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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
  • 签到天数: 3 天

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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 0 D  M) ^% \1 s  u3 ~2 k% g$ W" U% i

    5 h, J% J+ w) W1 i  m解释的不错
    % o2 |! N! |2 x) R* ?& V8 D: y: x$ z) _/ r
    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。
    & F# p! o* p. ]0 ^. j7 ^
    ; j# W  X/ ]; [ 关键要素
      P, L8 h% {0 ?/ W. c. p1. **基线条件(Base Case)**2 d6 ~" S. a. @, g) c0 V: F1 ]
       - 递归终止的条件,防止无限循环3 W% `- v1 H" d
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 13 W. C$ |* ?: `. r
    & x; Z% E; \% g- H+ f, E3 {) j
    2. **递归条件(Recursive Case)**
    * ]+ K( }$ f6 `   - 将原问题分解为更小的子问题
    / U$ ]# m7 ~. L9 P1 S   - 例如:n! = n × (n-1)!
    0 C" w' s3 G8 \2 k* w) W, N( R1 h% d
    " g' k5 E4 R" K 经典示例:计算阶乘  L/ D8 L# ^4 O- {. Q
    python+ R; i0 N! w9 q+ D
    def factorial(n):/ T+ n, B' r+ @/ E/ s9 y( _8 A
        if n == 0:        # 基线条件: ?5 C8 k+ V7 I- m" |
            return 13 E0 V6 C/ E9 K# I4 y1 a2 P3 N0 p  f
        else:             # 递归条件4 N: l" V; f. W" Z  T" h
            return n * factorial(n-1)  }8 _0 M& r  k1 q
    执行过程(以计算 3! 为例):% q; V: I, k* i3 E4 d
    factorial(3)# j8 Q; ~6 ^% J4 L
    3 * factorial(2)
    0 G. g( o/ n4 A3 * (2 * factorial(1))) N) A" u1 C8 V* |) c% q: A
    3 * (2 * (1 * factorial(0)))* C% u4 w. l- k: k8 F+ k
    3 * (2 * (1 * 1)) = 6  |. M/ Y/ h1 _; f1 C. g6 f0 |
    * K0 X/ C1 t6 }2 J7 d7 e
    递归思维要点  R5 @1 Y) [# j$ d
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    6 B3 g9 J5 ]  f# R8 N+ K4 P2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    * `! i9 _( C2 m+ [! T' H8 ]3. **递推过程**:不断向下分解问题(递)' T( n# G% A! {
    4. **回溯过程**:组合子问题结果返回(归)9 l0 _! a' j* u0 w' y( K/ y

    " l/ b: k( T% t7 {# ]$ s 注意事项
    % g% ?2 s7 I$ j. y  ?6 ?; R必须要有终止条件9 y5 U4 a: o, _
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    - t4 Z7 F5 @& c" o% i: U7 ?某些问题用递归更直观(如树遍历),但效率可能不如迭代
    6 ?+ U+ L) W& ~$ i3 z/ x' y尾递归优化可以提升效率(但Python不支持)8 r# A& n, o1 }

    : E9 Z% V% w+ J. _0 J5 X 递归 vs 迭代
    7 P1 ]/ a8 d7 u3 p|          | 递归                          | 迭代               |: X3 A7 x4 v* M! P9 F
    |----------|-----------------------------|------------------|
    + Q8 X+ ?# w. q/ d/ K| 实现方式    | 函数自调用                        | 循环结构            |
    - ?) b9 q, U. Q6 r4 y/ e( || 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |* U$ `' i5 X  d) @1 a' M- r8 w
    | 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         |
    ' h( A) ^& N2 B| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |) \3 X0 D9 g  X9 D( E; q/ y
    0 Q! R7 N" C  k% F
    经典递归应用场景
    & H/ h- |3 l* c2 s" C- D9 i7 C1. 文件系统遍历(目录树结构)
    2 \" H8 D# W9 p3 |$ t2. 快速排序/归并排序算法: B! C* H# y  G! T( d: j# G
    3. 汉诺塔问题9 Z8 I# |4 @6 D  ^" \) g! s- ^
    4. 二叉树遍历(前序/中序/后序)
    - ~2 _1 {* ^5 i6 x/ Z5 Z. \5 t8 K5. 生成所有可能的组合(回溯算法)
    ) @1 I; ]% ~9 X7 s& H, I0 @$ K
    . P" w& S" V0 I+ p8 b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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  • TA的每日心情

