space graphs). Conclusion The Transform-and-Conquer paradigm is a powerful problem-solving strategy. The key to success is to be able to view the problem from different perspectives and to be able to see similarities between the problem to be solved and some other problem, not necessarily in the same area. The more you know in general, the better your chances to successfully apply the transform-andconquer paradigm.