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[科技前沿] 突然想到让deepseek来解释一下递归

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    2025-9-8 05:08
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     楼主| 发表于 2025-1-29 14:16:55 | 只看该作者 回帖奖励 |倒序浏览 |阅读模式
    本帖最后由 密银 于 2025-1-29 14:19 编辑 * w8 g: O& C% v0 u5 j$ _# E
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    解释的不错
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    递归是一种通过将问题分解为更小的同类子问题来解决问题的方法。它的核心思想是:**函数直接或间接地调用自身**,直到满足终止条件。- a3 Q8 p1 ]; S

    & p2 I3 s; N+ A# g% e( y' W0 I+ q 关键要素
    ! f  G  L7 i( _: [! a$ q# {1. **基线条件(Base Case)**
    0 B; d4 ]- z' t0 v9 {9 k! S  ?   - 递归终止的条件,防止无限循环" P* T" U+ \9 |9 P" n4 w, s" \2 v
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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    2. **递归条件(Recursive Case)**: \3 F+ k5 k& Z2 x# [+ e
       - 将原问题分解为更小的子问题4 \# Z* W+ n& f6 z
       - 例如:n! = n × (n-1)!0 I% L8 @4 k, k3 y

    5 r* n! C0 R. k8 x- G 经典示例:计算阶乘
    7 {* v% ~. K5 G& Z; {' Ppython
    % O# h2 a( a: _0 pdef factorial(n):
    3 v+ F# p. p$ {1 U4 e. D/ y    if n == 0:        # 基线条件
    3 m+ L' W  a8 B# u: w; X8 o        return 15 ~, e7 I1 C+ x4 `0 e
        else:             # 递归条件! I1 [& z5 g# b
            return n * factorial(n-1)0 L2 S7 Q; L* y8 U1 R( G: A- C
    执行过程(以计算 3! 为例):
    1 ^2 W: e3 E- ?factorial(3)
    ( ?2 q: c! @. V* r' c1 K1 Z: A3 * factorial(2)
    - [9 Q8 o" _/ ^( _9 Z+ h, g1 L3 * (2 * factorial(1))
    3 u* D4 _. ^; h+ r/ t# v3 * (2 * (1 * factorial(0)))# H0 U$ j" L/ D: \- o# H
    3 * (2 * (1 * 1)) = 6
    ) ~2 m: e. ^8 \0 S, l* }: y) n# c; K2 R- y4 k
    递归思维要点
    3 \5 O6 [1 b( {$ w4 \0 [* p. `: U1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # `) ~2 M! c. J  @7 e2. **栈结构**:每次调用都会创建新的栈帧(内存空间)2 d" i/ `: h4 {& e
    3. **递推过程**:不断向下分解问题(递)6 {, ^9 N& Q" w4 C
    4. **回溯过程**:组合子问题结果返回(归)+ S( u9 ^- y+ o8 u

    . H& y; @$ [, N; z: `$ _ 注意事项% B: F9 a4 {: a! w: u, ~; Y
    必须要有终止条件
    $ y; I3 j# T; i3 L, A$ F递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  G8 W# R4 F! w& x5 K  E
    某些问题用递归更直观(如树遍历),但效率可能不如迭代; X( A  L* ^& f4 L
    尾递归优化可以提升效率(但Python不支持)$ V2 }- i- i7 R; e

    $ z( A9 p: s7 O7 L 递归 vs 迭代7 F* _0 `6 A8 u( X9 X: I6 W7 O
    |          | 递归                          | 迭代               |
    ' u4 ]: F8 Q8 n  h( [|----------|-----------------------------|------------------|
    : U9 T- K# Z) }4 \| 实现方式    | 函数自调用                        | 循环结构            |
    0 e2 b8 w+ I% `1 \( V4 a- q| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. [' t) r- `' W. j3 q
    | 代码可读性  | 对符合递归思维的问题更直观                | 线性流程更直接         |% q/ k% D% P( r% }8 I, O" Z
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |& h# P( M% @8 U! u" }1 g* R
    - _; I* T% n: T) {) R
    经典递归应用场景/ K' l. U' @! V2 @; j( u) X) [2 W
    1. 文件系统遍历(目录树结构): Y; y/ f. S7 v6 b: E
    2. 快速排序/归并排序算法
    ! P& ~, I" K* S+ M/ x$ d3. 汉诺塔问题* F0 U+ @, \* }5 b
    4. 二叉树遍历(前序/中序/后序)! E3 O% v6 r7 E- T$ k! X
    5. 生成所有可能的组合(回溯算法)
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    试着用递归思维想象:你站在一面镜子前,镜子里有无数个逐渐变小的你在照镜子,这就是递归的直观体现。但记住每个"分身"最终都要有结束的时刻,这就是基线条件的重要性。

