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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 ' n- F1 w* k3 ^' [
    ; ?& h, m: F8 ^" g
    解释的不错! S/ Z4 ^8 e3 L$ V$ f
    + @  @) B! v5 S
    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。
    / Z' A9 G. j6 z' m0 s! v1 G" I, H* ]- ^
    关键要素; ?" }# V) {0 B( B+ L
    1. **基线条件(Base Case)**
    - Y. ^2 X# G2 {! w5 i% y   - 递归终止的条件,防止无限循环
    ; ^( ?3 R! q  y- U   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 12 A  L* d5 v! D* u% y
    5 D# ?2 h* X6 `+ h& J
    2. **递归条件(Recursive Case)**" O6 o# U, l; v- V" C
       - 将原问题分解为更小的子问题9 y  ~) F4 V! Q$ e  s) W& I
       - 例如:n! = n × (n-1)!
    5 f2 r; t  V5 h! m: O2 ]# h" x; G/ W- y) Y% g7 D
    经典示例:计算阶乘/ L6 S4 R; k; S. D) Z6 J4 O9 V
    python- Y3 B7 k! a8 p0 O4 c4 |' S" t
    def factorial(n):
    ; ?3 t# o  R+ }0 {    if n == 0:        # 基线条件
    % M% h/ _& p. ?        return 1
    6 w: ^$ P$ U/ K6 \6 W) A    else:             # 递归条件
    * a" F% q8 ?: Q$ d( b% D' t. |        return n * factorial(n-1)6 I9 m* T2 v; E' ?3 D$ d  I7 |
    执行过程(以计算 3! 为例):
    " y" f& o- W- O. \. Ffactorial(3)
    ) e/ S9 w2 M3 J( F, k/ o  G3 * factorial(2)
    , l7 A! J9 R  N. A+ ~9 r3 * (2 * factorial(1))1 z6 I4 A  s5 v
    3 * (2 * (1 * factorial(0)))0 Y/ L: l4 V. m0 s
    3 * (2 * (1 * 1)) = 6
    % s7 y+ d. v$ \! q. I0 d# \4 r6 j& O
    递归思维要点
    " k  z2 n9 Q. I( K7 R/ s, e, _1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 s/ q, Y. }( t, Z( ~4 S9 ^
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ! r6 Y- K8 c% [3. **递推过程**:不断向下分解问题(递)
    ; Q" A9 u* G. X8 J, v& Y% n4. **回溯过程**:组合子问题结果返回(归)! C# u9 b& t  _. t

    3 @3 B9 I0 X. ` 注意事项
    " U5 P5 X* n; m. A& t3 Z必须要有终止条件
    1 a' I2 g* v; N6 R递归深度过大可能导致栈溢出(Python默认递归深度约1000层)+ h3 X' z" N' k: f, c) [/ I
    某些问题用递归更直观(如树遍历),但效率可能不如迭代  x8 i& r3 }; j2 w7 L; S1 R- |
    尾递归优化可以提升效率(但Python不支持)
    0 s1 _, X; q. c
    ! M  o6 m8 r/ Y3 u/ p 递归 vs 迭代. h* |1 x0 f: I3 q, Q
    |          | 递归                          | 迭代               |; Y, g1 a8 e& ^( W/ H$ r4 m
    |----------|-----------------------------|------------------|
    ! O& D8 h7 j2 t| 实现方式    | 函数自调用                        | 循环结构            |
    5 k+ `/ O. _% C2 [2 _  _| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    3 ^6 }) b! P5 [| 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         |
    ) T3 ^: I/ f$ f3 Z| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    ' @. i% y  v& A! C% L
    " x$ p) m: K8 Y 经典递归应用场景$ a! d- [4 r' f0 A6 `9 [/ b: V4 k. z
    1. 文件系统遍历(目录树结构)2 V0 p" B, ^/ D: E
    2. 快速排序/归并排序算法; Z/ x6 c1 S* g" w" {( \& O
    3. 汉诺塔问题' W/ i* n& ^4 m7 b
    4. 二叉树遍历(前序/中序/后序); P+ f- O8 l" g. a8 t$ p0 z& w
    5. 生成所有可能的组合(回溯算法)
    ( V; H9 M+ N, F3 Y( w) f, z
    / G5 z3 P5 o: z7 X# ~* c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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    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,: w  ?! h) P- E  ?/ c
    我推理机的核心算法应该是二叉树遍历的变种。
    $ \/ s. h  h: g/ c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:
    # E4 ]* q) l8 A- C; eKey Idea of Recursion5 z8 w1 c. K% m. z8 X+ O7 v
    2 E5 t0 m5 @( I! W  ~5 ]
    A recursive function solves a problem by:6 e* W( L9 i! P6 H* Y# Y9 o
    & V3 q. W8 `, E4 |
        Breaking the problem into smaller instances of the same problem.. C" K0 P/ Z9 o2 p

