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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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  • TA的每日心情
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    2025-9-8 05:08
  • 签到天数: 3 天

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    楼主
     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 3 J" H; S' x8 |' Z) O& P1 J( W7 {, _
    # ?' O' X. A9 {
    解释的不错. B6 V. I; j3 p/ N, z0 \9 ?

    : I; i! e) S4 `; R( p% B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。* a- E* x$ a- N, U6 s* s, G

    $ S3 ?  P$ e7 \. g; j& x 关键要素
    3 `5 T: q# f3 T# F9 k1. **基线条件(Base Case)**
    ' w5 g& l( q! E5 {: G- `* P, c   - 递归终止的条件,防止无限循环8 F/ ^2 ~3 _4 D7 d$ h5 F) `
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    9 j' E3 n( P/ L8 a, P. d( v8 N/ ?: K( }8 M. ~2 |
    2. **递归条件(Recursive Case)**
    ' {! @( q7 S5 o, o7 ?   - 将原问题分解为更小的子问题7 J- u1 U" K8 B. F& W& d6 M
       - 例如:n! = n × (n-1)!
    3 {0 c* X2 X7 P5 @( U5 o$ d3 g% A* K) N& H& [' U2 |0 j7 K
    经典示例:计算阶乘! ~# `) U2 h" Q' n0 W
    python, R, J; e8 t, l+ _  O$ Y4 [
    def factorial(n):. o  l$ x7 t1 y% H( }
        if n == 0:        # 基线条件$ @0 p3 e4 d' D
            return 1) Y$ [2 p0 ~2 {% |1 _' ]8 w0 g& I
        else:             # 递归条件
    ; W) {# e2 @+ \# s3 O        return n * factorial(n-1)7 p# ?3 P. n; c, w, Z
    执行过程(以计算 3! 为例):5 a/ U+ }4 p, p, g+ Z1 n
    factorial(3)# V  ?% b% l. j9 t7 _
    3 * factorial(2)- n7 s& e7 [' Y; g8 {
    3 * (2 * factorial(1))3 b) W) y# v8 \1 B& Z( f7 x. t8 Y2 W
    3 * (2 * (1 * factorial(0)))
    ! F+ D) Q: M+ r5 F1 m7 b3 * (2 * (1 * 1)) = 6
    4 p7 {2 ?! X6 s. x0 m6 r: E8 V  ^( Y3 d, K- H
    递归思维要点
    4 F* e5 y! Z9 f7 J% D1 h6 ^1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 x! I; @) @, _5 X& H0 C
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 p$ s- i- U* Z6 ~9 Y3 ~4 Q! R
    3. **递推过程**:不断向下分解问题(递)2 U& t# Y9 `9 E+ E) H
    4. **回溯过程**:组合子问题结果返回(归)- a: P, h% t# r3 O; q6 {+ h0 D( P& |

    # @' ?2 o0 Y! d 注意事项" ^- J) C6 ~* p' f
    必须要有终止条件4 i+ d9 H0 M# f
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    ' u( f2 A& ^" K: q2 D( I某些问题用递归更直观(如树遍历),但效率可能不如迭代
    ) i9 H  G$ s! a* r% i( R, i尾递归优化可以提升效率(但Python不支持)
    - W2 x! c' g! |
    # @" I$ \# |/ q' X 递归 vs 迭代8 i! v/ ]$ G2 n* D$ e: l
    |          | 递归                          | 迭代               |
    " p# w  s2 |- _# v$ A- _! l% F6 G|----------|-----------------------------|------------------|+ q: H+ E6 r) }. z; v" D- l
    | 实现方式    | 函数自调用                        | 循环结构            |" J8 E; k$ l, B' \8 C
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# Y  y2 B  a9 R" @5 M
    | 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         |1 l0 @* Q# J2 o
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    % m( q* r0 k  \9 m
    & }5 e6 L! J1 o8 {; u6 R 经典递归应用场景
    6 V1 [" F* t! y9 `* f1. 文件系统遍历(目录树结构)  f. Z6 l' U: v5 A% D- m) s4 T
    2. 快速排序/归并排序算法
    % I5 |4 E5 k, v7 L* H- w3. 汉诺塔问题5 W+ X! F: v2 z/ m; s9 G
    4. 二叉树遍历(前序/中序/后序)' c/ z+ v/ ]: ?. @6 j3 K
    5. 生成所有可能的组合(回溯算法)
    4 I! S! d/ D( q) \7 j
    4 p: h) O, V+ B& {2 T5 X) {; j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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    前天 07:11
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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,. L2 f$ N% m2 V) }9 {  _! l1 \
    我推理机的核心算法应该是二叉树遍历的变种。
    & U8 W. m; G8 Q- d$ c: j1 F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:
    2 |5 r$ ^( W- O1 Y# ]$ c- A) fKey Idea of Recursion4 g) v- [/ _9 {% q
    ' @% B1 @, Z: k3 G& ]  h$ p3 P
    A recursive function solves a problem by:
    & e) S8 w2 v, Z8 p, M$ z3 E# W- i$ r* ^" p8 E! L
        Breaking the problem into smaller instances of the same problem.6 U6 n; M6 T3 n0 r% h

