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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
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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 9 n: z8 B0 Z- h+ j( P3 H

    8 L( ]  R$ x0 d, M3 D% S解释的不错+ k% q4 \+ E1 t, Q  }- u

    8 O! S/ ]% ^  g$ v# P7 @; h! X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。
    1 ?% D# A1 u# u" V6 {, Q$ t3 W7 w1 N* p# i4 |" c
    关键要素
    ) ]$ i4 W. @. o) [3 [1. **基线条件(Base Case)**
    ! p8 a6 g, g9 j9 u) s( J   - 递归终止的条件,防止无限循环- h- N& \# W  j" a/ x
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1' c% q7 B' R8 y1 O# D
    * n- o, u- l% v( ?" ?& R5 f7 G
    2. **递归条件(Recursive Case)**
    ) ~: k% R" t9 a# ^9 _; n   - 将原问题分解为更小的子问题
    + u5 v' v5 M/ r. f( C$ Z- Y4 f   - 例如:n! = n × (n-1)!0 C) E0 z0 v5 e1 q# {0 k  p

    1 x* P# |8 D6 N0 ^. D 经典示例:计算阶乘
    $ S1 q7 Z! K3 s+ q5 E. M9 Lpython
    ! D/ Q- U; B* l0 z5 ~: b9 n2 ^def factorial(n):
    2 B" h0 P: T' o0 u9 X& A# X3 h    if n == 0:        # 基线条件
    3 [* _( v* h; i5 C        return 1" n$ U8 S* |: ~( P; ^: N: o
        else:             # 递归条件
    ) J  ]0 p2 }( a1 I! @        return n * factorial(n-1)
    3 c! t: v6 W" L/ h8 C执行过程(以计算 3! 为例):' [0 |  l* B: T8 j* y0 Y8 g7 s
    factorial(3)
    ) b& h) _. G$ o2 G% E. r7 R3 * factorial(2)
    * k  A* `' ?1 r5 V" I' \3 * (2 * factorial(1))
    " N0 o, A/ V: n. S) i# Z/ f* a" I! G3 * (2 * (1 * factorial(0)))- r. e8 L9 w4 l: h: D, j2 c
    3 * (2 * (1 * 1)) = 6
    ) y" y$ m6 i3 L5 I' ^* @  H' P7 q( Y0 D
    递归思维要点1 r! m5 H' N4 E0 A) P! J
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    : E- ?8 i9 B6 ^3 ]3 h2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    & q* g" B0 ?, q# \9 E4 ~2 G3. **递推过程**:不断向下分解问题(递)
    * O5 n3 K9 a4 h4 M4. **回溯过程**:组合子问题结果返回(归)
    ! U" b% ]4 V$ H3 I% z- n0 n, X9 H; g2 n5 S9 z9 Y# D' |7 f9 b
    注意事项
    7 v  f- G4 z4 h$ t5 q必须要有终止条件+ j% }" X# S2 e$ }( N, t
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层), P2 p: }" S$ `3 U9 i  a. O
    某些问题用递归更直观(如树遍历),但效率可能不如迭代' o. [- @5 I1 Q  u
    尾递归优化可以提升效率(但Python不支持)$ d1 u9 Q* e  j; q5 e9 z) y

    - o, I: n/ `7 o4 U5 a 递归 vs 迭代8 x) y/ Z1 Y2 `
    |          | 递归                          | 迭代               |
      ]' T% [# n3 `+ A& ?8 l|----------|-----------------------------|------------------|
    - R; H$ C( n/ ^- h| 实现方式    | 函数自调用                        | 循环结构            |2 h% H, z5 v: Y4 m! G
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |, ]9 _8 G6 Q6 E7 k5 l
    | 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         |
    * d6 G7 U+ t8 U8 {( }# I# h8 G" C| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    % K) a/ b8 W  U% j1 Z
    2 `  T" P5 N9 v4 e1 ` 经典递归应用场景
    4 p7 x, y/ @/ n1. 文件系统遍历(目录树结构): d9 j/ v# E. C. t. y- I
    2. 快速排序/归并排序算法
    6 c5 d" W8 G+ b9 N( W+ D3. 汉诺塔问题
    : w; g; a( K; g+ l4. 二叉树遍历(前序/中序/后序)
    ! ]. A2 t& I0 s" g2 b: q9 _5. 生成所有可能的组合(回溯算法)
    1 U5 Z. r6 i) s  s" r6 x0 @$ _3 b; f9 y. {6 ^
    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
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    前天 01:20
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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,/ E7 |! c7 ^5 v5 ]2 o3 ?
    我推理机的核心算法应该是二叉树遍历的变种。! d- P6 e1 t4 |2 b7 y" y1 e5 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:
    ; Z- `7 _; e5 ?* I: pKey Idea of Recursion' v; z, f' N7 Z, {6 C; k

      m, T- O! y. @+ y1 WA recursive function solves a problem by:: u& O: A/ @( g; k  V! U/ m