    8 a8 a( \& b3 I" D6 E1 ]    Solving the smallest instance directly (base case).
    % c5 y" S( S  ^" C, P
    & D6 Q' L9 \, W2 B( A, ~( C% h- B    Combining the results of smaller instances to solve the larger problem.5 R" ?: B% R& Z; G8 M1 ^' C
    , x( c+ u3 o  Y1 {- |+ Q5 }  }, \
    Components of a Recursive Function
    4 F' j8 g/ D4 Y" v( S" B8 b' F$ R2 y. T6 P& u/ \( Z0 B
        Base Case:9 T$ L/ B) A2 o; l+ i' D
    8 n: T6 |% h5 S+ P6 v2 E0 o
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    + M' D' U/ T. a/ X8 i5 P: t5 O6 c' L+ a" l0 j/ d7 l# [
            It acts as the stopping condition to prevent infinite recursion.' D, L, ^0 N1 }; B; C
    & i% A9 ?0 x2 p4 p% G' I8 U$ p
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    ( L; a2 B% |  c- ]# x( b( U+ C9 {, J
        Recursive Case:
    0 r% C+ k0 _7 b* U  `- a$ L# \/ r! n- R
            This is where the function calls itself with a smaller or simpler version of the problem.
    # V9 \  `8 e5 R% G
    " O# j; F' J: d. l9 ~5 y4 A        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).. ~0 a8 S# G, V; I  `0 X5 U

    4 M& I1 [" R" \+ H8 hExample: Factorial Calculation
    2 q3 ?2 n1 _- q* C" b- i* x5 R1 i7 N. c- K1 C
    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:' Z! V- v/ g, c8 T9 g
    . E" w# M2 s" X/ c1 a% [/ v* R
        Base case: 0! = 1+ r' Q3 t8 w& W& u, _. [
    6 q  i7 ~1 T- Z% j( S% o1 V
        Recursive case: n! = n * (n-1)!+ Y# y# C; i  ]# n. a. m  r
    ( I" R# \3 K9 W6 J$ I0 t
    Here’s how it looks in code (Python):0 N, u/ k, ^1 f* z
    python
    ; T+ Z; n! \4 q* \. Y
    9 A8 T( S( j" `* X( C) x: A1 [9 W' S& U& H# `/ K$ g3 H: V
    def factorial(n):% _2 m1 j( y: S1 ~& Q7 Y, H
        # Base case
    4 r5 s6 J: a; D+ }+ }7 B3 c    if n == 0:
    / z, l) s: j( Q: t+ m7 V) l% ]        return 1
    # }1 j! a% q2 M* u6 Q1 r    # Recursive case
    & i& K3 F7 N7 J    else:) Q  \( L) h  n+ R2 B: S9 }
            return n * factorial(n - 1)
    ' a' I  U' e* K1 Q& g. O, @( y1 [- a0 u
    # Example usage% s. n+ L5 a$ l
    print(factorial(5))  # Output: 120
    # g" G# C. p) I- R8 C4 M
    . Y" F+ M, k7 x4 n" CHow Recursion Works7 G! n( c/ a- M. P' ?
    1 j3 Z4 [) e1 M% g, d
        The function keeps calling itself with smaller inputs until it reaches the base case.
    . g' ~5 D% Z2 u2 E# |1 {
    / D5 P6 R  l% `  M    Once the base case is reached, the function starts returning values back up the call stack., A: [8 t4 G7 y* K* S' e3 M' }# A: Z
      ^0 i& |& `* b0 y) a) S' o
        These returned values are combined to produce the final result.+ }3 M% N0 V, D+ G* F
    0 y8 X, z2 Z8 K' t7 y9 F) B
    For factorial(5):
    4 s+ X& {8 q4 I& H8 N! u2 \
    ; w" v0 z% z4 G, Q2 F' g7 S& ]* p# |% t; p) K
    factorial(5) = 5 * factorial(4)
    8 z" f# \2 `$ Q1 Y# `factorial(4) = 4 * factorial(3)9 Q* ]" |, D' |/ H) m5 Z# X
    factorial(3) = 3 * factorial(2)6 A; k- b. ^9 c8 N# Q
    factorial(2) = 2 * factorial(1)) P3 P8 e  r& ^2 O) x# f
    factorial(1) = 1 * factorial(0)
    3 I  |9 K3 i$ t1 x7 tfactorial(0) = 1  # Base case4 a% ]7 ?8 j- V( o9 d3 a* q

