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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
      Z. y" o) B( J0 A4 J, F2 ?5 p- E5 q3 N* P& _1 @  I, J1 r
    解释的不错- U4 l$ |* w4 s7 T) o6 ?# Z0 B6 y
    8 p% M- R7 P3 Z- n$ {$ I. W
    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。) ^& u: _' o! k3 q% y( g
    1 T* W& M7 v/ C, v
    关键要素# s6 A( X9 y) a- o
    1. **基线条件(Base Case)**2 \- ]( n7 S4 e& V
       - 递归终止的条件,防止无限循环' Z' n4 f/ R- a  X. R. {
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    4 E1 a4 L' i4 f7 {' Q7 D
    $ c* I0 t1 J2 f+ u' n0 R, `* u2. **递归条件(Recursive Case)**$ [, b5 \6 q% A! U- u% ]
       - 将原问题分解为更小的子问题6 F' j9 W3 H3 r& x
       - 例如:n! = n × (n-1)!
    2 q3 K  L; h! \' D3 {7 W( ?! v9 D/ \# R& I5 Z: b) t( l9 u
    经典示例:计算阶乘
    + o( z- n5 j& q$ G2 Y; Opython
    6 O4 N* @" k, y* s# cdef factorial(n):
    4 j; K, D1 C' e0 O6 N0 N) }3 k5 |: H4 e4 c    if n == 0:        # 基线条件6 I/ K( v; T# [4 ]' S  Y0 P/ l
            return 18 A) H6 j9 f% R: S
        else:             # 递归条件9 V4 i) {* B1 u5 z. X5 R
            return n * factorial(n-1)
    7 E3 M% n9 a' j1 T% D5 [执行过程(以计算 3! 为例):% f7 F6 `* t" Z& G5 B+ {+ w
    factorial(3)2 m! B  P1 S" S6 K
    3 * factorial(2)
    ) x" a! h' g) {0 `; z) P. _1 Z5 n3 e3 * (2 * factorial(1)), v" B! t4 y8 Q! Y
    3 * (2 * (1 * factorial(0)))/ F. Q+ C( ]- Y- S% X
    3 * (2 * (1 * 1)) = 6
    & f$ q3 j4 v$ n( f7 v$ n7 d
      x4 Z7 r' ~+ s' K% G 递归思维要点/ ^' X$ L% {; i, \2 l9 c
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑7 P+ s- C" s9 R  i8 d
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)4 K' d5 c. x/ V
    3. **递推过程**:不断向下分解问题(递)% ~4 Q5 a( S- R  U
    4. **回溯过程**:组合子问题结果返回(归)
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    1 H' `! ^8 w, s& T  z' ~ 注意事项
    : f; k, a$ n% i8 ?' l" c必须要有终止条件
    $ x5 f& a% Q. ~% \, R( P递归深度过大可能导致栈溢出(Python默认递归深度约1000层)  O4 D6 a9 b1 t" C3 ]% z- \4 X
    某些问题用递归更直观(如树遍历),但效率可能不如迭代0 D4 T7 P& H+ z$ k
    尾递归优化可以提升效率(但Python不支持)9 I: H0 L' g; K9 w
    5 W7 X% I: d  Y
    递归 vs 迭代
    + T7 \5 @% t8 ^' i" |# ^|          | 递归                          | 迭代               |& v4 F+ n% N6 K& M- f1 _
    |----------|-----------------------------|------------------|
    ; ?( P" R: b0 Q+ H5 _9 X| 实现方式    | 函数自调用                        | 循环结构            |  a' b; ]$ o: h& {
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    / N' u- V5 a# F# n8 J* M* z% i| 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         |
    8 H; \/ R, e% c# }* V: e| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    0 Y+ ]: k" }# n9 m+ J0 \% h
    - s0 @& y7 t$ A* z8 e+ l! V/ ^ 经典递归应用场景6 ]4 j- n8 D- l: V  r+ Y0 ^2 T) Q
    1. 文件系统遍历(目录树结构)
    8 ?6 d( X* K. s) E/ ?2. 快速排序/归并排序算法$ f+ I/ C6 K- ]' x& t
    3. 汉诺塔问题
    6 V# B" i% v2 {/ {* Y& S6 S4. 二叉树遍历(前序/中序/后序)1 O6 X+ S  o( z1 j
    5. 生成所有可能的组合(回溯算法)
    4 F" w# k4 V/ X) r7 B) M( |3 \" j& Y2 q) _* r1 S0 u
    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," |- v, b  |3 u7 X( R2 D6 N7 X
    我推理机的核心算法应该是二叉树遍历的变种。
    0 |$ K0 ]6 X3 |3 y: b2 v; I; {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:7 M- ~5 F- u0 M/ T7 ^/ B$ Y' e( \. U
    Key Idea of Recursion
    3 X+ t8 z' Y8 C6 a* b8 o& b7 P; U' e0 t; ]$ T- r! |8 t- \' y
    A recursive function solves a problem by:
    9 p- C# e1 a6 U. j' w% T
    & K1 J8 a# A2 c" @9 H4 M- x3 c    Breaking the problem into smaller instances of the same problem.
    ' R: Z( G" `* w7 b. a' H+ q( k1 N% L2 t; T; G6 f  p
        Solving the smallest instance directly (base case).
      h; c' b2 A* c$ e; y
    ; d+ B. j! r1 V5 K    Combining the results of smaller instances to solve the larger problem.
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    ) k% @  D' k% }# }$ t* xComponents of a Recursive Function: _9 O" E9 H- E! H5 |* z$ |
    9 ]' C/ t. m+ w, R7 Z# L/ B0 S
        Base Case:
    % r! g+ L$ e+ D$ A2 Q* X) O9 R$ x4 U( R. n
            This is the simplest, smallest instance of the problem that can be solved directly without further recursion.: Z, Q5 D2 ^1 i( G7 k
    3 L4 q& v, W5 b& K7 W# F. N3 R) w1 E7 w
            It acts as the stopping condition to prevent infinite recursion.
    % }/ x7 Y" p, W* J! h
    $ p/ l4 ^* L0 S! X' @7 \8 I; b        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.' E, X3 i" b' I6 x, `5 J
    7 K: D0 l+ ?+ X! @
        Recursive Case:
    # s! w! [* m- n  D; O) ?2 M, O% J0 U5 P
            This is where the function calls itself with a smaller or simpler version of the problem.: m8 M+ j1 h3 A7 x' E& _! k# k