    % O: O: A, Y+ o: p    Solving the smallest instance directly (base case).
    ; ]. A! ?6 [$ F, D+ K5 }9 Q; N3 \- n. P2 D8 ?3 Q% h
        Combining the results of smaller instances to solve the larger problem.
    ! B$ ^; T$ ^  _
    0 j7 d! T/ B$ n6 UComponents of a Recursive Function4 l1 k  f; j, L2 w$ E: M" Q2 O
    8 M1 y7 R- t2 q" ?) |
        Base Case:
    ) E- Y9 b3 \3 p$ f5 T( X) V! _: c  i* N9 o; d+ b# J& m: `- W' s
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.6 ^$ h2 }) @2 |9 q) K" v

    ( E( t) {0 b  t/ ~, A3 _) I        It acts as the stopping condition to prevent infinite recursion.7 v1 ]/ J# E- I% k, Y. E
    % n1 Q+ f: N! e
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    8 i+ k4 L. E$ s! i5 T1 T+ M3 t. r# s1 D  A6 F5 }+ Q7 O
        Recursive Case:" D2 E0 l* t+ @$ h* q0 e& B
    * _+ C4 A4 R0 N+ w* v
            This is where the function calls itself with a smaller or simpler version of the problem.
    $ s3 |! G7 |1 E. h7 B3 i7 X& x& C0 n9 x4 P! |$ O. A+ R8 s2 Y  K) y
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).7 X! d- r' W  z, v# ^6 K# G4 o+ G# W. n
    , {6 }( b' _+ f$ Y- U2 n/ n$ o9 W, k  P0 [
    Example: Factorial Calculation0 a* R" M- x2 [  u
    : O4 j1 q" Y$ _6 L0 _
    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:
    7 {7 S: ~0 l9 n# _- z; ]. s* t& P* p
        Base case: 0! = 1
    ' h1 r3 [5 v) ~$ z
    5 u7 [/ Y$ Q9 e+ P6 k4 ]# l% j5 n# s    Recursive case: n! = n * (n-1)!
    $ g$ D0 t9 ]8 }4 _/ \, Q
    6 i: R" N8 O, oHere’s how it looks in code (Python):" M5 t2 {6 z* z& ?8 z9 q( f
    python
    ) K/ `# w$ V* ^+ M: e% @4 q* \$ ~# Z* [* B4 S) D
    - `! w, [( q  y; {9 R# T2 n% }
    def factorial(n):& s5 k; n3 }  }& N
        # Base case% \1 V3 n1 H* J$ B  b& W' c
        if n == 0:
    ( y9 V( U5 c% K! |& W5 _% E: Y        return 1
    & x/ h6 u1 j1 e; Y" j    # Recursive case
    4 e6 g( e" ?6 n5 L' o- [7 x7 b9 o. w: m' g    else:
    . V/ A" ]  u# X        return n * factorial(n - 1)
    ' k! n( e) B& i8 @5 c
    % Q1 w4 j% B. q1 C5 N& y6 w# Example usage( Y1 f0 s+ U% {. n: ~
    print(factorial(5))  # Output: 120! w  j9 M! q/ S( e3 \

    - i- v6 S: d/ K+ yHow Recursion Works: n, B. A6 R; |7 L1 B) X

    ! x7 d8 l( p# z6 w3 y0 {5 Q$ `    The function keeps calling itself with smaller inputs until it reaches the base case.
    ; s) m2 b7 B- t; C- o$ \+ S3 _; G9 b) j; I
        Once the base case is reached, the function starts returning values back up the call stack.6 i1 w5 B# Q: Y  J$ o7 q7 W+ B

