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def binarySearch(a, key, start, end): 
	size = end - start
	if size <= 0: 
		return None 
	center = (start + end) / 2
	if key == a[center]:  
		return center 
	elif key <  a[center]:  
		return binarySearch(a, key, start, center) 
	else: 
		return binarySearch(a, key, center+1, end)

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def binarySearch2(a, key): 
	if len(a) == 0:  
		return None 
	center = len(a) / 2 
	if key == a[center]: 
		return center 
	elif key <  a[center]:  
		return binarySearch2(a[:center], key) 
	else:  
		res = binarySearch2(a[center+1:], key) 
		if res is None:  
			return None 
		else: 
			return res + center + 1