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Data Structures with Generic Types in C++

Amortized Analysis of Spreading and Gathering

Explore the amortized cost of the spread and gather operations in SEList, a linked list variant. Understand how these methods impact the performance of add and remove operations, and learn to analyze their efficiency using the potential method in amortized analysis.

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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:

C++
void spread(Node u) {
  Node w = u;
  for (int j = 0; j < b; j++) {
    w = w.next;
  }
  w = addBefore(w);
  while (w != u) {
    while (w.d.size() < b)
      w.d.add(0, w.prev.d.remove(w.prev.d.size() - 1));
    w = w.prev;
  }
}

Here is the implementation of the gather() method:

C++
void gather(Node u) {
  Node w = u;
  for (int j = 0; j < b - 1; j++) {
    while (w.d.size() < b)
      w.d.add(w.next.d.remove(0));
    w = w.next;
  }
  remove(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 b 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’ll 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 b−1b − 1 ...