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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 1 d/ Z. W' \1 h' r
    * r1 N- R- R8 X) L) i( h4 K
    解释的不错
    % j7 y3 P( {) i: P1 p! V
    * b" Q$ V7 p& S* ]) q! m) g1 {' o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。# f1 Q0 }1 E0 `  P: u

    " C" j8 ~0 X* R/ ]: f 关键要素
    ! J, E1 L8 k4 F5 J: V- S1. **基线条件(Base Case)**
    . \) {  V$ F% J7 Q. @( `   - 递归终止的条件,防止无限循环
    9 ~# z  u9 l( G* E1 M# _" F/ W) X! D   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) f6 t# s. g4 d4 p( h3 A4 n1 L4 w: Y, n
    2. **递归条件(Recursive Case)**3 E/ @, w; y; _
       - 将原问题分解为更小的子问题
    ) N8 }6 t  m3 c& d3 Q* y5 _   - 例如:n! = n × (n-1)!/ l1 v$ W! S1 o; ]* N+ U
    , [- s) J# ]# `+ o6 o
    经典示例:计算阶乘3 D! d& p' _4 p4 \# W
    python
      r7 f) t" x9 W( I2 vdef factorial(n):# K3 l4 O- p/ N. L( t
        if n == 0:        # 基线条件
    6 L' n1 E5 h) f: F8 m- [! G        return 1
    + C1 H1 C. a  a2 R& A/ ?    else:             # 递归条件: F  I/ Y/ ~( I- |. f( {/ R7 j
            return n * factorial(n-1)
    ) s  p& Y9 V! H9 a5 E4 Q" g2 m执行过程(以计算 3! 为例):6 Q  j9 R3 n1 i# n7 M( P4 R
    factorial(3)
    9 f- D3 b  R& K; t! `9 R! e+ K3 * factorial(2)+ k6 q% j! x& ?- ]
    3 * (2 * factorial(1))
    ! O! M( ]0 w: ]$ ?# q3 * (2 * (1 * factorial(0)))% w- y+ [) s% [* g6 o, u+ c$ ]
    3 * (2 * (1 * 1)) = 6, L' f7 D" Q; G; ~& I

    + V6 @  s" K3 F* {, K7 S( M2 N 递归思维要点2 U7 g1 _' z2 x6 E
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑4 b& ~, w+ Y* e. h* ~' j+ {
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)% r. C5 _: ]+ t9 G) }* E8 A( Z
    3. **递推过程**:不断向下分解问题(递)9 J" D# F9 s4 o3 p; r' K
    4. **回溯过程**:组合子问题结果返回(归)
    , V3 q% X3 I" A' }" c9 M. b- d* V* d7 q
    # |2 ?2 ], a( s/ ?0 k$ d 注意事项
    % A1 v5 ~1 ^( g! _% J+ X必须要有终止条件/ U+ O& D  O! L0 P# L
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    - h& U3 L! R% K2 y2 L3 S6 P某些问题用递归更直观(如树遍历),但效率可能不如迭代
    & f% k6 ?4 H) _8 v1 h尾递归优化可以提升效率(但Python不支持)
    6 v* f8 G! `+ G9 M4 X7 B) ^9 }7 e7 x2 p6 u2 U. |
    递归 vs 迭代
    5 d; M! i8 y/ C9 \$ P7 b+ o0 z0 f|          | 递归                          | 迭代               |
    6 [: k0 Y! \+ `) s/ ||----------|-----------------------------|------------------|
    ; Q+ o' J! I0 H| 实现方式    | 函数自调用                        | 循环结构            |
    6 z2 n& @$ c7 o5 K| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |; ?" N& x+ m! H2 j0 b$ d5 o
    | 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         |
    5 l( N4 F  v" ^, D/ `| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |+ g/ E( K* @3 A, i+ q* e

    " l1 I2 g+ W# `: v& ? 经典递归应用场景
    * |2 w& R! r/ v; M3 U% e1. 文件系统遍历(目录树结构)
    : M% ~) q7 j4 v0 U, [2. 快速排序/归并排序算法
    + y+ Z3 v1 D: u/ B3. 汉诺塔问题
    ; R9 A( F9 g" d1 m; A* x' m, P4. 二叉树遍历(前序/中序/后序)& }! M) u" Z, [9 Z% G3 k. Z
    5. 生成所有可能的组合(回溯算法)
    1 |3 Q' c2 E; d) n" ^
    # s8 s' f' I3 D2 _' f1 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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    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,  z' z* h/ G: k2 d# D& P
    我推理机的核心算法应该是二叉树遍历的变种。- {' Z9 i5 j  v; i5 F- d" p
    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:
    ; ^" r' I3 [* V1 ]9 {4 k+ M) dKey Idea of Recursion
    ( T, ^3 h1 `0 ~5 t
    " j4 ]7 c0 ^+ H5 ?* k4 `A recursive function solves a problem by:; Z* e# V' S, N7 ], k" l, ^$ Z

