Dynamic Programming Problems经典问题集

Author posted @ 2012年3月05日 09:21 in Algorithm , 18381 阅读

1. The Integer Knapsack Problem (Duplicate Items Permitted):

You have $n$ types of items, where the $i^{th}$ item type has an integer size $s_i$ and a real value. $v_i$

You are trying to fill a knapsack of total capacity $C$ with a selection of items of maximum value.

You can add multiple items of the same type to the knapsack.

Solution: Let M(j) denote the maximum value you can pack into a size knapsack.

We can express M(j) recursively in terms of solutions to smaller problems as follows:

$M(j)= \left\{ \begin{aligned}\overset{}\max\{M(j-1) , \max_{i=1...n}M(j-s_i)+v_i \} \quad j\ge 1\\ \\ 0 \quad j \le 0 \end{aligned}\right.$

Computing each M(j) value will require $\Theta(n)$ time, and we need to sequentially compute $C$ such values.
Therefore, total running time is $\Theta(nC)$ . Total space is $\Theta(n)$ .

The value of $M(C)$ will contain the value of the optimal knapsack packing.

We can reconstruct the list of items in the optimal solution by maintaining and following “backpointers”

2. Maximum Value Contiguous Subsequence.

Given a sequence of $n$ real numbers $A_1,
... A_n$ ,

determine a contiguous subsequence $A_1, ... A_j$ for which the sum of elements in the subsequence is maximized.

Solution: Let $M(j)$ denote the max sum over all windows ending at $j$

$M(j) = max\{ M(j-1)+A[j], A[j]\}$

It only takes linear time $O(n)$ to run the program since it only need to solve $n$ sub-problems and each takes constnt time.

3. Making Change: You are given $n$ types of coin denominations of values $v_1 < v_2 ... < v_n$ (all integers).

Assume $v_1 =1$ , so you can always make change for any amount of money C.

Give an algorithm which makes change for an amount of money C with as few coins as possible.

Solution: Let $M(j)$ denote the min number of coins required to make changes for amount of money $j$

$M(j)=\min_{i=1...n}\{M(j-v_i)\}+1$

It takes $O(nC)$ time because we are solving $C$ subproblems,

each of which requires minimization $O(n)$ different terms. (Similar to integer knapsack problem)

4. Longest Increasing Subsequence. Given a sequence of n real numbers $A_1,
... A_n$ ,

determine a subsequence (not necessarily contiguous) of maximum length in which the values in the subsequence form a strictly increasing sequence.

Solution: Let $L(j) denote the longest strictly increasing subsequence ending at position $j$ , therefore,

$L(j) = \max\limits_{i<j}\limits_{A[i]<A[j]}\{ L(i)\}+1$

to find the solution overall, it has to find the maximum over all potential ending points $j$ of $L(j) :

$\max\limits_{j=1...n}\{L(j)\}$

since the longest increasing subsequence could end anywhere.

The algorithm takes $O(n^2)$ time to run because I have to sort $n$ subproblems and each of them takes $O(n)$ time.

5.Box Stacking:

You are given a set of $n$ types of rectangular 3-D boxes, where the $i^{th}$ box has height $h_i$ , width $w_i$ and depth $d_i$ (all real numbers).

You want to create a stack of boxes which is as tall as possible,

but you can only stack a box on top of another box if the dimensions of the 2-D base of the lower box are each strictly larger than those of the 2-D base of the higher box.

Of course, you can rotate a box so that any side functions as its base.

It is also allowable to use multiple instances of the same type of box.

Solution: constraint : can only stack box $i$ on box $j$ if $w_i<w_j$ and $d_i<d_j$

(without loss of generality, assume $w_i\le d_i$ ).

First sort the base area by decreasing order:

then let $H(j)$ denote the tallest box stack I could form with box $j$ on top, so

$H(j) = \max\limits_{i<j, w_i<w_j, d_i<d_j}\{ H(i)\}+h_j$

The algorithm takes $O(n^2)$ time to run because I have to sort $n$ subproblems and each of them takes $O(n)$ time.

At the end, the solution is found by taking the max of all potential top boxes $j$ of $H(j)$ .

(Since we are not sure in the optimal solution, which box is on top)

6. Building Bridges.

Consider a 2-D map with a horizontal river passing through its center.

There are $n$ cities on the southern bank with x-coordinates $a_1... a_n$ and $n$ cities on the northern bank with x-coordinates $b_1 ... b_n$ .

You want to connect as many north-south pairs of cities as possible with bridges such that no two bridges cross.

When connecting cities, you are only allowed to connect the the ith city on the northern ban to the ith city on the southern bank.

Solution: consider the south bank city (below the river) $i$ ,

let $x(i)$ denote the index of corresponding city on the north bank ( $x(i)=3$ in the figure).

So we could compute the $x(i)$ by sorting in $O(n\log n)$ time.

Now we want to find the longest increasing subsequence through $x(1),
... x(n)$ , so it is identical to problem 4.

7. Balanced Partition:You have a set of n integers each in the range 0 . . .K.

Partition these integers into two subsets such that you minimize $S_1 - S_2$ ,

where $S_1$ and $S_2$ denote the sums of the elements in each of the two subsets.

Solution: Use $P(i,j)= \left\{ \begin{aligned}\overset{}1 \quad \text{if some subset of } \{A_1, ... A_n\} \text{has sum of j}\\0 \quad \text{otherwise}\end{aligned}\right.$ where $i \in [0,n]$ and $j \in [0,nK]$

therefore, $P(i,j)=1$ if $P(i-1, j)=1$ or $P(i-1,
j-A_i)=1$

dynamic programming formula is:

$P(i,j)=\max\{P(i-1,j), P(i-1, j-A_i)\}$

this procedure takes $O(n^2K)$ time to compute.