      ^9 \2 C& ?: x        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    4 V% q" a/ j" i" ~7 i; y5 k% b) \" V5 T, l3 a
    Example: Factorial Calculation
    * h) V6 i! t8 H1 b; @. ?/ `% \
    ! t+ t5 G& W4 yThe 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:
    0 M/ P# V( y' @$ G6 h) W4 i0 Z* Y/ W5 g. v
        Base case: 0! = 1) L% J* t; X/ ~( W

    ; W; S. Q9 p9 W, Z; Z" l+ G2 z    Recursive case: n! = n * (n-1)!
    8 l& V! V1 ^  M; B+ I* \& z
    - g8 i! E; R6 o* D% q2 eHere’s how it looks in code (Python):
    * F0 M$ d) `& s# Mpython/ Z# l" z, G. O/ f

    ( e2 Z7 Y8 k9 T- ?. w: v  G. U! b5 g- x5 b1 o
    def factorial(n):
    1 |) X% L9 G! O    # Base case
    6 N( X( |* }# m: i7 k' c# f    if n == 0:8 M8 d5 _) [6 s7 L( B# a% g
            return 1
    & j; e, f$ ]; _    # Recursive case
    , _: t' G- {0 @8 e/ Y    else:
    5 y' d% F7 w! n; u        return n * factorial(n - 1)
    2 g2 q. c' R! l$ N# ^
    , [4 K0 i1 U5 s5 v& f% k# Example usage; @1 y  B* W5 ~' j1 n6 t# r
    print(factorial(5))  # Output: 120
    . ^4 I7 l* b& q/ K2 g, O& G3 Q/ w/ D+ w8 b( U
    How Recursion Works
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        The function keeps calling itself with smaller inputs until it reaches the base case.
    " k9 b6 p& P  ^' J9 f) T: @) o  V3 v6 y3 B
        Once the base case is reached, the function starts returning values back up the call stack.
    2 S4 d: X. L8 y
    ' P) I6 r  f% a) N) p* R4 R    These returned values are combined to produce the final result.2 s; p4 A  H5 v! P