    2 a! o( X) {/ ^2 H( O) RThen, the results are combined:
    2 G( H5 e/ P. X0 D' V  \" b9 B7 t* c4 c/ w5 }9 e1 K4 I

    7 I. O  Y! Z. o* jfactorial(1) = 1 * 1 = 16 y: l$ n- l& {. m& B
    factorial(2) = 2 * 1 = 2
    9 j# d' C7 A8 ^4 s7 V" Tfactorial(3) = 3 * 2 = 6# }6 h/ Q/ [- c' p% m% t
    factorial(4) = 4 * 6 = 247 s0 v2 {7 |+ P2 c5 }+ c
    factorial(5) = 5 * 24 = 120
      j7 p* W6 t+ n" c
    % L8 W2 x) w2 I# G0 DAdvantages of Recursion
    9 J9 \( t; Y3 o9 e! ]$ t: s5 G5 e, n( }( k) J" u7 J. v) O: 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).
    ! A6 E" h6 O6 G' {
    6 h2 e5 j, u5 }    Readability: Recursive code can be more readable and concise compared to iterative solutions.) T3 g, X; D% }3 n5 `

    6 h, y- L9 z  d5 |4 v. ~Disadvantages of Recursion& }7 F; q0 a) W* H- ~8 c

    $ r/ F; L% G6 V! Q, \3 X    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.
    % k* a; u8 h5 e! f, t' G1 U( M! A; L0 ?' F$ F% j; X  {
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    ; g: W9 U" k0 Z+ x4 H2 h1 ?3 ?
    : G1 G& P9 d, p+ \When to Use Recursion
    , }" i9 T* c, `2 i
    3 [$ I7 b) L, h8 U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    . f; v0 h' O/ X$ c5 l/ ^6 d( F% a8 |- T
        Problems with a clear base case and recursive case.. C' J* w0 b, m

    . M) e% D2 k9 ~0 @/ n( ?1 u+ uExample: Fibonacci Sequence5 L8 S; H& K  D# ]0 V

    & q# a/ ^. e- m, S: d9 D# XThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    , [. F7 Y3 T7 u; q2 A% b+ r7 v  _* A5 f  T  b) A0 ^& i6 d' Y
        Base case: fib(0) = 0, fib(1) = 1
    , N- K! z3 N  B& ^
    9 L$ W/ ^. K3 X4 K" I( u8 B# H3 f  Q    Recursive case: fib(n) = fib(n-1) + fib(n-2)8 g- E0 y4 d7 P! N( r- a
    ; B7 m: a+ x: W1 }
    python& d0 S# S2 ]0 Q
    $ `" G( `7 D9 J

    # ?4 ^- A, g: u1 Ydef fibonacci(n):& z  H* A+ \0 H4 h8 @
        # Base cases, i9 X7 N, s0 [/ A+ i. R6 q" ?; y
        if n == 0:% V  H- W: q4 M/ \
            return 0
    : o7 b6 t; W, [1 [+ t# w! f2 `+ n    elif n == 1:6 \+ `5 q! q0 ?; S/ a
            return 10 M. `5 \2 w; o- v4 M
        # Recursive case
    / y* @+ T" }& |- Y- t    else:
    6 w9 L* q8 v, n) d; I9 V! k/ y% N4 f! Q        return fibonacci(n - 1) + fibonacci(n - 2)
    + u8 M! ~- c. a2 X/ h. F2 Z$ u# \  z) _! \
    # Example usage7 G1 x4 _) W1 B. l. X
    print(fibonacci(6))  # Output: 85 e2 Q9 ^2 s& D% Q1 W4 F' x

    4 R/ Q6 ?$ I6 H. \* ^5 U- rTail Recursion5 |6 c4 ~8 A" ^( e1 d5 a; |

    ' ]& @, f$ ]$ V. w, y1 e, i. wTail 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)., {* y4 `  ~* R; G4 U. N: t% u( z0 e

    " g* v% P4 |! o4 `4 e9 r) ^4 AIn 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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