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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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    2015-11-30 11:11
  • 签到天数: 2 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 . H+ I) L; q$ i% E. N# \8 w- ]
    ( x+ Q* ~- q2 l' _7 j+ ]
    解释的不错
    / m/ Z3 Z4 L- M  u# j
    4 w# \2 w8 z- 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。2 f) V% Q! _) ^
    9 Z' f, X/ U4 w; b6 C$ x5 ^! L, z
    关键要素! C" E+ V: W1 r% o3 g
    1. **基线条件(Base Case)**- L0 B) ]' k/ O  U2 g- x. \
       - 递归终止的条件,防止无限循环
    9 ~, V& b5 C. f1 @  h   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1% Y2 y# L" C  R# v( X; [& _/ k1 h9 e

    3 `2 S) B8 T0 g) O( H9 g$ I- [2. **递归条件(Recursive Case)**5 Y0 M) k+ b. v: ]' c% o/ K
       - 将原问题分解为更小的子问题; J, n1 t4 b. v& p5 o1 U& ]
       - 例如:n! = n × (n-1)!1 n0 M9 M& _' B$ S/ q! N- Y9 G

    ( J1 L$ [* R  T5 Y; j 经典示例:计算阶乘
    # S! r6 Z' H6 w9 S9 n5 gpython2 O" D6 a! M4 @2 h. a/ f: ^3 L
    def factorial(n):4 ^4 z1 e% P  z, m5 w7 K8 Z" F
        if n == 0:        # 基线条件  x( g! j4 |: w% r& o" d& i
            return 1
    / [7 H7 @" {: k# ?+ r    else:             # 递归条件
    ; t) z' @" |1 T' x" y        return n * factorial(n-1)& d1 D( g8 h$ g5 ]0 a, Y" C
    执行过程(以计算 3! 为例):
    ) N2 O( b  Q" E4 C# I4 ]+ Ofactorial(3). }% C5 o0 y5 X- l% V- c
    3 * factorial(2)
    4 V+ V3 O! E) H1 K: P: \3 * (2 * factorial(1))
    : m; |) v6 A" m2 u  G& H3 * (2 * (1 * factorial(0)))2 A6 k: P# b) N1 G: ?: s2 l. l
    3 * (2 * (1 * 1)) = 6
    4 ^  H" J6 u0 {& w. K0 |+ i; @9 X
    递归思维要点
    % n* c8 }9 w% m* l; A) T1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 f/ p4 q* ]. o: P% q
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    ! A9 o: |+ ^+ H3. **递推过程**:不断向下分解问题(递)  l6 ?$ V. G) r1 Z2 X  B' [
    4. **回溯过程**:组合子问题结果返回(归)" X+ g9 z$ u! X, K' |
    , x* p# j, L8 U7 X! C  c
    注意事项9 f5 _4 x* s) g& H/ S
    必须要有终止条件
    4 U$ X4 N. ]3 c/ j! n递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    4 f7 R+ h! {  O" T某些问题用递归更直观(如树遍历),但效率可能不如迭代
    / O4 X8 G* U! q1 ^. [$ @尾递归优化可以提升效率(但Python不支持)
    $ T- A3 {0 m; L, P
    # O2 P3 t! ?* I: n% B 递归 vs 迭代
    , v% z& l, Q) O! G2 ~|          | 递归                          | 迭代               |# K( e; B/ p+ e* H) U6 O
    |----------|-----------------------------|------------------|9 r3 W) h3 J& i) T9 i% l' m
    | 实现方式    | 函数自调用                        | 循环结构            |
    4 U% ~$ a6 Y  f5 o8 G8 |; f| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# ?2 Y- C& r2 F) w3 q- 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         |1 D! Z0 K9 _, t. r1 `9 ~9 w
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  b# f% v' X7 c, Z3 G, U
    5 c% d( T# e4 c& A5 f/ m
    经典递归应用场景" N9 ]/ H* d0 X2 i/ ?
    1. 文件系统遍历(目录树结构)4 T9 Q) B  |2 T9 s
    2. 快速排序/归并排序算法
    * A8 I8 W9 V; b! I/ f3. 汉诺塔问题
    : ]6 X; a% K7 p4 S; g9 e* a4. 二叉树遍历(前序/中序/后序)
    # U! u8 m0 x+ l* R2 ?, Z5. 生成所有可能的组合(回溯算法)
    9 r. Y7 S8 [2 ~& Q2 j, `. R1 J9 v. p; ~; ]
    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,6 M& f; r3 j. m' u. \% T' E  C% v7 |
    我推理机的核心算法应该是二叉树遍历的变种。. K. c- I4 i; N3 d
    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:
    " c  I" X0 C! A8 Q4 wKey Idea of Recursion. [/ ?2 N0 k( e0 c" L

