WEEK11: recursion --------------------------------------------------------------- F: binary search and merge sort BINARY SEARCH: - this algorithm works well with recursion, since at each level we eliminate half of the list and repeat the proceedure on the smaller remaining list - here are two ways to code binary search using recursion: def recbinsearch1(x, y, low, high): """ return True if x is in y. use binary search, assume y is already sorted... """ if high < low: return False else: mid = (low + high) / 2 if x == y[mid]: return True elif x < y[mid]: return recbinsearch1(x,y,low,mid-1) else: return recbinsearch1(x,y,mid+1,high) # ----------------------------------------------------- # def recbinsearch2(x, y): """ return True if x is in y. use binary search, assume y is already sorted... """ if len(y) <= 0: return False else: low = 0 high = len(y)-1 mid = (low + high) / 2 if x == y[mid]: return True elif x < y[mid]: return recbinsearch2(x,y[:mid]) else: return recbinsearch2(x,y[(mid+1):]) - what are the differences between the two versions?? - run timeBinarySearches.py and see if you can explain the timing results

Here are some interesting merge sort links:

MERGE SORT: - merge sort uses a divide-and-conquer method to sort a list: given a list, split it in two (if the size of the list > 1) sort each of the two halves of the list merge the two sorted halves back into the original list - this is easy to express using recursion: def mergeSort(L): n = len(L) if n > 1: L1 = L[0:n/2] L2 = L[n/2:] mergeSort(L1) mergeSort(L2) merge(L1,L2,L) - that last step assumes we have a function called merge that merges the two sorted lists back into one big sorted list YOUR TURN: - copy /home/jk/inclass/timeSorts.py and view the mergeSort function - run timeSorts.py and see how long mergeSort takes for N=5000,10000,20000 (you may want to comment out selectionSort and insertionSort for the N=20000 run) Here's what I get: N 5000 10000 20000 ----------------------------------------- L.sort 0.0053 0.0114 0.0102 selection 7.1494 14.5595 51.2008 insertion 11.7596 19.9309 77.8990 merge 0.1288 0.1150 0.2420 So merge sort is much better than selection or insertion, but not as good as the default .sort() method. - how does mergeSort's time depend on N, the size of the list? - how many "steps" does mergeSort take to finish it's work? Each level of recursion requires copying all N values from the halved sub-lists back to the bigger original list. So that's N copies * how many levels of recursion. Each level of recursion requires splitting a given list into two smaller lists. So "how many levels of recursion" is really just "how many times can we split the list in two?", which we already know is log(base2)N. So mergeSort's time = N * logN, which is better than NxN. See /home/jk/inclass/pylabLinePlots.py for a graphical comparison of NlogN vs logN vs linear vs quadratic.