    昨天 07:21
  • 签到天数: 3249 天

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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,
    ! X) v6 Y; n$ x5 x% x, _1 ?我推理机的核心算法应该是二叉树遍历的变种。
    - o, E% G% J) `! T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:
    3 S& H7 v* h0 ^5 Y0 wKey Idea of Recursion
    ! P$ B+ r7 {0 @, m1 u5 u2 _( j( \9 w
    A recursive function solves a problem by:) I- L" x6 ?5 o

    5 t0 m- p- Q0 K$ M    Breaking the problem into smaller instances of the same problem.' g; v5 w/ \  L- F/ L; [" k! w
    # R+ Y  n9 R( k9 R
        Solving the smallest instance directly (base case).
    * Y$ E  g$ g3 O8 A! g3 m
    1 d+ s# Z9 _' e+ A; S% f. ^    Combining the results of smaller instances to solve the larger problem.
    * ~* J( q6 H6 G+ n, `+ j' L  s) x7 M& J8 I" c
    Components of a Recursive Function
    % x+ m  c; X1 w  g! O9 _- {# S+ k
    - f1 U/ @. m- `# Z6 U    Base Case:% i  {! e+ y- N% d* E' Z$ |

    # t" N3 \: T( }  N9 Q8 @        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.( L% M7 `/ i3 m8 n+ w
    / Z1 i; p: `$ O% W" C& d
            It acts as the stopping condition to prevent infinite recursion.
    ( P  x0 P( M! d. c1 v
    ) k0 M! k4 B  u  j( t7 Q/ T        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    + ^: R5 n7 }; ?" A  g  C* ?( O( G6 T. z  y7 v
        Recursive Case:! ]4 _$ U5 o  l( I4 S) |2 ~

    * K/ L$ j) x5 L        This is where the function calls itself with a smaller or simpler version of the problem.
    # H$ t. A  i; U2 r6 j- M
    & P- ?8 L; D' |3 o2 O        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. q7 l1 x& r, Z* m% y
    ) [' c/ H0 F: r2 I" ?7 T7 w
    Example: Factorial Calculation
    8 M; h4 H+ M) M9 `) ~, E) A/ `# f; ]0 W; M4 g! U6 V
    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:! v6 a9 |4 V) c- \' S

    ! P% Z/ P/ P. x7 A    Base case: 0! = 1
    9 {# Q9 a6 }: b6 l% ~2 d# r' w* F7 m
        Recursive case: n! = n * (n-1)!8 y3 \, s3 @0 H

    6 |; j! f; T  ]! j- y: T4 LHere’s how it looks in code (Python):" l% L8 P) p$ V' G; @% m
    python; ^" {* K3 p/ p, v

    : h$ c6 L9 D8 w0 g
    ! T5 |5 S3 W# L9 Gdef factorial(n):
    , q, f/ ^  \8 t) ]    # Base case
    4 }8 B7 |) o$ _0 s7 |, x    if n == 0:
    - |; O1 k, q* z2 ]$ z) i" l        return 1% f+ r6 ]% U' c# M- B' k' ]% O0 \
        # Recursive case
    - M' w; ^, P6 K    else:7 l9 Y9 B$ n5 S3 O3 s
            return n * factorial(n - 1), E$ c0 @% Q  I% A+ e

