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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 & }' [; |5 W" D- ^+ V5 O; Q- Q8 F, R
    6 j1 Y- k& `, b- o& B
    解释的不错
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    ; A" f7 T* }( m# u( j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。! {8 P; {- \% ^% s; O

    0 ~: ~+ L" w, t9 d+ g. J7 l: ~ 关键要素5 s) ^0 `' r  u4 \
    1. **基线条件(Base Case)**9 E9 v  r$ `  ~5 I2 }* y
       - 递归终止的条件,防止无限循环- U; v' k) P% B8 y& K5 D
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1$ I/ r3 w' b/ t! z; |1 L% l: R# s
    . o' v% h# {* U0 u
    2. **递归条件(Recursive Case)**
    3 `: ?2 A; r. u) N   - 将原问题分解为更小的子问题9 n' W! G& _. R& A  G: I1 M$ `
       - 例如:n! = n × (n-1)!+ W& S+ ]9 q' Y; }# `/ k8 _
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    经典示例:计算阶乘
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    def factorial(n):+ B3 n" ^* ^+ S7 t
        if n == 0:        # 基线条件
    # V' v' d! D- I, e& L- X$ n9 l        return 1
    9 W% K" K1 Z/ n    else:             # 递归条件( Y3 U6 P) O+ X0 _, b- \
            return n * factorial(n-1)
    . C+ A2 M2 ]; Y7 ?. v* ]执行过程(以计算 3! 为例):
    8 q+ @  ?+ y0 w! `0 ~. cfactorial(3)
    3 N9 A; I# z2 h% A# m3 * factorial(2)
    9 f, M- Q1 M) K" R. o' B) Y3 j5 [% r3 * (2 * factorial(1))
    1 Y  |8 A: o' t% R! N* l8 X3 * (2 * (1 * factorial(0)))
    1 \1 Q$ L) F5 g$ L1 {2 s5 S/ a) q3 * (2 * (1 * 1)) = 6
    . F% F2 F9 {1 Z5 u; O7 l# A( w" P  k0 {6 T
    递归思维要点, F3 f* }/ x: r! |6 F* ]& l" ]/ `
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 u9 y8 S# o& Y3 B  P
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间): L- m2 ^2 V) d" N1 u  F- ~
    3. **递推过程**:不断向下分解问题(递)
    , o. c' |- r' e" s/ ^4. **回溯过程**:组合子问题结果返回(归)( ?- l9 E( ]' ^  n  _+ i
    $ @* h7 Y  s2 l# V4 u/ g
    注意事项
    4 Z& Q6 h0 u0 h! _必须要有终止条件- |4 c" Y; X" X/ Q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  }% F; J9 z3 T0 g. u9 B% I3 Z" ^
    某些问题用递归更直观(如树遍历),但效率可能不如迭代$ ]/ a7 ^1 T# g) Q- L9 r/ ]
    尾递归优化可以提升效率(但Python不支持)
    ! P! L, x' U" ~+ M9 b' R
    $ T# V1 m) W! l* B* b/ `4 F 递归 vs 迭代: W! C. n& w7 h" Z: Q
    |          | 递归                          | 迭代               |
    * Q9 W6 ~9 E) }3 ], T- A|----------|-----------------------------|------------------|  K8 o9 k9 c4 i8 s/ U% M! \
    | 实现方式    | 函数自调用                        | 循环结构            |
    " s  L5 Z( \1 e. O! q* s! z) s| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |5 [: X& c1 f" h& `
    | 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         |
    $ R! c5 ^  F) `7 X| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |( x1 I) |! M) }
    * s, S6 a6 ]9 K4 R9 R
    经典递归应用场景" ~, ?. y9 R. m0 u/ r& N6 L$ E
    1. 文件系统遍历(目录树结构)
    4 ]6 b$ L. ?1 e9 q/ G2. 快速排序/归并排序算法
    ! M3 @& k2 K$ M5 |7 I3. 汉诺塔问题
    3 a% c8 m  A4 ]5 ~4. 二叉树遍历(前序/中序/后序)! |5 R3 u5 Q9 Z1 [8 b4 ~
    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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    3 天前
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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,3 V, _5 M6 q* e2 T7 q% X9 M
    我推理机的核心算法应该是二叉树遍历的变种。
    2 L: H0 ], q/ L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:0 ?# Q* q. t8 `) B" f3 Z
    Key Idea of Recursion
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    A recursive function solves a problem by:, ?1 G* ?2 W8 m1 V

