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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
     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 4 }# Y. h; j3 U" i" w3 K
    % z- F( u9 d! l& V
    解释的不错9 g$ p8 G5 ^% z' n; s7 b8 J6 T( T% Q
    # w) R. J  m" @& C' B0 e3 {
    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。- ?9 X- a  p( @
    9 G! ?! D" x, w: a% w. \/ q" ~( I
    关键要素
    6 r8 Z* z6 j, L3 N5 m' A5 L; p1. **基线条件(Base Case)**
    0 f* Y' y5 y' l6 s4 _   - 递归终止的条件,防止无限循环# U0 h* O1 l8 R) {6 C5 s
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    # z. p) \. y) r  O- w
    7 s6 q/ z7 m* p- Y0 T2. **递归条件(Recursive Case)**, G- D! V  U7 X2 }4 ^6 m
       - 将原问题分解为更小的子问题1 U% N3 @+ k# y- S! j7 n
       - 例如:n! = n × (n-1)!1 J7 \; Y8 q: R7 h# J

    . _. J" f6 d$ g 经典示例:计算阶乘
    % K% a6 p5 J1 l6 [0 N; Fpython
    2 o& c) a7 O* ]8 Odef factorial(n):/ {( x: {/ X0 p, B* [& J
        if n == 0:        # 基线条件* \7 ~% C* D' @% f* [4 T  m
            return 1
    + |' d7 V7 b. v# z# V$ Y1 X8 U    else:             # 递归条件
    ; ^: i/ x- P( r0 e8 u  ~" \        return n * factorial(n-1)9 s4 o6 k5 Y! E
    执行过程(以计算 3! 为例):
    8 n, `+ N1 D) I0 V% T+ W( dfactorial(3)
    " s3 }1 x* H" t4 ~1 |9 T9 f3 * factorial(2)
    5 D7 _+ ~0 U6 x& d3 ]3 * (2 * factorial(1)), i* _) f$ A& @; [0 y3 c
    3 * (2 * (1 * factorial(0)))
    + H9 X0 a8 i4 w, J/ D5 @0 g" L3 * (2 * (1 * 1)) = 6
    " M8 a2 @8 I# {
      L  @9 w5 G3 v- C; Q 递归思维要点
    + o& \9 W; s0 q4 c' ^+ X; E- R1. **信任递归**:假设子问题已经解决,专注当前层逻辑9 c- ~  S, _4 r
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)) e% O6 h9 |$ e% H" ]
    3. **递推过程**:不断向下分解问题(递)# X% y) V& J2 Y2 ^9 e7 f
    4. **回溯过程**:组合子问题结果返回(归)( v- J. N" ~& M! `0 ]& F9 ?

    9 t8 E) D' R# m& ?. q) a 注意事项" ]! Q+ x: l! k7 W9 p' h" `% P3 t
    必须要有终止条件9 s. ]9 h+ [3 R  Q
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)( }9 w, {8 q) u' H; F5 \9 p
    某些问题用递归更直观(如树遍历),但效率可能不如迭代
      x9 }3 t( z% E尾递归优化可以提升效率(但Python不支持)1 H" \' S" T2 e& M0 O

    & y3 n* E, m9 ^- b6 O. k 递归 vs 迭代1 k5 [4 m1 s1 A% d. ]
    |          | 递归                          | 迭代               |$ B( \0 A- |7 z7 D7 C
    |----------|-----------------------------|------------------|6 i, v, G: k8 o# f
    | 实现方式    | 函数自调用                        | 循环结构            |& y5 H. P% h, A( a, Q$ L6 r
    | 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |# e; f2 |( F+ ~+ n
    | 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         |! c3 p; \0 g9 `+ n* p
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |4 ~6 i! K8 H, O" p% N0 z# x: _
    ' n7 [$ v' d! M
    经典递归应用场景7 }/ C1 Q$ L9 m: |" K8 K  M
    1. 文件系统遍历(目录树结构)
    / I  M8 Z. {7 w: l8 d2. 快速排序/归并排序算法
    7 [/ |* A; m: G1 c0 r: Y3. 汉诺塔问题
    : S0 x, H  Z8 B- g0 z# N4. 二叉树遍历(前序/中序/后序)
    4 `# W' a, p- ?* U3 d  {+ l5. 生成所有可能的组合(回溯算法)
      x+ m- B- u0 ^9 {& q! O$ Y" Y' v
    ( V8 W9 [$ Y0 P, m( C/ w- m7 W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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    前天 06:39
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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,; A8 A* z4 x) O2 m' `
    我推理机的核心算法应该是二叉树遍历的变种。! V, n- o8 e' |  D2 h# G5 Z
    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:8 L2 j4 z+ A( E
    Key Idea of Recursion
    8 x- Z* g! Y2 [6 p1 e4 f+ C4 S4 y  m
    ) C& d2 K9 n1 e$ @3 x  NA recursive function solves a problem by:1 V& u$ y% |7 }& S
    ; c/ c5 J9 l" y9 O% I8 S7 I
        Breaking the problem into smaller instances of the same problem.5 Z  n% U% N( e* x' D9 i
    . A/ @0 d7 t  h" x0 J
        Solving the smallest instance directly (base case).2 {# Y6 b- ~9 T7 _, D

