设为首页收藏本站

爱吱声

 找回密码
 注册
搜索
查看: 1588|回复: 3
打印 上一主题 下一主题

[科技前沿] 突然想到让deepseek来解释一下递归

[复制链接]
  • TA的每日心情
    开心
    2025-9-8 05:08
  • 签到天数: 3 天

    [LV.2]筑基

    跳转到指定楼层
    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 / a0 r8 ]/ S; n0 ^

    6 f3 E; L" \0 ^1 M  K1 I0 K解释的不错
    $ H& n% V, V/ P# K) t* z1 j8 {/ R7 i
      h5 Y! h1 h& ?0 @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。( o' D6 j$ i& e$ O' [& A& r4 A( _; ?
    ; W  m8 Y- P% ]+ K; B" K
    关键要素& T7 d5 b" P; m
    1. **基线条件(Base Case)**! {5 O) c; d2 M& `% M; z
       - 递归终止的条件,防止无限循环# u3 j' s- N8 O9 O& s7 Q- x& v
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
    ) ~8 }% I" s5 _1 G# }+ ^) ~( @* j: W. q" h7 ^7 L
    2. **递归条件(Recursive Case)**% H/ m* h: D2 l
       - 将原问题分解为更小的子问题
    ( [+ K3 \6 L3 Q0 j+ {& X; q   - 例如:n! = n × (n-1)!* M, I: l. b5 ]  b# K
    ) f$ T0 ]% F+ B/ H) ?0 w2 ~4 d& ?0 c
    经典示例:计算阶乘
    # J4 D! o1 a2 P6 ^1 z8 ppython
    1 z/ A( I1 P$ r, tdef factorial(n):& E) z5 U8 q9 s& g
        if n == 0:        # 基线条件
    9 c, a$ G+ d- @9 `% N% R        return 1
    6 h5 q" A, ]+ l- a( ^    else:             # 递归条件
    $ w; b' F5 ?$ L& L9 D# l7 U- }! }        return n * factorial(n-1). D7 R* f7 S2 ]
    执行过程(以计算 3! 为例):  f' H7 C3 _9 l6 G- L
    factorial(3)# S4 ]  H$ F: q9 t
    3 * factorial(2)
    2 i% |7 @! M5 P" l% W3 * (2 * factorial(1))0 m2 ^7 Q" w9 W
    3 * (2 * (1 * factorial(0)))
    7 I* M. R, H# e$ l3 J3 * (2 * (1 * 1)) = 6' {: {' C8 u  a# O  I

    * L5 Q, d0 b4 w3 M 递归思维要点
    9 N& q4 R8 v; A+ x) h7 C: o. P1. **信任递归**:假设子问题已经解决,专注当前层逻辑+ W$ S7 N. ]8 |) g8 C4 G
    2. **栈结构**:每次调用都会创建新的栈帧(内存空间)
    + K6 D, j, S5 T2 |8 n# ?2 F3. **递推过程**:不断向下分解问题(递)
    2 i0 ?7 G/ Y- ~  b3 I+ ]: |4. **回溯过程**:组合子问题结果返回(归)
    . P% H! B$ b# ?. z& e) l
    : `& D* i3 M& x8 I" b 注意事项6 ^4 i& T3 p  v0 y! C! a. }
    必须要有终止条件
    . W, u  h% f, P+ 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 U) F/ k! T& y- R4 X& x' d- B( H0 Q某些问题用递归更直观(如树遍历),但效率可能不如迭代- \% Y/ W& m) M6 D. }7 m
    尾递归优化可以提升效率(但Python不支持)
    0 P( c7 [" }0 k- e3 }/ W- K* |( }$ O6 f( @7 Z3 u, j
    递归 vs 迭代
    ! S5 R3 v9 x( o$ n: |8 e|          | 递归                          | 迭代               |, @; h! J7 w2 d  j7 c, N/ l
    |----------|-----------------------------|------------------|- W' A/ _# R( V- J
    | 实现方式    | 函数自调用                        | 循环结构            |
    . x& k$ g0 l% H| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |. R( ~/ w5 D2 ^1 }4 l7 P- X, i: k
    | 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         |2 b$ I, h3 g4 \2 `" I& u/ _
    | 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |
    , q- |0 m8 m& |5 a8 t& z
    & U2 O5 c% Q- V. o, j; ~ 经典递归应用场景
    2 C4 q& Q& {: M- D& z" O' n8 h- }1. 文件系统遍历(目录树结构)  t0 ?4 z1 o/ x7 Q% X6 n
    2. 快速排序/归并排序算法9 O7 \8 {. h) K
    3. 汉诺塔问题! {0 }1 D3 Z) |8 h( z; o5 C
    4. 二叉树遍历(前序/中序/后序)
    1 [6 k: i9 t2 h; l5. 生成所有可能的组合(回溯算法)1 ^- N- z* S$ P