    , O: ~" ^0 {4 t) I2 F; D. C2 \    Breaking the problem into smaller instances of the same problem.- u. }  |) B- F' v+ M+ W
    & _) s* X" E0 E8 v6 A- X+ Q5 \  B
        Solving the smallest instance directly (base case).  D7 P7 Y% S1 G- q$ J

    - Z) d" v$ n0 G/ g    Combining the results of smaller instances to solve the larger problem.
    & G6 D1 `" ]/ K; l3 R2 w7 t( {7 y* i+ A  a, `* o* V# X
    Components of a Recursive Function
    , m' H3 a1 H6 J6 j1 X% O% t" R. t, h) g( \6 w8 J, e4 ^& V6 a; m2 Y6 T
        Base Case:
    % j+ b. C9 B, M  I/ Q" j5 ?0 N; B" w. K1 e
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    1 A5 E% K! x% Y9 i3 E. l
    $ j/ r; a7 N! i* v        It acts as the stopping condition to prevent infinite recursion.' E' V- X- T; E2 Q% E# o5 g$ v

    ; D  i  u/ C( `- w7 |8 q        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; _  g: P1 c6 F# R6 W! R) Y" i

    & A3 R( K, V+ J! K% F    Recursive Case:/ X$ ]. g% ~# W
    % N& c$ {8 V. l; W) g  H3 K2 h
            This is where the function calls itself with a smaller or simpler version of the problem.+ d, d6 I5 S2 T1 d
    % z9 J/ g2 ~) G8 Z' x( ]3 C
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    6 U& A( u1 K8 l% s( k& h6 U" v: ^4 T% Z6 {: ^7 D$ N
    Example: Factorial Calculation4 ]" \  o3 r+ E6 C. B  ?

    " L3 O  _2 D$ C  I" f$ V3 N& vThe 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:) f4 ]( l( D( Y$ _

    $ @) ^3 L' W5 F+ V3 ?    Base case: 0! = 1# P% h, r9 K$ ?; r

      o" Q! k8 _! F1 W, x    Recursive case: n! = n * (n-1)!
    / u! n" C) R. Y7 i& p' ]0 s6 h
    ' ?' o7 k0 ~9 ~5 `Here’s how it looks in code (Python):
    1 T2 r0 O5 }! H, u$ _' _6 @python: d- |: x8 j+ P+ K

    ( b& A+ ~2 a/ a) D9 Y+ I) x+ {* x7 M: Q2 G2 \; E
    def factorial(n):
    9 Q6 i3 ^/ U9 s9 e4 P8 o- r    # Base case# t0 U+ e: {6 U* E: t
        if n == 0:4 z8 J$ e; V2 Q/ ]
            return 1
    3 Q& K. D5 v+ R3 {% z    # Recursive case! {- N2 r  ^6 k% q- m
        else:
    2 _9 p8 Z+ U  z        return n * factorial(n - 1)  R8 l7 ?' `: |
    4 \7 D* [' D* @* w
    # Example usage: n- I9 ?' s+ K" v+ F( u
    print(factorial(5))  # Output: 120
    " h: Q* C( y# @# \6 f2 j! ^% j4 g/ k4 F+ R6 p( L, E8 f. A2 _
    How Recursion Works
    # W$ X$ t, b$ o& P: L; Q4 S/ K
    " N: z8 j, u5 k3 o) J/ h9 n    The function keeps calling itself with smaller inputs until it reaches the base case.
    ( c2 A- i1 a% h6 k, x+ y" g3 N6 m$ c! S3 l
        Once the base case is reached, the function starts returning values back up the call stack.6 K1 N$ O. n, P
    0 {1 z2 d4 z) v( e7 P4 a# |2 N: \
        These returned values are combined to produce the final result.; m% @. p5 x0 `$ |
    % m$ g5 b7 \% I3 B% F
    For factorial(5):6 v$ m- R" K9 H7 l) F+ \6 H$ c