    1 y5 v+ i' M9 s$ x1 |' r7 f$ D2 W2 SA recursive function solves a problem by:
    ' Y- o; s; ?9 H, {+ b9 Z3 {
    0 {  \. C, A/ s- B3 i5 l$ |: e5 K    Breaking the problem into smaller instances of the same problem.
    - K* K5 q9 @( s8 R6 [0 D2 `' c. N0 T8 T4 V* S
        Solving the smallest instance directly (base case).
    ( A  ~7 d: G( x' ~6 b* a5 g% @0 P; F3 {- D9 X
        Combining the results of smaller instances to solve the larger problem.( |. D+ C) }8 @* e9 E6 Q
    5 y5 Q' Y4 I/ _
    Components of a Recursive Function
    7 [7 P2 Q+ f! w; ]$ W1 b. d
    ' s1 D% [# S& t4 G% E1 z5 O    Base Case:) }& |, `- o% c0 e

    3 J" Q& b2 t* T        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    . e  [) |* k7 O1 P: s: H/ {3 z" m1 T
            It acts as the stopping condition to prevent infinite recursion.
    5 y$ l. d3 H1 `) m* F! Q2 u# n. i: P+ z9 F2 s; Z9 Y" F4 \5 |
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.: U7 m: w. U$ u) g1 }& b& l

    * H; j& M/ B7 j" l; D2 F    Recursive Case:
    # n9 c$ E' P. B3 p
    ) L  D* p% M- e9 e        This is where the function calls itself with a smaller or simpler version of the problem.
    ) }! N" p' Y4 B
    , d9 @" u- ]! _: W3 c. t- y1 k        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    $ L* s8 `; w' z6 u+ F0 [. L& d2 ~- b8 r0 G& n4 m
    Example: Factorial Calculation
    & _& ?0 T/ b2 G: U5 ?6 \
    , `. r0 ?4 @- X  EThe 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:. p" d3 r) M1 ?1 p! Q* {
    - w# j0 {0 t6 V! f+ y
        Base case: 0! = 1
    + w8 ?! e0 P4 ^% I6 P4 c( n
    $ ]* `5 g( B5 v& w$ B. t% H6 Y; v( Y    Recursive case: n! = n * (n-1)!
      I+ P" \: _+ {' t& p. S
    9 S$ O$ j  K3 WHere’s how it looks in code (Python):* Y) G! l1 I5 J# ^3 q
    python
      n7 O! }9 Q3 y8 x1 l7 x# H/ }6 \8 X( C
    , V$ H( E; D/ l7 y! V, v1 m
    def factorial(n):# y9 |+ X. O) u" I! p% f2 Y
        # Base case( A! Z9 J$ M& Y% h8 B6 O
        if n == 0:
    3 \* O! M2 l( X; i) B; y3 i        return 10 i! b4 f8 \: o( N  B7 b4 y8 B
        # Recursive case7 |( l; [( e6 p& A! q% l: e- Q
        else:: Q/ s) f( @% s" l4 ~$ T# x
            return n * factorial(n - 1): G  \/ Z7 c' x. u) d

    & a5 z5 M" B5 s8 S1 ~) W* p4 }# Example usage& b+ g. M6 f( t# T5 T/ Z% q
    print(factorial(5))  # Output: 120( J: p1 k+ R7 L8 C/ x0 ], q

    1 X; `& o) O( }) Z& EHow Recursion Works7 ^% a; s7 j4 K" O

    8 r* X7 [  ^* b# v    The function keeps calling itself with smaller inputs until it reaches the base case.
    3 Y6 e0 P8 K9 ?( ]6 {& m
    # D4 v" {, a2 S* `1 g9 j( S    Once the base case is reached, the function starts returning values back up the call stack.
    * D/ R1 ]5 w# ^
    ) _2 f* \$ d/ o( [- m" E    These returned values are combined to produce the final result.' ]& }' b7 U. B8 K
    ; A) z& q, z) v) C4 U. Q- k
    For factorial(5):, Z& @# h; D' w% _

