10.3: Discussion and Exercises
- Page ID
- 8479
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.)