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Data Structures with Generic Types in Python

Amortized Analysis of Spreading and Gathering

Learn about the analysis of gather and spread methods.

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Next, we consider the cost of the _gather(u) and _spread(u) methods that may be executed by the add(i, x) and remove(i) methods. For the sake of completeness, here they are:

Python 3.10.4
class SEList(BaseList):
    class BDeque(ArrayDeque):
        """A bounded-size deque"""
        def __init__(self, b):
            super(SEList.BDeque, self).__init__()
            self.a = new_array(b+1)
        
        def _resize(self):
            pass
        
    class Node(object):
        def __init__(self, b):
            self.d = SEList.BDeque(b)
            self.prev = None
            self.next = None
            
    def __init__(self, b):
        super(SEList, self).__init__()
        self.b = b
        self._initialize()
        
    def _spread(self, u):
        w = u
        for j in range(self.b):
            w = w.next
        w = self._add_before(w)
        while w is not u:
            while w.d.size() < self.b:
                w.d.add_first(w.prev.d.remove_last())
            w = w.prev

The implementation of the _gather() method is:

Python 3.10.4
class SEList(BaseList):
    class BDeque(ArrayDeque):
        """A bounded-size deque"""
        def __init__(self, b):
            super(SEList.BDeque, self).__init__()
            self.a = new_array(b+1)
        
        def _resize(self):
            pass
        
    class Node(object):
        def __init__(self, b):
            self.d = SEList.BDeque(b)
            self.prev = None
            self.next = None
            
    def __init__(self, b):
        super(SEList, self).__init__()
        self.b = b
        self._initialize()
    def _gather(self, u):
        w = u
        for j in range(self.b-1):
            while w.d.size() < self.b:
                w.d.add_last(w.next.d.remove_first())
            w = w.next
        self._remove_node(w)

Amortization

The running time of each of these methods is dominated by the two nested loops. Both the inner and outer loops execute at most b+1b + 1 times, so the total running time of each of these methods is O((b+1)2)=O(b2)O((b + 1)^2) = O(b^2). However, the following lemma shows that these methods execute on at most one out of every bb calls to add(i, x) or remove(i).

Lemma: If an empty SEList is created and any sequence of m≥1m \ge 1 calls to add(i, x) and remove(i) is performed, then the total time spent during all calls to_ _spread() and _gather() is O(bm)O(bm).

Proof: We will use the potential method of amortized analysis. We say that a node u is fragile if u’s block does not contain bb elements (so that u is either the last node, or contains ...