Chapter 3 Stacks and Queues - 3.2

problem 3.2: How would you design a stack which, in addition to push and pop, also has a function min which returns the minimum element? Push, pop and min should all operate in O(1) time.

At first, I thought it was impossible to impement such a stack. However, when I turned to the answer page, I found that I ignored the most important property of stacks, FILO (First In Last Out). Each element in a stack can keep properties of all elements beneath it, for all elements below won't be changed before the element is removed. It means that we can record the minimum value of all elements in the element above them. When an element is at the top of the stack, the minimum value it keeps is the minimum value of all elements in the stack.

MAXIMUM_VALUE = 1e7

class node_with_min:
    def __init__(self, value = None):
        self.pre = None
        self.value = value
        # Each node keeps record of minimum value in the stack,
        # when it is at the top of the stack
        self.min = None

class stack_with_min:
    def __init__(self):
        self.bottom = node_with_min(MAXIMUM_VALUE)
        self.bottom.min = MAXIMUM_VALUE
        self.top = self.bottom
        self.length = 0
    def push(self, value):
        min_val = min([value, self.top.min])
        new_node = node_with_min(value)
        new_node.min = min_val
        new_node.pre = self.top
        self.top = new_node
        self.length = self.length + 1
    def pop(self):
        if self.length <= 0:
            return False
        to_delete = self.top
        to_return = self.top.value
        self.top = self.top.pre
        del to_delete
        self.length = self.length - 1
        return to_return
    def min_element(self):
        if self.length <= 0:
            return False
        return self.top.min
# Test cases
if __name__ == "__main__":
    stack = stack_with_min()
    stack.push(0)
    print "push 0, min = ", stack.min_element()
    stack.push(1)
    print "push 1, min = ", stack.min_element()
    stack.push(2)
    print "push 2, min = ", stack.min_element()
    stack.push(-1)
    print "push -1, min = ", stack.min_element()
    stack.push(-2)
    print "push -2, min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()
    print "pop,", stack.pop(), "min = ", stack.min_element()

The lesson is: you can achieve quite high performance if you sacrifice space. If you cannot achieve performance that is high enougth, you didn't sacrifice enough space.