    6 U& a9 T/ P0 B+ N6 e7 U    Combining the results of smaller instances to solve the larger problem.
    9 j% B  S# _2 {) `8 B& O. P5 ]; |
    Components of a Recursive Function
    ! B% x4 d7 J' U$ c% M. F7 f
    6 w* X1 a1 K7 }% A  h/ `7 z! u) ~    Base Case:4 m- L7 e2 x4 c+ {0 y' g

    2 I+ d9 e. p& t( r$ H7 M" D, J5 M        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ' A, R7 [: z8 K, _" }* M2 n6 ^6 j$ ?, [
            It acts as the stopping condition to prevent infinite recursion.& y" c* ^/ E/ m, l

    " j6 s, {0 a9 N  y( y) h+ U- ^        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
    # h0 a' C! I* O% m3 k9 j7 ?$ L2 ~& o# f2 U* u% e
        Recursive Case:" a, d3 z) q8 ^* r& A
    . F8 S5 A* W& ^$ ~: \
            This is where the function calls itself with a smaller or simpler version of the problem.
    , x- c6 S0 `" |+ n
    1 l2 Y& K$ J8 P" f7 e" D& I0 n* M        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).+ H: {+ d9 E, \% H- P1 ~$ h; e
    ) [) G& c$ `% j6 v
    Example: Factorial Calculation( _% ^5 h4 p+ n, `5 k$ G$ o

    ! [( p1 X7 f  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:
    7 ?. O* F$ a, r, N3 {6 O- i$ T
    + }, l8 C% W: z8 ]$ B  \    Base case: 0! = 1
    5 z1 m  Q" \! b" P, ~! {9 B0 J9 p% i
        Recursive case: n! = n * (n-1)!
    0 B, `% A2 Q# b  H: y' C/ B  T/ U' v, o% v
    Here’s how it looks in code (Python):
    - l6 ^' p0 e4 o7 e# L- z4 P6 W0 q" {python) o% \0 Y1 G, I+ h0 M" {! r2 o
    , i; G# R( t* z9 y7 m
    5 a: y7 c4 S/ O2 O% o
    def factorial(n):6 u5 h$ x/ t& _" M6 M
        # Base case* F: ^1 g) f; f2 w! b
        if n == 0:
    3 ]- s) c% L. L6 t8 L# s        return 1& _3 S6 P0 r7 o
        # Recursive case
    - A4 S- k- }! X6 w$ \    else:( o! ^/ H, l( a% f* [) B; ?
            return n * factorial(n - 1)) E& i: Y' ~. g5 z+ R

    , d( J. n5 u+ R* P( Q* Z. z) X# Example usage
    . S. D3 \% t) Q; H( U& Yprint(factorial(5))  # Output: 120
    - J+ u) _/ r4 A2 x& V0 W% a5 q' B$ l8 \
    How Recursion Works
      E1 J2 s7 u. u4 T3 F# i+ Z5 U. T% Y. H( @& f
        The function keeps calling itself with smaller inputs until it reaches the base case.
    & `: R4 ]9 E/ b7 ~9 z" k. ^- u
    . Z$ j* N- r$ c* d- U1 y6 Y. y    Once the base case is reached, the function starts returning values back up the call stack.# R2 X; r3 Y* R

    " r! A2 ?4 a. ]6 O1 m' o, g% l    These returned values are combined to produce the final result.
    & V) |, E- x/ N7 p- ?; j7 c& V" S- D0 J2 L% s+ _. t
    For factorial(5):& Q7 j1 f6 c: {" j$ y
    : F/ }/ P) N! B* x) ~# e; g

    5 G3 H, Q4 ~8 U* \+ y( efactorial(5) = 5 * factorial(4)
    ) R. W; V8 D6 N4 d3 g4 Vfactorial(4) = 4 * factorial(3)) i6 @" r5 J5 E1 ?0 O" V8 p) ~; C
    factorial(3) = 3 * factorial(2)7 C& y: w; R& k2 W  F$ m
    factorial(2) = 2 * factorial(1)( D5 r& `" K5 q1 W+ p' Z/ _
    factorial(1) = 1 * factorial(0); `) b& R' ?" m7 C" C& r9 f! U7 k
    factorial(0) = 1  # Base case3 ^( X  b. l( [3 I8 j. l3 g; ?

