```
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,
"""
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,
"""
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

- 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.

```