TAOCP 5.2.1: Sorting by Insertion
Section 5.2.1 exercises: 37/37 solved.
Section 5.2.1. Sorting by Insertion
Exercises from TAOCP Volume 3 Section 5.2.1: 37/37 solved.
| # | Rating | Category | Status | Time |
|---|---|---|---|---|
| 1 | [10] | simple | solved | 13m25s |
| 2 | [11] | simple | verified | 1m15s |
| 3 | ▶ [30] | hard | solved | 9m01s |
| 4 | ▶ [M20] | math-medium | verified | 16m17s |
| 5 | [M27] | math-hard | solved | 32m26s |
| 6 | [23] | medium | verified | 1m27s |
| 7 | [M20] | math-medium | verified | 14m42s |
| 8 | [10] | simple | solved | 4m47s |
| 9 | [20] | medium | solved | 3m37s |
| 10 | [22] | medium | verified | 1m29s |
| 11 | [M10] | math-simple | verified | 2m59s |
| 12 | [M20] | math-medium | verified | 2m46s |
| 13 | ▶ [M16] | math-medium | verified | 2m55s |
| 14 | [M28] | math-hard | solved | 4m47s |
| 15 | ▶ [HM3] | hm-simple | solved | 1h32m |
| 16 | [M2] | math-simple | solved | 53m18s |
| 17 | [M21] | math-medium | solved | 9m01s |
| 18 | [HM2] | hm-simple | solved | 7m44s |
| 19 | ▶ [M25] | math-medium | solved | 20m21s |
| 20 | [M22] | math-medium | solved | 41m49s |
| 21 | [M25] | math-medium | solved | 1m51s |
| 22 | [M30] | math-hard | solved | 19m32s |
| 23 | ▶ [M22] | math-medium | solved | 54m26s |
| 24 | [M43] | math-project | solved | 23m21s |
| 25 | ▶ [M22] | math-medium | solved | 14m14s |
| 26 | [M35] | math-hard | solved | 59m13s |
| 27 | [M41] | math-project | solved | 5m07s |
| 28 | [15] | simple | solved | 1m48s |
| 29 | [40] | project | solved | 1m19s |
| 30 | [M23] | math-medium | solved | 2m42s |
| 31 | [25] | medium | solved | 48m18s |
| 32 | [10] | simple | solved | 4m23s |
| 33 | [25] | medium | solved | 43m30s |
| 34 | [M10] | math-simple | solved | 22m |
| 35 | [21] | medium | solved | 7m11s |
| 36 | [18] | medium | solved | 29m30s |
| 37 | [M25] | math-medium | solved | 11m18s |
TAOCP 5.2.1 Exercise 1
Yes.
TAOCP 5.2.1 Exercise 2
Let step S4 in Algorithm S be the comparison step that determines whether the current key $K$ should be inserted before $K_i$ or whether $K_i$ should be moved right.
TAOCP 5.2.1 Exercise 3
No.
TAOCP 5.2.1 Exercise 4
The running time of Program S is $9B+10N-3A-9.$ By the results stated in Section 5.
TAOCP 5.2.1 Exercise 5
**Exercise 5.
TAOCP 5.2.1 Exercise 6
Let $R_1,\dots,R_N$ be the records, and let their final order after two-way insertion be determined as in Table 2, where each new record is inserted either to the left end or the right end of the curr...
TAOCP 5.2.1 Exercise 7
Let $a_1 a_2 \dots a_n$ be a random permutation of ${1,2,\dots,n}$.
TAOCP 5.2.1 Exercise 8
A correct solution must address stability in the sense of TAOCP: records with equal keys must preserve their relative order after the entire Shellsort process.
TAOCP 5.2.1 Exercise 9
The flaw in the previous solution is that it never uses the data in Tables 3 and 4.
TAOCP 5.2.1 Exercise 10
In step D3, Program D performs three actions before any comparison outcome is known: it sets $i = j - h$, it loads $K = K_j$, and it loads $R = R_j$.
TAOCP 5.2.1 Exercise 11
Let \pi = (1,2,5,3,7,4,8,6,9,11,10,12).
TAOCP 5.2.1 Exercise 12
A 2-ordered permutation $a_1a_2\cdots a_n$ satisfies a_i<a_{i+2}\qquad (1\le i\le n-2).
TAOCP 5.2.1 Exercise 13
Start from the standard lattice representation of a permutation used in Section 5.