    / V: }! V1 x  _0 Z' N% Q4 ~    Breaking the problem into smaller instances of the same problem.( u! B' P7 w: @4 f/ f! d0 Y
    2 a- ?0 @9 q6 b* W5 h  U. }0 h
        Solving the smallest instance directly (base case).
    # T9 z, E9 V% ^% V/ u
    & d: v& H( b3 g+ e+ z! c" ]0 A    Combining the results of smaller instances to solve the larger problem.
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    ( `6 n0 u2 B& |. m2 jComponents of a Recursive Function, E( h  d) L. F; G1 L1 h
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        Base Case:8 B$ t: n$ j8 A; j% k( P
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            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ; d! {% z( K: i7 p$ _7 Q
    & L0 T: O) c( H- B        It acts as the stopping condition to prevent infinite recursion.
    # k9 ~" L! }7 G- L' R7 g
    4 }* n2 E/ r9 z, F: C* f. e1 v4 X        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.1 Z9 E& H8 a* w! o% p- w5 t7 y
    ( Y2 @, w& [% U* M+ A' O6 s7 ]" P
        Recursive Case:
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    / a3 R1 m' G0 @9 i! ]2 R        This is where the function calls itself with a smaller or simpler version of the problem.
    9 G/ K4 i$ r* `1 t* ?2 p  k) p& x$ @0 p$ P
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    " ?. D, M. N3 G; f6 C& g
    1 E" P# S3 y$ C( d$ x9 QExample: Factorial Calculation
    6 e% g& p1 Z; m% b6 b7 [# H$ i; e' [- M1 K3 a% O
    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 w  b* k8 K; J
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        Base case: 0! = 1
    - p9 M9 q! J" `1 s( v! B9 h6 ^7 a: W/ v3 f9 o5 n/ ^
        Recursive case: n! = n * (n-1)!
    3 }! x  ^4 t# n" w3 A. c) C/ m* F; }* L9 ~- J5 ~' E) i: K
    Here’s how it looks in code (Python):0 a% J2 R; \% A! B6 _
    python, H7 p, k3 ^# d* |
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    4 r& ~. F5 N0 ]; q! c4 b/ `
    def factorial(n):
    : J$ H; D6 L7 k* _; q    # Base case; K  V* M8 }6 _) G; U
        if n == 0:
    9 b  ]7 i7 I; E& _) e        return 13 J' {, w$ N$ v8 `
        # Recursive case- n2 }  r* Z9 s+ G" q4 m' M
        else:
    7 V6 R  Q  r% W5 F2 J) H        return n * factorial(n - 1)
    % a( _7 l3 J% D- i8 o$ R! u9 u
    / o" Y9 k3 f8 \! |# I8 p# Example usage
    . h* Y$ o4 P# N0 q: [% bprint(factorial(5))  # Output: 120
    % L1 e1 B- [& f% q) \6 g) z3 b$ [; {6 u  N
    How Recursion Works
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      u: y) v" g% \    The function keeps calling itself with smaller inputs until it reaches the base case.
    ) o" {) M& b. q% Y- P  L; \: K# U1 G
    ! U4 b/ Z* u8 Z6 ]- X3 Y7 f! j    Once the base case is reached, the function starts returning values back up the call stack.
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    0 p" S/ e; _$ y' A) U+ t/ X    These returned values are combined to produce the final result.  L+ G: J+ i  L. w9 Q
      X" C( z$ f! T! q* P3 q. `
    For factorial(5):2 |% C6 ^7 }0 c9 h! z. P/ d