    ' F$ S; A: c8 Z5 _( h* v* {1 @
    3 d5 D. H& [) z$ Ofactorial(5) = 5 * factorial(4)
    ! I. i8 d( _8 Z7 F/ z0 cfactorial(4) = 4 * factorial(3)
    1 h# Q0 k1 H- p0 Nfactorial(3) = 3 * factorial(2)# ?0 U8 s4 ?+ L. V  P
    factorial(2) = 2 * factorial(1)
    6 O/ O" e/ P6 h3 H! I4 ]- o+ N% Tfactorial(1) = 1 * factorial(0); j! o) E3 R2 T1 H) Z
    factorial(0) = 1  # Base case; `# y1 n4 l. b/ i
    5 r9 l0 D- f. z& i
    Then, the results are combined:, M' d  W/ U, x/ ^+ y' ?

    ) n- T0 o. ^8 Q8 z1 Z5 ^
    & l# @! {+ q8 q6 Ifactorial(1) = 1 * 1 = 1
    ; @  \' H2 z) a% _: Bfactorial(2) = 2 * 1 = 2; N3 s7 o2 d  F# k8 ]4 l
    factorial(3) = 3 * 2 = 6
    $ ^3 [) |* ~2 i, i. [: ?; z, hfactorial(4) = 4 * 6 = 24) V  ]( P; e  X7 ~  z; _
    factorial(5) = 5 * 24 = 120
    $ P; {: I, I- ?* O- E5 l; I3 L' F5 ^, [5 L5 _
    Advantages of Recursion
    0 }- [; ?: n" l1 m. Z& T
    5 n- Y. ]8 R, n! f) q5 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, l0 _! X* D! ^: q) n- {
    7 w6 W" X* {2 v  |
        Readability: Recursive code can be more readable and concise compared to iterative solutions.( R! B9 z& P  M% K/ ?
    2 M3 r# l9 x2 F
    Disadvantages of Recursion: y, K( t2 X2 F0 O! M
    : S: V+ `  N; p  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.  c/ v9 U- a& L0 l

    % f; o% T0 ?- z, k: |3 H% a    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    9 u3 ]8 ]% x( {& e/ [% t3 u
    6 \' ]3 G  \( [' yWhen to Use Recursion
    " O' Q% w" g/ O# I# p" \5 b
    ) L( C5 v; H+ k" k: a    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).2 H1 s9 {3 u" T5 y* b1 o

    ( H9 t2 F( i4 f0 B" Q    Problems with a clear base case and recursive case.
    1 M( G4 O9 w! C/ U# d
    & Z* r3 ], Y9 J' bExample: Fibonacci Sequence
    + K5 _; q, w, v- q4 N1 z- F/ O6 q( |9 V
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    + c9 c7 I8 p* A" B) B. B& H0 J; F* I# h2 v& `4 L
        Base case: fib(0) = 0, fib(1) = 1; W: v2 B  x. S7 c/ x5 J

    $ Y* r+ y; ?1 B* \1 f# T+ y    Recursive case: fib(n) = fib(n-1) + fib(n-2); j; t% E$ V6 D2 {
    5 e: y! R' R0 |% K) l) f
    python* j% }& U2 o( p8 S0 s, @

    3 o, L4 H9 x2 g" q; X% a" S! h" F6 v( c0 o# a. N0 J
    def fibonacci(n):7 T& }$ l. J" }! D$ j& g
        # Base cases
    0 m$ z* |8 V( A- O. L/ X    if n == 0:
    ; T! s7 p4 ?6 y0 `        return 0
    + U, }$ p6 X5 k$ ]" j6 h% t    elif n == 1:
    ' f* i# I1 z, y$ G0 y% V        return 1
    # l+ e- R8 f) }! _% T# n7 ?    # Recursive case; |2 c  l; E9 N& n6 ?1 e
        else:& p# D5 H0 ?  v9 n5 n; U
            return fibonacci(n - 1) + fibonacci(n - 2)$ y9 }: ^' t2 e

    ; ?* C% h& F" _5 s3 D# Example usage
    5 q) y4 x3 j3 rprint(fibonacci(6))  # Output: 8- ^; b/ z4 x, ^# a! R" ^6 \: C/ y
    ( z/ D! t* r, w4 W; |1 D3 _; p2 ]% A
    Tail Recursion. a2 a6 N5 {0 B: x  s

    5 x% `$ f" i, R0 X( t& l" LTail 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).
    & y+ x+ I5 }5 c1 N: P) v" t% E% ^, z% ]" q+ A
    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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