# 10.3: Discussion and Exercises

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The implicit representation of a complete binary tree as an array, or list, seems to have been first proposed by Eytzinger [27]. He used this representation in books containing pedigree family trees of noble families. The `BinaryHeap` data structure described here was first introduced by Williams [78].

The randomized `MeldableHeap` data structure described here appears to have first been proposed by Gambin and Malinowski [34]. Other meldable heap implementations exist, including leftist heaps [16,48, Section 5.3.2], binomial heaps [75], Fibonacci heaps [30], pairing heaps [29], and skew heaps [72], although none of these are as simple as the `MeldableHeap` structure.

Some of the above structures also support a \(\mathtt{decreaseKey(u,y)}\) operation in which the value stored at node \(\mathtt{u}\) is decreased to \(\mathtt{y}\). (It is a pre-condition that \(\mathtt{y}\le\texttt{u.x}\).) In most of the preceding structures, this operation can be supported in \(O(\log \mathtt{n})\) time by removing node \(\mathtt{u}\) and adding \(\mathtt{y}\). However, some of these structures can implement \(\mathtt{decreaseKey(u,y)}\) more efficiently. In particular, \(\mathtt{decreaseKey(u,y)}\) takes \(O(1)\) amortized time in Fibonacci heaps and \(O(\log\log \mathtt{n})\) amortized time in a special version of pairing heaps [25]. This more efficient \(\mathtt{decreaseKey(u,y)}\) operation has applications in speeding up several graph algorithms, including Dijkstra's shortest path algorithm [30].

Exercise \(\PageIndex{1}\)

Illustrate the addition of the values 7 and then 3 to the `BinaryHeap` shown at the end of Figure 10.1.2.

Exercise \(\PageIndex{2}\)

Illustrate the removal of the next two values (6 and 8) on the `BinaryHeap` shown at the end of Figure 10.1.3.

Exercise \(\PageIndex{3}\)

Implement the \(\mathtt{remove(i)}\) method, that removes the value stored in \(\mathtt{a[i]}\) in a `BinaryHeap`. This method should run in \(O(\log \mathtt{n})\) time. Next, explain why this method is not likely to be useful.

A \(d\)-ary tree is a generalization of a binary tree in which each internal node has \(d\) children. Using Eytzinger's method it is also possible to represent complete \(d\)-ary trees using arrays. Work out the equations that, given an index \(\mathtt{i}\), determine the index of \(\mathtt{i}\)'s parent and each of \(\mathtt{i}\)'s \(d\) children in this representation.

Exercise \(\PageIndex{5}\)

Using what you learned in Exercise \(\PageIndex{4}\), design and implement a `DaryHeap`, the \(d\)-ary generalization of a `BinaryHeap`. Analyze the running times of operations on a `DaryHeap` and test the performance of your `DaryHeap` implementation against that of the `BinaryHeap` implementation given here.

Exercise \(\PageIndex{6}\)

Illustrate the addition of the values 17 and then 82 in the `MeldableHeap` \(\mathtt{h1}\) shown in Figure 10.2.1. Use a coin to simulate a random bit when needed.

Exercise \(\PageIndex{7}\)

Illustrate the removal of the next two values (4 and 8) in the `MeldableHeap` \(\mathtt{h1}\) shown in Figure 10.2.1. Use a coin to simulate a random bit when needed.

Exercise \(\PageIndex{8}\)

Implement the \(\mathtt{remove(u)}\) method, that removes the node \(\mathtt{u}\) from a `MeldableHeap`. This method should run in \(O(\log \mathtt{n})\) expected time.

Exercise \(\PageIndex{9}\)

Show how to find the second smallest value in a `BinaryHeap` or `MeldableHeap` in constant time.

Exercise \(\PageIndex{10}\)

Show how to find the \(k\)th smallest value in a `BinaryHeap` or `MeldableHeap` in \(O(k\log k)\) time. (Hint: Using another heap might help.)

Exercise \(\PageIndex{11}\)

Suppose you are given \(\mathtt{k}\) sorted lists, of total length \(\mathtt{n}\). Using a heap, show how to merge these into a single sorted list in \(O(n\log k)\) time. (Hint: Starting with the case \(k=2\) can be instructive.)