    ) ?% V6 I* Y+ U8 k# Example usage
    ( n3 E8 ]" r, q3 a4 |( Nprint(factorial(5))  # Output: 120: b; O* N4 v! }: {9 P" f# n
    0 E, j6 h$ w/ S& u; O2 j- Z
    How Recursion Works
    ! y6 S. s; f/ L) \2 [) b& _4 |% _
    # c9 H8 D! j$ Z    The function keeps calling itself with smaller inputs until it reaches the base case.+ r% x) @8 n7 k4 \; W
    ! d6 J4 ~7 q8 r" A- ?& w
        Once the base case is reached, the function starts returning values back up the call stack.
    1 `+ X- a, K* b; [" u' \3 n/ }+ ?# j+ c
        These returned values are combined to produce the final result.
    7 G! n' f% s( \6 k- K
    , V# x$ }5 R7 T1 d5 p& eFor factorial(5):4 }: y0 U0 p) A6 P1 J+ T
    . P1 }7 L9 v, s0 f7 M) W
    + ^7 A- X$ D1 a7 g
    factorial(5) = 5 * factorial(4)/ J1 u* Z: c4 u0 b  X% K9 G, j( O7 `0 W
    factorial(4) = 4 * factorial(3)
    $ D1 S5 ~. h( u# }, Pfactorial(3) = 3 * factorial(2)7 p, G: ~5 R4 o' I: O
    factorial(2) = 2 * factorial(1)3 l+ m; w2 f, n2 z; @4 D
    factorial(1) = 1 * factorial(0)5 o* d4 v* u8 D" o* B# C; R
    factorial(0) = 1  # Base case
    ' J! g9 G. ?; A- X4 V1 k/ K! J. v! D: Q1 R3 O8 G/ M
    Then, the results are combined:
    $ Q# s& V+ }9 S  G, U! D
    6 Q1 B' Z- J  R" y
    ! T- D1 _: O- v$ r, x5 y. \0 qfactorial(1) = 1 * 1 = 1
    ) j6 ]) ]4 H) @+ H3 p* qfactorial(2) = 2 * 1 = 20 _4 K- T4 z5 C3 s3 u
    factorial(3) = 3 * 2 = 6) a/ G) B$ c3 W# r4 W
    factorial(4) = 4 * 6 = 24+ h( J, m0 x0 i7 X
    factorial(5) = 5 * 24 = 120
    0 S: B7 A! J4 f! o
    & O' Z9 F) {: m) O# L% dAdvantages of Recursion" W2 ~: O, `( q" ^+ e6 X
    4 n8 t- g$ t9 ^3 L! Y0 z+ e6 O& Q
        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).
    / p) R) S& {- Y/ J0 l. o" o- J& u2 N
        Readability: Recursive code can be more readable and concise compared to iterative solutions.3 X, Y" h; s0 B( p5 _

    0 G+ i+ i: B# w, v0 gDisadvantages of Recursion
    0 N5 W; g. C, B$ i9 w! L) U+ v2 P9 E6 a, V
        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.
    $ r8 o% e3 M+ c
    $ [6 I& j) H5 {8 @& N    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ! k' [  }6 i( ^/ y; U. y9 P4 V
    5 ]$ ?2 e' n* ^) {8 [2 Y* yWhen to Use Recursion
    % _4 f1 f5 M+ f0 b# w) f6 L3 P& h8 i1 P0 ~5 L$ Z1 V  V
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 J6 y' p/ R/ {! x- Y

    % b0 G8 |! F' o+ k9 d5 w- u    Problems with a clear base case and recursive case.
    + K0 `( c1 H* i
    1 o/ O3 h# z! {  w* s6 x: QExample: Fibonacci Sequence( U5 M$ P  t% C* d; ?7 y
    7 f% o3 S( W9 F. j
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    4 a! X$ u* T6 k
    7 W% \5 [; P( b9 }! Q; [    Base case: fib(0) = 0, fib(1) = 19 J  Q; L  p4 [! T7 u; U

    / F. I7 u- E2 W$ g0 z' n    Recursive case: fib(n) = fib(n-1) + fib(n-2)1 T; n( r) D, K' {* O, _+ Z
    6 ^! g, v5 ?$ b, z: O
    python
    ' g, B: t/ V7 `* E* @1 D
    - n$ w: p$ h  v' S8 h: R* r4 C5 _* R& j( E. N7 E
    def fibonacci(n):  S3 |; T* [5 i3 i
        # Base cases
    . t3 [; S0 ~1 [9 \+ v3 V% R4 P8 n    if n == 0:
    , e: B" A+ J, j% g8 z        return 0
    * G: F  V! h" Z) d+ Z    elif n == 1:$ X/ b8 m4 W1 L
            return 1
    3 o2 T' a( r: x6 r+ ?4 o( f    # Recursive case
    " h! O( r9 c& Z% \( V/ u6 B0 b    else:# U0 c! Z1 {! y. M8 O: L) e
            return fibonacci(n - 1) + fibonacci(n - 2)  C* B0 |1 K' V' S% ~

    9 i7 j+ _3 l+ Q" n- y+ }; Q5 P( l# Example usage/ f# g  u0 q4 _1 h+ |# U: K2 Z
    print(fibonacci(6))  # Output: 8( D9 f* P# w+ c% b2 B; K( \: N
    % Q$ Y& r  k7 V) V5 J- k! Y
    Tail Recursion
    ) S) ]1 V: {- H/ j. a+ p
    . h, B+ c$ f; r$ s  m, qTail 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).
    $ g! Z2 [$ F5 p% X9 ]+ g$ g& z; h
    % `  B+ j/ R2 P, jIn 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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