    ' G! C2 n. |  pThen, the results are combined:
    : G  {  E" U! X: B; E
    ! T4 }3 R6 a5 n4 a" W" Q+ p6 F% D8 k$ I. _& z; g: S* I- w/ M
    factorial(1) = 1 * 1 = 1( w9 @" C; `8 W' A' i1 W/ V
    factorial(2) = 2 * 1 = 2
    $ J! ]2 r3 P  i% E# ?factorial(3) = 3 * 2 = 62 c! J$ @# d! ^( x" K8 I
    factorial(4) = 4 * 6 = 245 O2 f' p" X) t' x$ u! {+ o9 @
    factorial(5) = 5 * 24 = 120, |- `7 b# V1 M" S& }  U/ e

    4 d$ e" c4 I+ D' q" w0 o3 b6 J7 J  ZAdvantages of Recursion# C/ w  S  j' g$ C- K0 H! S0 y( J, A

    ) P$ f* M( Y1 W( o4 ^" b) q4 N7 S  \    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 ^! {, J0 i5 a, I( l

    ' \0 o4 B; H$ M3 Q) z* c    Readability: Recursive code can be more readable and concise compared to iterative solutions.7 i% k6 z! D" U, ]/ A* y

    3 L" Z& w" Y& g: ^' Y  ZDisadvantages of Recursion
    ( {6 q2 B  l9 ]5 i" Q3 G+ U6 R4 ]! p/ c  Z0 v
        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.
    ! P" K4 M8 c1 D+ }# Y' J& T) s3 W+ C% f
        Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization)." l- D1 r1 J, c8 U

    , ?4 a3 m) C1 Y  u! r8 {+ [3 NWhen to Use Recursion
    5 t+ ?. d5 b) r( e! r' A/ E0 X
    5 j! @4 ^; c  H" Z8 ]/ f$ A; N    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).8 d$ i3 x" {, s8 ]& S

      \, o$ f) E, u6 V! u" u8 X; D    Problems with a clear base case and recursive case.
    6 t+ i0 G# P1 z( _* D! m7 u8 ^+ k* n& {$ o
    Example: Fibonacci Sequence
    9 M$ Y6 M$ D. A0 E, l2 G+ B  v* }; \' o- \" j' s  q* _
    The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:& [. z0 s: P: I2 a* C1 |: l0 S5 p+ ~

    ( t2 D+ m8 h" n( o0 ]; d    Base case: fib(0) = 0, fib(1) = 1
    4 n. x7 J. J) O) T; V. t9 z' X3 y7 O- p$ b5 R: {
        Recursive case: fib(n) = fib(n-1) + fib(n-2)
    % H2 b" N/ U* j6 r6 \. V0 H1 v5 `# q. B! `/ r
    python# R- i! v4 E; N' M5 k$ o

    " X% t. Z6 d/ o7 v% v
    3 p0 U+ D1 ~: z# D4 L2 Cdef fibonacci(n):  J  `9 `# S. b) d' W0 }0 O& r& @+ a
        # Base cases
    . V! Y3 d7 H* `3 y1 K/ I    if n == 0:6 E2 _9 M! {/ r' P8 y- e. J
            return 0' T6 T: C( L. h1 i
        elif n == 1:# H( q, E" I. V  a) w
            return 11 Q% w4 [+ Q! w8 O8 m* s5 H$ ]  Q
        # Recursive case7 x% N) z6 q0 p' M# o2 ~% j
        else:+ b5 y. A2 e5 l" Z
            return fibonacci(n - 1) + fibonacci(n - 2)2 S+ S$ h" ~. O/ X8 J! n' m
    6 F, _9 v) g- j$ e* ?# h) d. u
    # Example usage
    , S: u- \2 v- d, }, |) Qprint(fibonacci(6))  # Output: 8
    ! d( w- V# K5 k" I6 q0 u: V0 b4 O7 L
    Tail Recursion& u& Z3 h0 k( k, z3 P5 X4 o

    7 K% ~) V3 I; ^! h6 b8 hTail 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).8 Z* `. F/ A4 o2 I
    6 O% x- w- Z: b7 G- |- Y# c# m
    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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