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    昨天 06:39
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    沙发
    发表于 2025-1-30 00:07:50 | 只看该作者
    挺好,递归思维要点与我能够回忆起来我当时写递归程序的思路很一致,,或者被它唤醒,7 ^2 ~" r0 v  E' ~9 k6 N) `4 d
    我推理机的核心算法应该是二叉树遍历的变种。
    5 q9 b7 ~& l5 S. V. k另外知识系统的推理机搜索深度(递归深度)并不长,没有超过10层的,如果输入变量多的话,搜索宽度很大,但对那时的286-386DOS系统,计算压力也不算大。
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    板凳
    发表于 2025-2-2 00:45:59 | 只看该作者
    Recursion in programming is a technique where a function calls itself in order to solve a problem. It is a powerful concept that allows you to break down complex problems into smaller, more manageable subproblems. Here's a detailed explanation:7 s7 ?1 y9 j" ?. X
    Key Idea of Recursion) T  L8 V# L, m5 F$ F  l
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    A recursive function solves a problem by:
    % v$ c& h, r0 P7 T) e! C9 E% P2 f$ @- w
        Breaking the problem into smaller instances of the same problem.
    ' k% O7 S0 h) W2 P# W/ \8 h6 w0 v
        Solving the smallest instance directly (base case).
    / s3 p! d) O7 a2 H# G
    0 [0 P/ y: g5 @- w    Combining the results of smaller instances to solve the larger problem.
    : {2 A/ B1 ^, t- o8 K+ _( ~: I3 Q) S
    Components of a Recursive Function
    ! `% D" C! m' |& Y1 @9 I
    9 z( j2 w+ Q* i4 U2 Q3 H    Base Case:- W# ?/ S. ?' V1 L
      M0 _- @& c8 _$ H- D1 }
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
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    $ n& ?. c2 q% j- {' P% b% C" M) s        It acts as the stopping condition to prevent infinite recursion.
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            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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        Recursive Case:2 J  F! z1 ^1 F7 U

    * l7 [0 v+ Z- E7 C/ y6 X        This is where the function calls itself with a smaller or simpler version of the problem.
    + @  q$ x+ i4 R9 I( l" s: s/ r; z
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    5 f$ P* F1 W( A5 e& O' ?6 L. i$ e/ m5 [" Q/ X: P" ?
    Example: Factorial Calculation
    ; Q6 l# y; G. z5 M2 J& Y4 }
    $ E0 `7 ?) C3 j! O; d+ M- n! ^The factorial of a number n (denoted as n!) is the product of all positive integers less than or equal to n. It can be defined recursively as:
    2 C7 ~/ C4 x! n% P; E
      G3 i  Q. H$ [( b3 e- t& Q    Base case: 0! = 1  l( @7 M3 P3 F6 W( p0 e5 @( M$ P

    1 Q$ k; k* B& m1 [7 V* y6 }    Recursive case: n! = n * (n-1)!
    * @1 z5 e4 e3 o$ s! E* l1 R; [
    ) f5 }9 _" G. x* wHere’s how it looks in code (Python):) e1 B$ d. q) N: Z, n" O
    python: c& D% a0 _3 Y6 [- t
    ' z4 @9 v& \; ^$ J$ d2 d
    / e& j4 U+ X$ r
    def factorial(n):+ m) |+ V7 ^8 Y8 ~; d8 y. l! Z
        # Base case4 n9 t) }7 @' t! f* O. @, [7 p: D
        if n == 0:8 {+ C! Q. L3 j" d
            return 1# N( I6 a+ w1 f% e2 b
        # Recursive case6 V1 y/ N+ g' L& z
        else:# V1 j! w# J7 R+ T5 u
            return n * factorial(n - 1)
    + v3 k8 N+ H' }4 J- N# L. p" w4 @& c- J( C( T0 ~# e9 ?
    # Example usage
    8 m( j! @/ K* ?& d; h4 uprint(factorial(5))  # Output: 1207 I2 `. @) l7 g" Y+ j3 w6 N, b* U

    3 {. a/ s- l/ z) j# NHow Recursion Works
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    ! I; M/ s7 V  @# h+ Z    The function keeps calling itself with smaller inputs until it reaches the base case.3 l5 v  @  x1 ^+ D4 z, [# p

    ; o. n  \, a8 e    Once the base case is reached, the function starts returning values back up the call stack.
    7 T1 [5 A. {3 x( S) T2 A1 H3 W  y) `0 p) T8 a
        These returned values are combined to produce the final result.! D# R/ x; L9 x5 Q& F3 A8 P) }