    1 l1 q4 A3 E2 l' s' Z' m7 y    These returned values are combined to produce the final result.  v! ~( b; D3 c$ i. C5 M; E
    4 |" b: a% E2 \5 p2 v% W( }! w
    For factorial(5):
    2 H! T- j; A$ n! N
    " K# x% T9 c% B! {5 {; k- y
    1 r3 K( o0 [3 Ffactorial(5) = 5 * factorial(4)# P( S  C+ m6 s/ o- i
    factorial(4) = 4 * factorial(3)3 e" h/ h( C" s; i9 f) T' p2 w, P
    factorial(3) = 3 * factorial(2)
    ; |/ {: ~) c7 ]3 h5 |factorial(2) = 2 * factorial(1)
    2 a8 i6 _% I) W8 U  g& b# Lfactorial(1) = 1 * factorial(0)& g! @) B3 M5 s# j* n4 |
    factorial(0) = 1  # Base case
    ) h0 T3 b" |5 f- r% ?* c1 ]/ ^+ Z# k$ |1 A! `
    Then, the results are combined:
    6 w, o! K5 a+ x5 P  m; v' K( i2 I- H/ V9 R. ~

    & K. k: j. x0 A2 Q* g$ C. q! \3 Zfactorial(1) = 1 * 1 = 1) L* S% ?9 v$ R  S# I
    factorial(2) = 2 * 1 = 2# e: G6 {  f  }' A9 `
    factorial(3) = 3 * 2 = 6
    # A' o& _& _: H" D' tfactorial(4) = 4 * 6 = 24
    0 y8 f9 h+ o& p5 U, C( F3 Bfactorial(5) = 5 * 24 = 120) V3 ~" Z8 U) y) V- d
    . I, B5 I% n9 J" C+ ?
    Advantages of Recursion$ T' }8 y" v- [: b: w! O$ O  K
    . T" t5 B' A# u  z4 h: p. K  e1 i
        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).! ?9 |6 `/ y$ r6 z) o! y$ A( ^1 j5 u

    0 ^3 T9 J3 g! @    Readability: Recursive code can be more readable and concise compared to iterative solutions.' y2 u( V3 I' B1 v! l

    0 J) ?; x" h: n6 H" D/ jDisadvantages of Recursion
    & q# X' l7 e* J  |( g4 G' J5 g+ [
    , E" Z4 s! Q6 `4 i- \. N    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.0 ]; ~4 J2 S) r0 g4 s  b

    & l, `8 f  F0 D    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).- n0 U0 j3 U% ~' ~' @# M

      E2 E% B- l" _3 y1 cWhen to Use Recursion2 D6 W6 l( ^8 {2 j
    9 r8 E" e! {: O* o5 u% a. {
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ) N+ |2 r5 x: Y
    * V3 {+ [$ ]7 b7 C    Problems with a clear base case and recursive case.
    % }- E; @2 _: W* L5 k8 S
    9 ^& h7 E  Q+ i0 UExample: Fibonacci Sequence
    ) H, g( Y+ z, j1 R$ B: q& t7 [3 p& }% O" R- a3 p8 j' u3 ~
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 E- N5 Q8 J3 [6 h4 ^

    : r1 I7 g8 ^  D# A3 X    Base case: fib(0) = 0, fib(1) = 1
    ' K- i# b. r; W( r: A: c6 V% b; s" i  J1 L) e* O
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    * ]4 z, I) \+ H5 d& I6 N
    ; v3 L4 m. }* hpython% j, F6 v. V0 _

    . c4 E% N1 T) D3 A
    ) X+ o) N* k5 O: Fdef fibonacci(n):! i7 w+ z3 H; w3 t7 w4 _5 v
        # Base cases
      z9 |5 C; y4 M# f9 H- h9 ?    if n == 0:; Z4 }* ?9 l) g( U0 q
            return 0% B9 i9 }" j4 _( e6 F! m8 }. e, j
        elif n == 1:
    4 J5 A, m9 W4 i        return 1. h8 J& A1 j3 K
        # Recursive case
    : A' V, j' T6 |3 Y( O* S2 F    else:0 k  @$ n4 s9 E( g1 D0 O( _* E( O
            return fibonacci(n - 1) + fibonacci(n - 2)* |1 k/ e2 r7 J% \/ h) q" J+ j
    ) E5 m: ], b8 z- K0 i/ j
    # Example usage8 O# o2 m" k3 N- A
    print(fibonacci(6))  # Output: 80 E8 i  u5 b. l' E4 y# F' @
    1 b+ G/ U& r4 |5 F  X8 j
    Tail Recursion
    , u9 }6 p) G4 {, |
    9 p) }  C2 U% P' f; sTail 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).
    # p" k6 s! c( U5 a8 L* W% Y) d
    % d; a# b( p( s/ _+ K2 WIn 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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