    ; |" E/ ?4 G8 q" t! b3 t( J    Breaking the problem into smaller instances of the same problem.
    ; @, n1 L# w3 U+ X; s4 E! J- H. c" {9 C  A- M8 I
        Solving the smallest instance directly (base case).
      S& _. [$ p3 _8 P, E% o; f0 |0 Z1 F; ]* t+ U# c- V
        Combining the results of smaller instances to solve the larger problem.
    . |+ x( T2 O' ?8 \  Y/ O* H5 z( H, n2 r
    Components of a Recursive Function
    : ^4 n" g3 c$ ~( p9 ?8 c. @3 p8 ~
    5 ?3 |9 s+ v2 e, C. Q1 u: M, C    Base Case:4 R2 p! ?, D( ]% E

    * A( K8 i3 b$ K& t! u& G- ]. d# G3 E        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.& [7 N2 L2 i& r/ \# h* p  D
    & g) w) I5 H* ]! N. y1 v" O  B4 m
            It acts as the stopping condition to prevent infinite recursion.5 c0 ?6 C$ L" g8 g' ?; M6 b" |' d- o

    & s8 D8 T2 h5 n( d# S        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 w. _3 P0 @# u+ J9 }1 v) {
    5 b" v& \& j3 J$ a3 E" z& D3 b$ ^3 H
        Recursive Case:- w) B  A5 |& T2 j

    ( z6 U' [5 g. e' H* _! h        This is where the function calls itself with a smaller or simpler version of the problem.
    " y3 }6 Z. ~4 p5 ^2 U: A/ n% E1 k1 F0 B4 _1 A) ^" w  s! o  e0 e& G
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 E3 U; _: C9 W( w

    7 \$ ~- D& G+ X' P+ o" ?1 DExample: Factorial Calculation
    ' U- \+ n' W' p
    ! M) Q) Q" O2 `7 b& b  ?! s3 r9 u; ?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:' y* W0 r8 ^  L% T
    . c9 d; Q0 F: ~4 K* ?
        Base case: 0! = 1" z% J9 l* q) t

    6 ~& l8 V1 y& C" A, p! J+ Q! {4 r    Recursive case: n! = n * (n-1)!) M0 Z! l- p7 r+ r$ F) @( Q' t

    ! ^  z: A. S) G, N3 b' ~' X4 KHere’s how it looks in code (Python):( ?0 j, w6 g/ q% M
    python
    & F  W' H" a2 C" O
    1 K9 W! u9 W% x  T2 u- Z6 M4 w* O) m7 f+ j6 ~* S
    def factorial(n):4 }2 Z. G) n% ]
        # Base case0 N& y. x4 i, [3 l
        if n == 0:
    5 F* Q! ~$ A- Z+ S* }        return 1
    & h$ _; c- k- ^" i1 b) u$ r, F    # Recursive case
    ; G* f4 j+ j: W  C7 t2 w; Q    else:; u2 z5 b) P4 l+ ]  X6 X' u
            return n * factorial(n - 1)1 {7 e9 R2 n5 l* a

    - r- ~3 ^$ f) G8 f, k- b+ |# Example usage
    , ?# t; d8 w3 X8 `0 C5 T$ Mprint(factorial(5))  # Output: 120
    8 Z( `) `$ a9 a1 i8 Q* F+ T4 N; s: x
    How Recursion Works
    3 t0 H$ v3 |( R! r5 m  Y8 C5 T6 Q" W
        The function keeps calling itself with smaller inputs until it reaches the base case.
    4 g' h( I  T/ I1 `# X! O6 `7 k* w4 p3 r, ?9 h  n9 \
        Once the base case is reached, the function starts returning values back up the call stack.# q2 u6 ^4 q/ b. A3 f+ o: m& \
    " V4 b3 \6 w, R3 ?+ n9 W/ t
        These returned values are combined to produce the final result.# @2 M; L* C: }7 g2 E
    5 J6 b9 C5 @  g2 D9 |- y
    For factorial(5):
    9 ?9 o/ i: w7 b% t8 `. w
    % n/ k9 h8 B% U- s+ E0 i1 x6 t1 c+ V+ v: M, B$ t, t
    factorial(5) = 5 * factorial(4)3 Y% {0 L3 D4 T/ b  ^) g! ?7 [
    factorial(4) = 4 * factorial(3)& y1 g1 L& @* \9 L+ ?' L7 l: N7 G
    factorial(3) = 3 * factorial(2)
    3 i  t9 R& X; M) L: ?factorial(2) = 2 * factorial(1)4 e+ ~' z: b' _8 u% K
    factorial(1) = 1 * factorial(0)3 S% x* _  a* p$ Z: @
    factorial(0) = 1  # Base case
    ' u+ w% D1 p' w
    * X3 _9 l+ o& D: dThen, the results are combined:8 t8 A) j3 h: A
    - E4 G" I" O2 L' r