    6 {( ^1 j- j" g) m' ~* ZFor factorial(5):
    ! F9 ?6 B+ w2 b' U, }. x2 `9 y0 Q- A) z' t5 {1 V# A
      B2 @% Z* h8 c6 R% t; O3 s
    factorial(5) = 5 * factorial(4)
    " T2 l/ T2 ^5 R: Y+ R+ gfactorial(4) = 4 * factorial(3)
      O% m4 t& D" C0 jfactorial(3) = 3 * factorial(2)0 S% ]. @6 l9 ~* J# s/ h9 I- B. E
    factorial(2) = 2 * factorial(1)
    $ U  {% m, o7 W+ mfactorial(1) = 1 * factorial(0)
    ; b7 Y# O, a* e$ R) qfactorial(0) = 1  # Base case
    7 [& X6 f( c  q6 k  d% N- v* e/ r/ l6 Y
    Then, the results are combined:
    / j) k' [0 p7 {0 i7 c1 x
    - f4 c& e; \6 O0 w! D: m3 ^- v  O) A) A0 F. m. m: {
    factorial(1) = 1 * 1 = 11 c/ H1 @( c& p# N- ^! s% f
    factorial(2) = 2 * 1 = 2
    ( ]3 J3 A9 e$ g; s, }" y% D  v% p% h# Dfactorial(3) = 3 * 2 = 6
    1 j4 @" V- u! Y6 e( E" `factorial(4) = 4 * 6 = 24: x) O0 m6 L; v% o) C9 J2 Z  u7 G
    factorial(5) = 5 * 24 = 120( q& c  J0 T  H& m# e9 Z

    ! ]3 |3 a, J- Y7 {! F$ ~: W& [Advantages of Recursion0 d% v1 J! j9 G- ]! ~; v

    ; L7 ^% `$ w5 D    Simplicity: Recursive solutions are often more intuitive and easier to write for problems that have a natural recursive structure (e.g., tree traversals, divide-and-conquer algorithms).( [/ D+ b, C/ H: Q% I. i
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        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! ?" u9 j; l8 t' |" L9 ?  n+ S6 G( j6 {" [$ n. U0 T6 e
    Disadvantages of Recursion& o+ x! x& Z- G1 b. ^
    % A' h* ^: X% y$ K# \' L, ~
        Performance Overhead: Each recursive call adds a new layer to the call stack, which can lead to high memory usage and potential stack overflow for deep recursion.
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        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    # L* k& ~: }& W+ Z" J
    " H3 O& ?7 c- b+ X: e+ bWhen to Use Recursion3 c$ ]: G9 `: \: P

    ! H" U6 Y+ B1 B' \' {& h" {    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    : F! ~+ ?6 t( W3 a
    2 @& B/ Q* v  X0 h) D    Problems with a clear base case and recursive case.# J0 s# `7 w% E+ B
    . N2 c+ w! h1 {# N
    Example: Fibonacci Sequence7 |8 I! [# j2 b3 Z- z; V
    ; N7 ]9 G) b5 `8 x" W. B! T
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:$ L! [. L" B1 I( D

    4 H  z3 ~  W" V& n5 L7 x    Base case: fib(0) = 0, fib(1) = 19 V0 V$ M1 s" |' b
    / }/ I4 e1 E1 Y1 S4 J' V  t
        Recursive case: fib(n) = fib(n-1) + fib(n-2). ~  T* G9 U( o" @4 r

    - ]! g) x9 _$ k7 ]4 ?python
    1 z4 Q, Y: s+ d9 c- c2 `4 c1 o! z1 D) P5 d+ b% E3 w& e4 Y3 j% E
    7 h9 P. A- g  e5 n
    def fibonacci(n):
    ( k0 Z' W0 O4 |! b    # Base cases
    0 o+ d5 s" L. }% |1 M# `    if n == 0:
    4 O1 |+ e5 D- c& X        return 0
    % d! `! C: X1 \; m+ P    elif n == 1:
    ' H3 |8 `: E' r+ h6 f0 v3 {        return 1
    $ ~9 |  N# h( b5 a1 @/ d" i    # Recursive case
    6 d8 D7 I: o; a# K1 ^! m1 b1 Z    else:: p' T: u1 D) x  t! ]* G, g
            return fibonacci(n - 1) + fibonacci(n - 2)
    7 E3 B! G) ~) M5 c3 i) h  V4 _4 `7 m
    # Example usage; }$ J$ n4 _: ^
    print(fibonacci(6))  # Output: 8
    & p8 `. a5 U$ i
    . A; P5 c8 o+ b1 ~& ?' CTail Recursion
    1 q+ W6 n. P( @$ O# W; d+ n+ |  k) u1 @, j" X3 m9 d
    Tail recursion is a special case of recursion where the recursive call is the last operation in the function. Some programming languages optimize tail-recursive functions to avoid stack overflow, but not all languages (e.g., Python does not optimize tail recursion).
    3 l# Z% T0 }7 N4 X2 H/ o
    2 A% _8 S1 s5 U. s) }In summary, recursion is a fundamental concept in programming that allows you to solve problems by breaking them into smaller, self-similar subproblems. It’s important to define a base case to avoid infinite recursion and to understand the trade-offs between recursion and iteration.
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    地板
    发表于 2025-2-2 00:47:27 | 只看该作者
    我还让Deepseek 给我讲讲Linux Kernel Driver 现在的开发流程,让一个老同志复习复习,快忘光了。
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