TAOCP 5.2.1 Exercise 14
Let $A_{i,j}$ be defined by Eq.
TAOCP 5.2.1 Exercise 15
The generating functions are defined by the following recurrences.
TAOCP 5.2.1 Exercise 16
Let $n$ and $h$ be positive integers.
TAOCP 5.2.1 Exercise 17
We consider Shellsort (Algorithm D) with $N = 2^t$ and increments $h_s = 2^s$ for $s = t-1, t-2, \ldots, 0$.
TAOCP 5.2.1 Exercise 18
We minimize the asymptotic estimate for the sum (6): \[ E = \frac{N^2}{4h_{t-1}} + \frac{\sqrt{\pi}}{8}\,N^{3/2} \sum_{k=1}^{t-1} \frac{h_k^{1/2}}{h_{k-1}}, \] where \(N\) and \(t\) are fixed, \(h_0 =...
TAOCP 5.2.1 Exercise 19
Let the increments be \(h_{t-1}>h_{t-2}>\cdots>h_0=1\) and assume they satisfy the divisibility condition \[ h_{s}\;\text{is a multiple of}\;h_{s-1}\qquad(1\le s\le t-1).
TAOCP 5.2.1 Exercise 20
We first recall the statements of Lemma L and Theorem K from the text.
TAOCP 5.2.1 Exercise 21
Let \(h\) and \(k\) be relatively prime positive integers.
TAOCP 5.2.1 Exercise 22
Let $s \ge 2$ be an integer.
TAOCP 5.2.1 Exercise 23
After passes with increments \(h_{s+2}\) and \(h_{s+1}\) the file is \(h_{s+2}\)-ordered and \(h_{s+1}\)-ordered: \[ K_i\le K_{i+h_{s+2}}\quad(1\le i\le N-h_{s+2}),\qquad K_i\le K_{i+h_{s+1}}\quad(1\l...
TAOCP 5.2.1 Exercise 24
Theorem P (as proved in the text) states that if the increments are \[ h_0=1,\qquad h_{k+1}=2h_k+1\quad (k\ge 0), \] so that \(h_k=2^{\,k+1}-1\), then the total number of moves performed by Algorithm ...
TAOCP 5.2.1 Exercise 25
A permutation \(K_1K_2\ldots K_N\) of \(\{1,2,\ldots,N\}\) is *2-ordered* if \(K_i < K_{i+2}\) for \(1\le i\le N-2\), and *3-ordered* if \(K_i < K_{i+3}\) for \(1\le i\le N-3\).
TAOCP 5.2.1 Exercise 26
We consider permutations of \(\{1,\dots,N\}\) that are \(3\)-ordered, \(5\)-ordered and \(7\)-ordered, i.
TAOCP 5.2.1 Exercise 27
Let the sequence of increments used by Algorithm D be \(h_{t-1}>h_{t-2}>\cdots >h_0=1\).
TAOCP 5.2.1 Exercise 28
Table 6 in Section 5.
TAOCP 5.2.1 Exercise 29
Let $N=1000$.
TAOCP 5.2.1 Exercise 30
Let the set of increments be \[ S = \{\,2^{p}3^{q}\mid 2^{p}3^{q}<N,\;p,q\ge 0\,\}.
TAOCP 5.2.1 Exercise 31
**Algorithm.
TAOCP 5.2.1 Exercise 32
The list insertion sort (Algorithm L) applied to the 16 keys from Table 1 (503, 087, 512, 061, 908, 170, 897, 275, 653, 426, 154, 509, 612, 677, 765, 703) produces the following final link array \(L_0...
TAOCP 5.2.1 Exercise 33
We first recall Program L (list insertion sort) from Section 5.
TAOCP 5.2.1 Exercise 34
Formula (14) gives the running time of Program D (Shellsort) as \[ 9B + 10Nt - 3A - 10\sum_{s=0}^{t-1}h_s + 13t + 1 \quad \text{units}, \] where \(t\) is the number of passes, \(h_s\) are the incremen...
TAOCP 5.2.1 Exercise 35
* There are \(M \ge 1\) lists.
TAOCP 5.2.1 Exercise 36
The sixteen keys of Table 8 are (in the order they appear) \[ 560000,\;087000,\;512000,\;061000,\;908000,\;170000,\;897000,\;275000,\; 653000,\;426000,\;154000,\;509000,\;612000,\;677000,\;765000,\;70...