    ) ]( u" M4 @/ d; L, X; ?' G: n# a; {5 \9 }3 @
    factorial(5) = 5 * factorial(4)
    $ r: N; X6 G9 W0 D; ifactorial(4) = 4 * factorial(3)+ C( i( O) t. F3 |0 E' i; E' i
    factorial(3) = 3 * factorial(2)
    + R/ n# s( o3 R3 {0 {' Ofactorial(2) = 2 * factorial(1)
    & B$ C) X* n* |factorial(1) = 1 * factorial(0)4 ~" N' P: j% K9 I
    factorial(0) = 1  # Base case
    ' i+ A# Z% e3 H% ?: h5 b3 {4 T
    ! n: k: s) H, D; w- a" ?2 `6 cThen, the results are combined:1 M& k6 ~4 ?" Z7 h6 g
    ; V/ s3 i% S. s" z% Z9 W6 N& K
    + g1 V. [. }! q4 }& ?
    factorial(1) = 1 * 1 = 1
    * Q' W+ B$ N0 L8 {0 P6 f+ ?: v% pfactorial(2) = 2 * 1 = 2; J/ e4 F" H8 h4 h# C! j; ?5 b
    factorial(3) = 3 * 2 = 6
    # ?+ d- u  o3 L. h3 M) P% _# `! lfactorial(4) = 4 * 6 = 24
    9 s8 n: ~! B/ w; P" `) ^factorial(5) = 5 * 24 = 1202 q" k3 B$ Y" i# u" }: `" i3 G
    ! j6 ~6 p& n1 o$ v
    Advantages of Recursion; U$ ^. N  |, o- X" u6 ]5 B8 D0 {* j: u

    " p# R) I: f# c9 z    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).
    - c8 v/ p5 {' Q: D1 T, T7 A+ H
    * t" O* ~" j) ?" A) i' f* |    Readability: Recursive code can be more readable and concise compared to iterative solutions.
    , F5 V8 G7 L; [$ Y* F1 i# v3 q" u7 x5 \+ E6 [
    Disadvantages of Recursion' n6 v  M  m* }# j  E! a

    % J' Q" A- |5 t. ?6 {1 c) c    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." x7 P& f) s* I$ j9 L- l
    ' h( m- W# M1 |. k+ O( H
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).) c) H& l0 H. d* w2 m4 H

      P' p7 X# |: r# w* A3 B. `When to Use Recursion4 A* U( i3 I; R2 y) `  E* ?( K
    ' _" w6 Y8 [0 N$ J2 \
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).$ b9 u9 c4 T5 U4 T! V5 |/ K1 i) ]) B; ~
    ( x2 q2 m. r) H5 _
        Problems with a clear base case and recursive case.& l7 I9 v! O- z6 E8 q3 |% o

    7 t0 l7 u! n4 V" ^2 gExample: Fibonacci Sequence4 @( V( h, g- l0 Z) g/ T

    0 p+ d( b) v  V) l: oThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:5 R1 P2 @  }5 {) ^. w

    % G0 e3 B3 }4 W4 N9 m    Base case: fib(0) = 0, fib(1) = 1. @; `. D4 }( v( q4 h! F; e) i

    * ]2 G3 }4 i9 M- q7 K$ v    Recursive case: fib(n) = fib(n-1) + fib(n-2)8 m% r: o' s1 f( Y( z
    & x. S3 v/ J0 ~% I
    python" _) \7 c( X! j5 X# o. M

    3 w8 L; i& N4 P3 V. [1 K0 [& ?+ Y; `1 w: N3 O
    def fibonacci(n):4 m% S; P# C- `
        # Base cases
    8 u5 Z; J8 j$ R/ B/ \+ V    if n == 0:
    8 ]6 ?- }9 p3 [8 c3 j        return 0
    3 Y3 v& K8 o1 @6 s  r, _2 w    elif n == 1:
    ; k+ g* k4 l7 U& B. Y        return 12 Q+ c- ~3 K+ v6 u, P5 w; X
        # Recursive case3 M+ B6 ^1 z+ @* p" p: I9 S$ q' K: d
        else:
    . _) r( c/ H& n  P        return fibonacci(n - 1) + fibonacci(n - 2)$ g6 J; E* Y7 D- e- Z' d; O

    7 x  q( A" _, H/ x+ w2 o: ~# Example usage6 Z' b4 C: S7 f3 [. M
    print(fibonacci(6))  # Output: 8
    9 M- l2 l6 f, Q; Q& r- j( k6 i( M. B4 X: U4 c$ u+ Q/ B; C
    Tail Recursion1 l& I2 S2 D: B9 w8 o
    & A9 l0 ]; n( n  a. n8 U) f
    Tail 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).+ h6 u! a9 ~4 M8 Q# i

    ' U' f, @  b' ?8 S7 GIn 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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