To solve the original problem, let $S=\sum(A_i)/2$ , and the purpose is to make $S_1 - S_2$ close to 0 as possible.

what we want to find is:

$\min\limits_{i\le S}\{S-i: P(n, i)=1\}$

8. Edit Distance.

Given two text strings A of length n and B of length m,

you want to transform A into B with a minimum number of operations of the following types:

delete a character from A, insert a character into A,

or change some character in A into a new character.

The minimal number of such operations required to transform A into B is called the edit distance between A and B.

Solution: Use $C_i, C_d, C_r$ to denote the cost for inserting, deleting or replacing a letter.

The goal is to compute the minimum cost of translating A to B.

Let $T(i,j)$ denote the minimum cost of translating $A[1...i]$ into $B[1...j]$ , then

$T(i,j)= \left\{\begin{aligned}\overset{}C_d + T(i-1, j) \\ C_i+T(i, j-1) \\ \overset{} T(i-1, j-1) \quad if \quad A[i]=B[j] \\ C_d+T(i-1,j-1) \quad if \quad A[i] \neq B[j] \end{aligned}\right.$

It takes $O(nm)$ time to compute and final solution is stored in $T(n,m)$

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2024年1月02日 05:46

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2024年1月16日 20:24

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카지노사이트 说:
2024年1月16日 21:46

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호빵맨토토 说:
2024年1月16日 21:49

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2024年1月16日 22:35

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2024年1月16日 23:50

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토토사이트 说:
2024年1月16日 23:52

이러한 유형의 주제를 검색하는 거의 모든 사람들에게 도움이되기를 바랍니다. 저는이 웹 사이트가 그러한 주제에 가장 적합하다고 생각합니다. 좋은 작품과 기사의 품질.

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바카라 바카라 사이트 说:
2024年1月17日 00:21

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바카라 사이트 说:
2024年1月17日 00:22

감동 받았습니다. 아주 드물게 유익하고 재미있는 블로그를 접하는 경우가 거의 없습니다. 귀하의 블로그는 중요합니다 ..

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2024年1月17日 01:10

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2024年1月17日 01:11

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소닉카지노 说:
2024年1月17日 02:38

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밀라노토토 说:
2024年1月17日 03:42

와우 ... 정말 멋진 블로그입니다.이 글을 쓴 글쓴이는 정말 훌륭한 블로거입니다.이 글은 제가 더 나은 사람이되도록 영감을줍니다.

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스포츠무료중계 说:
2024年1月17日 19:59

와우 ... 정말 멋진 블로그입니다.이 글을 쓴 글쓴이는 정말 훌륭한 블로거입니다.이 글은 제가 더 나은 사람이되도록 영감을줍니다.

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바카라사이트 说:
2024年1月17日 21:11

와우 ... 정말 멋진 블로그입니다.이 글을 쓴 글쓴이는 정말 훌륭한 블로거입니다.이 글은 제가 더 나은 사람이되도록 영감을줍니다.

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먹튀사이트 说:
2024年1月17日 22:15

와우 ... 정말 멋진 블로그입니다.이 글을 쓴 글쓴이는 정말 훌륭한 블로거입니다.이 글은 제가 더 나은 사람이되도록 영감을줍니다.

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무료스포츠중계 说:
2024年1月18日 00:13

와우 ... 정말 멋진 블로그입니다.이 글을 쓴 글쓴이는 정말 훌륭한 블로거입니다.이 글은 제가 더 나은 사람이되도록 영감을줍니다.
Q

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2024年1月18日 00:32

나는 그것의 굉장한 것을 말해야한다! 블로그는 정보를 제공하며 항상 놀라운 것을 생산합니다.

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2024年1月22日 00:36

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Störsender-Inhalt ak 说:
2024年7月29日 17:29

Die Verwendung von Störsendern kann in verschiedenen Situationen äußerst nützlich sein, insbesondere wenn es darum geht, störende oder unerwünschte Signale zu blockieren. Die Funktion "Störsender-Inhalt aktualisieren" hat sich als äußerst effizient und zuverlässig erwiesen.

Effizienz: Die Aktualisierungsfunktion sorgt dafür, dass der Störsender stets auf dem neuesten Stand ist und somit die neuesten Frequenzen und Signalarten blockieren kann. Dies ist besonders wichtig, um sicherzustellen, dass keine neuen Technologien ungestört bleiben.

Benutzerfreundlichkeit: Die Bedienung der Aktualisierungsfunktion ist sehr einfach und intuitiv gestaltet. Selbst für Personen, die nicht technisch versiert sind, ist der Prozess leicht verständlich und schnell durchführbar.

Zuverlässigkeit: In meinen Tests hat sich die Funktion als äußerst zuverlässig erwiesen. Es gab keine Ausfälle oder Probleme während der Nutzung, was ein großer Vorteil ist, wenn man bedenkt, wie wichtig kontinuierlicher Schutz in bestimmten Szenarien sein kann.

Sicherheit: Die Aktualisierungsfunktion bietet auch erhöhte Sicherheit, da sie sicherstellt, dass der Störsender immer gegen die neuesten Bedrohungen geschützt ist. Dies ist besonders relevant in sicherheitskritischen Bereichen wie Regierungsgebäuden oder großen Veranstaltungen.

Insgesamt bin ich sehr zufrieden mit der "Störsender-Inhalt aktualisieren"-Funktion und kann sie jedem empfehlen, der auf der Suche nach einem zuverlässigen und effizienten Mittel zur Signalblockierung ist. Die einfache Handhabung und die hohe Zuverlässigkeit machen sie zu einer ausgezeichneten Wahl.

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