    ( k' _4 c# N8 s7 B: B5 }For factorial(5):
    0 v/ u( r  P" P/ f. Z
    8 W' f* C3 w$ V. \0 @5 w" u+ q0 x# b4 o6 b- [* A% F* Z
    factorial(5) = 5 * factorial(4)
    ! w$ B$ v( K" m& o# bfactorial(4) = 4 * factorial(3), K: S' r8 y/ c# S# ~9 N
    factorial(3) = 3 * factorial(2)
    ; Q! {, R9 w( o6 nfactorial(2) = 2 * factorial(1)
      V6 h. o  F. ]factorial(1) = 1 * factorial(0)7 ?% O' e3 x" \3 Z6 W! w  D
    factorial(0) = 1  # Base case# z+ O2 g' U  v0 M7 M, N1 |
    $ D9 T; K# G6 e2 l$ I
    Then, the results are combined:
      e6 }, l( A5 P3 h: w# V/ o
    ) Y+ v* P' U$ y0 u" o( G3 [3 a% ]2 }* m6 a- h( i, g
    factorial(1) = 1 * 1 = 1, U! b) v  n2 ~
    factorial(2) = 2 * 1 = 2
    $ [* w" K; b6 Z& @" ^5 {factorial(3) = 3 * 2 = 60 q1 A1 Y# e9 o7 y, Q/ s
    factorial(4) = 4 * 6 = 249 A5 b) b% U% a3 Q+ |
    factorial(5) = 5 * 24 = 120; L7 l) p5 `. Z. k' A4 [7 T
    . C1 F2 _( X4 x/ T: A. z- J' a
    Advantages of Recursion
    3 U) {0 w! ?% S! X( L3 N5 r
    ) w7 W5 y0 S! ^1 Z3 U' ~    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).7 g6 x0 Y1 c* w' Q5 H+ Y

    + J/ u- b2 M. T* b) d1 Y5 E: F  R( z; y2 k    Readability: Recursive code can be more readable and concise compared to iterative solutions.0 E. H8 ], i$ i4 X: Z2 z! S) D, a

    ) I7 g: l/ C4 Y6 P) ^Disadvantages of Recursion% _" |# e" |: z. b" p

    , Q$ |2 F( v6 r0 L    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.+ w$ L" R$ J! }

    ) l+ g3 A! p4 k8 |$ i1 E    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    4 z( ^$ P) P" |+ K
    4 @! d' ~* i1 ?/ b- e: Y; \When to Use Recursion* k5 c5 S. K1 V. }6 S8 J
    9 P7 T' W4 U3 Y$ i
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    % c' |5 z& l& `) {2 W# V7 I6 N3 ^# ?5 z. {6 _8 B. D; g
        Problems with a clear base case and recursive case.
    # L3 L* z/ Z& ]
    ) H0 k9 V( p1 nExample: Fibonacci Sequence
    . n7 @& ~' K( W( n$ S+ ~4 l& T3 a# {  g
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:9 g2 x$ y4 P' T6 E( k$ ^  H& t0 A
    ; v* ?' d) S0 ^- j. ?4 R( M: [' Y
        Base case: fib(0) = 0, fib(1) = 1& V9 g4 a: ?; d& M4 V& J6 o  N

    1 C: g! f5 q  W. j# D    Recursive case: fib(n) = fib(n-1) + fib(n-2). Q' q7 ^( \; e

    7 q  x, C" f/ Y7 j' {7 t% l8 V3 J+ Lpython
    0 I" q1 g% T, c, |
    , s7 I7 J2 W- b" c
    , l) O- }, O$ ldef fibonacci(n):! g9 c$ R! G6 t
        # Base cases
    + l, v" s5 N7 _4 ]) X. n    if n == 0:
    5 l3 z# I/ i7 g3 U        return 0
    2 y/ z1 v, O! Q: |; ?8 v    elif n == 1:- U5 ]2 t: M0 r( s
            return 11 b. S, j! ^( L. a
        # Recursive case
    ( i1 g, B3 R! E7 k3 i# Z3 Q    else:
    & _6 c2 @( L  f        return fibonacci(n - 1) + fibonacci(n - 2)
    ) q: a# c1 x2 c- H% u2 `/ y
    ; Y8 A3 ^; u6 C# Example usage
      W& }5 d4 L+ w7 @6 [print(fibonacci(6))  # Output: 8. v% i7 |$ N+ d; J- H  M4 V
    0 l; `% ?* o/ v2 p3 ?
    Tail Recursion4 m3 P( \0 s5 A8 d* H' D6 x
    9 A, Q  E) @1 w2 m
    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).- m1 s0 l1 Y5 M4 n% u: |& r
    9 i, E9 q- t, y/ z" p" I. ]3 Y+ [7 z
    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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