    $ ~; K  s" s) i" j: Rfactorial(1) = 1 * 1 = 1& t; s+ Z7 d7 q
    factorial(2) = 2 * 1 = 26 s' D5 C0 O7 l
    factorial(3) = 3 * 2 = 6
    3 f" V3 y) H/ r' v/ r1 E  Ifactorial(4) = 4 * 6 = 24
    - l+ ^0 O; p" K0 z1 A; [  r+ Sfactorial(5) = 5 * 24 = 120" K8 u1 T* {8 O( g$ n8 r

    5 y. R8 O1 r6 b& eAdvantages of Recursion
    - j' f( }) K3 O2 q4 N/ d0 h6 }1 S3 A( Q2 J/ c* p
        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).4 m# W1 e% P6 X# d* {5 f* b( i
    $ F* b: x% H* B$ f
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
      [4 `- _8 O; e
    $ u$ `. p2 G/ M0 ]6 XDisadvantages of Recursion
    ( S' g) c! z8 z' y, w3 G- ^
    " a; u: I  l# M) W# }* ~    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.
    ! Z. P' o  {3 D6 O* o3 M7 ?. \' R, B, Q, e  X7 F& V
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    & n- a' V5 ?4 d+ e' G: G, ], y+ T# G2 _* N  u0 U7 x
    When to Use Recursion
    $ F- e6 a! V! v8 k# l* o' j; s; c
    3 }: Q& J6 Q* U5 l) n% U    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).! I1 H  z3 @& [1 C1 R

    ( t: d3 B5 s1 D5 B  k    Problems with a clear base case and recursive case.
    ; `& z8 I. U& F8 a1 |$ W, \
    9 {" Y3 \* l, L, X' O# `+ NExample: Fibonacci Sequence! R/ e9 W! p* ], N. j
    2 Y+ G' [3 H* [& e% e( ~" |
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    - ^+ V8 D8 X* W0 W% \5 ?+ y+ [5 y' l" G
        Base case: fib(0) = 0, fib(1) = 1
    $ D' }8 U/ g; R7 I5 f% O
    + c! q  |7 ]4 t& F$ K. d    Recursive case: fib(n) = fib(n-1) + fib(n-2)6 B6 s, p% H; w. x8 h: Q6 A

    $ w4 T& k  E# E3 A, }- Wpython
    1 ]( }7 B7 m# q7 O& w8 E0 Y* p. j9 A  B

    6 E" [# N* U8 n/ q2 L# M& vdef fibonacci(n):
    - U8 V2 Z, X% q; Q9 F. L    # Base cases
    6 I( z0 d/ t) K  |( y# l) k2 }- m    if n == 0:
    4 V1 W$ x9 \8 b$ j9 c; _& I        return 0! [2 @# b; w  H
        elif n == 1:
    5 i* T; g1 z( \, M/ d        return 1
    6 W' Z% a7 L3 H4 }2 ?    # Recursive case" V7 \0 M. l6 E/ I# @4 ?
        else:
    8 D7 t% x0 n  F$ ^. Y: ^  `        return fibonacci(n - 1) + fibonacci(n - 2)- ^( f' I) ~( k0 \+ W% r

    3 i4 O) B, I: i. r# Example usage
    " S: o# ?% z0 \4 c' f: xprint(fibonacci(6))  # Output: 8# [/ _$ O0 P/ W# G+ {  N

    ; u; H2 C7 s9 b+ ~: i. y, ^/ ?Tail Recursion$ V( W# r) u' y: r2 v% u5 k

    ' D8 s1 g1 ^* m4 N% z0 YTail 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).# n% Y, J1 Y, {9 F; X
    ! l: w3 S, v. x' w2 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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