    ( o. p0 L' _3 U. A0 d: R+ ]& j# E9 G* M+ W
    factorial(5) = 5 * factorial(4): F1 y; |0 V" C7 Q
    factorial(4) = 4 * factorial(3)
    8 W/ g) [8 i1 n/ C( Gfactorial(3) = 3 * factorial(2)/ w: A) N0 u5 c6 o5 g" K+ P1 T8 u
    factorial(2) = 2 * factorial(1)3 n5 u5 I# N$ W
    factorial(1) = 1 * factorial(0)
    + N% _; f( f. X# Lfactorial(0) = 1  # Base case
    6 n% e' W* {" }7 ^' ]+ f  T6 p2 ^. M  P, Z# f- S
    Then, the results are combined:
    4 I$ g' c' s. B8 x7 V2 ]' B+ W  @  t- q) J" e4 r3 B

    0 [  m' L* R+ A; Lfactorial(1) = 1 * 1 = 18 f1 {  D7 t" f) @7 K, S6 m
    factorial(2) = 2 * 1 = 2
    # o5 c6 E8 E- P. I. qfactorial(3) = 3 * 2 = 6  z6 T: V" D" B9 i, P# M  u7 o
    factorial(4) = 4 * 6 = 246 k0 D. D& A% |8 ~- s# x" R
    factorial(5) = 5 * 24 = 120
    1 b" A+ Q- [8 y. a' E! A/ \1 w+ [5 X' R* F1 u
    Advantages of Recursion
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    # ?' A$ A. {8 T" s$ g' _    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).
    & W$ j- M" j6 M. ^5 w: O! ^. A/ ~2 Q- t5 U$ Y( F7 y# e: k/ M0 i( t
        Readability: Recursive code can be more readable and concise compared to iterative solutions.8 x/ z9 [# _# j5 ?

    " c+ h3 J% e3 I) y2 {Disadvantages of Recursion
      F  u, a  @' O4 h1 s# {" ^4 O" f$ v, U/ I% S
        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.4 N2 h) o  Z2 f0 m5 c) w
    * l% b0 W3 @* s- }; x
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).+ M% ~! m$ T5 o  q1 f5 E4 |

    " b- W1 ?& ]# \& n: oWhen to Use Recursion
    6 x9 A5 z. M  q" Q; X' M# ]1 W% d* O  v4 [
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).9 C2 ]1 B& s6 ?

    " v% e. T) M' |* Z7 |# k) m    Problems with a clear base case and recursive case.
    7 `0 T* L$ j6 z" ^$ p. B; y& k; ?- L, j0 [
    Example: Fibonacci Sequence1 E2 K4 P5 ~+ u* q, y

    & t" R( R2 ]3 M- gThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 Z8 P- g9 {& c' a) j
    0 Q' Q2 j$ x% D0 P3 ^; L
        Base case: fib(0) = 0, fib(1) = 1% i# x2 J; {) D9 D6 [! X

    2 P' [- j$ B: Y* B" a, A" G& ^    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    ! ?( Y) z1 ~5 M! h# w* ?
    0 C$ T5 _/ h: i/ ?+ O4 E  [! Rpython
    - v# c: y: C# O) N5 P2 L
    7 B% ^; P; f' Z, l$ v
    - x1 K! ~8 G6 I. |1 udef fibonacci(n):: G1 ]' t8 G0 F
        # Base cases
    " v' ]# y7 c, w) [    if n == 0:& {' p! M5 A  r9 b& A2 q% d+ h
            return 0
    & q# Q2 N; Z* D1 {1 e3 t9 K& i    elif n == 1:" q* E0 u7 u, I& r& g
            return 1
    & j0 q4 I, r0 E    # Recursive case3 u! V5 l4 y. o- m
        else:+ M! i. g8 c) D) E; @0 ~
            return fibonacci(n - 1) + fibonacci(n - 2)9 F: L0 R; x* L# _. N5 g& u

    8 g* P& {; `6 V# b- g! ?9 y0 T# Example usage
    # R2 O  r8 G  j! v) eprint(fibonacci(6))  # Output: 89 L7 F5 K- V4 K  t, r+ S
    , \9 A0 N, c6 d0 d! m7 Y
    Tail Recursion
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    3 @3 W" T; ^* p% ~9 ITail 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).7 ~9 j: G. r; y" {
    0 t# o2 {$ d# `, V  x
    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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