Searching (within) Data. Programming and Problem Solving Searching (within) Data, Advanced Complexity Theory. Searching (within) Data.

Transcription

1 Searching (within) Data Linear Search Dennis Komm Programming and Problem Solving Searching (within) Data, Advanced Complexity Theory Spring 2019 April 1, 2019 Linear Search Most straightforward strategy to search Works for unsorted data Run through array from left to right and compare searched element with current one Needs up to n comparisons on array of length n if searched element is at last position (or does not appear) Searching (within) Data Binary Search Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 1 / 26

2 Binary Search Binary Search Example Given an array with the first 8 prime numbers, find the position of Idea Use that data is sorted Two int variables left and right These specify search space Look at value in the middle (index current) If this value is what we searched for, we are done If this value is too small, then also everything left of it is too small left = current+1 If this value is too small, then also everything right of it is too large right = current-1 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 2 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 3 / 26 Exercise Binary Search Binary Search Implement binary search Use three pointers left, right, and current In every step, shrink search space as described int left = 0; int right = data.length-1; while (left <= right) { int current = (left+right)/2; if (data[current] == searched) { System.out.println("Found at position " + current); return; else if (data[current] > searched) { right = current-1; else { left = current+1; System.out.println("Not found"); Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 4 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 5 / 26

3 Time Complexity of Binary Search Searching Time Complexity of Binary Search At first, there are n elements With every iteration, the search space is halved After the first iteration, there remain n/2 elements After the second iteration, there remain n/4 elements... After how many iterations x does there remain only one element? n/2 x = 1 n = 2 x x = log 2 n Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 6 / 26 Time Complexity of Binary Search 100 Comparisons Linear search Binary search Searching (within) Data Subarrays Input length n Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 7 / 26

4 Maximum Subarray of Fixed Length Exercise Maximum Subarray of Fixed Length Given are n positive numbers Find a subarray of length k with maximal sum Write method that that gets an int array with positive numbers and an int k returns the subarray of length k with the largest sum has linear time complexity Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 8 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 9 / 26 Maximum Subarray of Fixed Length Maximum Subarray private static int[] maxfixsub(int[] data, int len) { int sum = 0; int start = 0; int end = len-1; int[] result = {0,0,0; for (int i=0; i<len; ++i) { sum = sum + data[i]; int max = sum; for (int i=0; i<data.length-len; i++) { sum = sum - data[i] + data[i+len]; if(sum > max) { max = sum; start = i+1; end = i+len; result[0] = start; result[1] = end; result[2] = max; return result; Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 10 / 26 Give are n numbers (positive or negative) Find subarray with maximal sum Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 11 / 26

5 Maximum Subarray Exercise Maximum Subarray Kadane s Algorithm Every subarray (also a maximal one) has some end position x Consider all possibilities For every x 1 Consider maximal subarray A x 1 with end position x 1 Either it is part of the maximal subarray A x with end position x Or it is not (then, A x is a subarray of length 1) sum(a x ) = max{sum(a x 1 ) + value of x, value of x Implement Kadane s algorithm Save current minimum Decide whether the maximum subarray that ends at previous position should be part of it The maximum of two number can be calculated using Math.max Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 12 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 13 / 26 Exercise Maximum Subarray static int maxsub(int data[]) { int maxendhere = data[0]; int max = maxendhere; for (int i = 1; i < data.length; ++i) { maxendhere = Math.max(data[i], maxendhere+data[i]); max = Math.max(max,maxEndHere); return max; Advanced Complexity Theory Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 14 / 26

6 Classification of Problems Alan Cobham (*1927), Jack Edmonds (*1934) Problems allow for different complexities Naive primality testing (solution of problem PRIMES) has complexity in O(2 n ) There is algorithm with complexity O(n 7 ) Mergesort has complexity in O(n log n) Kadane s algorithm has complexity in O(n) These problems are solvable in polynomial time Thesis of C. and E. Efficient algorithms are those with polynomial time complexity Time complexity is in O(n k ) for k N O(n 3 ) O(n 10 ) but not O(1.41 n ) Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 15 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 16 / 26 Efficient Algorithms Efficient Algorithms The term is independent of the computing model If some algorithm implemented in Java can compute something in polynomial time, then some Python algorithm can also do this, and vice versa We therefore call the class of polynomials robust Problems that can be solved efficiently The class P contains all problems that can be solved efficiently What about problems that are not in P? The other way around: Which are the problems for which we may hope to find efficient algorithms? For the halting problem, for instance, this is hopeless (because we cannot solve it at all) So what are good candidates? PRIMES P Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 17 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 18 / 26

7 Polynomial-Time Verifier Hamiltonian Cycle Problem (HC) Consider alternative model A polynomial-time verifier (PV) does not have to compute the solution to a problem itself Together with the input, it receives a (potential) witness The witness is a string As an example, it may be a potential divisor if the goal is to check whether the input is composite Witness is enormous help for computing the output Does a graph G have a Hamiltonian cycle? (traveling salesman problem) Algorithm receives input v 6 v 5 v 5 v 4 v 1 v 2 v 3 not efficient PV receives input v 6 v 4 v 1 v 2 and has to compute YES or NO and witness v 1, v 3, v 5, v 4, v 6, v 2, v 1 v 3 efficient and has to verify it with YES or NO Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 19 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 20 / 26 P versus N P P versus N P Problems that can be verified efficiently The class N P contains all problems N stands for nondeterministic PV have an obvious advantage compared to algorithms If there is no PV for some problem, then there is also no efficient algorithm Search for efficient algorithms within N P If we can compute something efficiently with an algorithm, then we can also do it with a PV P N P But how about the other way around? We still do not know whether P = N P or P N P It is conjectured that P N P There are likely problems in N P for which there are no efficient algorithms It remains open for decades to find such a problem Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 21 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 22 / 26

8 P versus N P Stephen A. Cook (*1939) What can we do? Identify hardest problems in N P N P-Completeness A problem A in N P is N P-complete if solving A efficiently implies the ability to efficiently solve all problems in N P Cook s Theorem There is an N P-complete problem If an efficient algorithm is found for some N P-complete problem, then there are efficient algorithms for all problems in N P P = N P Wikimedia, Creative Commons Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 23 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 24 / 26 Richard M. Karp (*1935) P versus N P Karp s 21 Problems There are 21 more N P-complete problems Thousands of N P-complete problems have been identified so far If at some point an efficient algorithm is found for one of them, then there are efficient algorithms for all of them P = N P Today, we assume the following relationship Wikimedia, Creative Commons Today we know thousands, and still we cannot prove for one of them that there is no efficient algorithm P N P N P-complete Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 25 / 26 Programming and Problem Solving Searching (within) Data, Complexity Theory Spring 2019 Dennis Komm 26 / 26