    8 e. @5 W" k! k2 _7 E2 t: 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。

    评分

    参与人数 3爱元 +26 收起 理由
    pcb + 4
    老票 + 16 涨姿势
    住在乡下 + 6 给力

    查看全部评分

  • TA的每日心情
    无聊
    昨天 08:28
  • 签到天数: 3123 天

    [LV.Master]无

    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,( W9 w  u' C# ?! N% P( V- y
    我推理机的核心算法应该是二叉树遍历的变种。8 ~; ^! s8 x9 n$ U2 u1 a0 j
    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。
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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:& u- w9 ~4 z: m! ~4 c% j% O
    Key Idea of Recursion3 K. x0 e7 `& X' A
    9 V, r& c9 g& W: i0 I0 J
    A recursive function solves a problem by:
    7 j: K; W4 C0 ]( p+ y& I! I# e# t5 D7 P# Y" E9 [; ?
        Breaking the problem into smaller instances of the same problem.
    0 u2 }( o$ C: Q, }: b' |, E! r! U# M
        Solving the smallest instance directly (base case).* J- w* l; z0 a; a
    5 z' I: _+ f. j! d
        Combining the results of smaller instances to solve the larger problem.8 x  F+ ]( d7 V- Z" B% S

    4 v7 b5 l& k: W0 j% EComponents of a Recursive Function; K# F. j. F- k% x

    7 e5 B! j/ Y, Y    Base Case:
    ' x! g! C$ I$ E
    5 m6 j, ?. e  M  j  j4 H0 q1 E. @        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    " p/ f1 @3 [, @& v- J. m; Y  k+ X
            It acts as the stopping condition to prevent infinite recursion.
    7 }) V% u! V6 W& ?' Q3 y0 S- a" z( V8 Q* w; ]) ?
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
      X9 `+ ?/ N2 L( H- S0 }$ U. O3 D7 N
        Recursive Case:4 @; R+ u6 d  [

    ) A1 t8 \; ?* |        This is where the function calls itself with a smaller or simpler version of the problem.
    6 X0 G( n! Y$ b, a( L1 Q1 l- S2 X: Y* C# y! Q, Z$ ?) f" H) W9 ]
            Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).% k5 {4 d" ]" ]4 j  ~. W& b

    : A# b1 N0 t* qExample: Factorial Calculation
    2 m6 s& o& T( A% j( s) {3 ?
    , K* i% ~: B9 Q$ P! @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:1 R( s1 Q4 Z: A1 m- m
    - p) v5 W0 D/ B
        Base case: 0! = 1
    : \5 d( L3 w/ m& a0 z, G  v) q9 H& }) `4 ^  d3 s
        Recursive case: n! = n * (n-1)!
    3 r, v/ O7 E, n8 I- C2 ?& }& G
    5 o4 F! p  m" n0 V* R5 _Here’s how it looks in code (Python):
      U7 u: W% T# E2 p- g! [python0 d9 z& E- q7 z6 L. }. j

      ^4 Z+ C% B. G+ D8 t' \9 z) }% T3 o% d# K0 ^
    def factorial(n):- [% h# |' z5 x
        # Base case
    & w3 `! r$ T  o, j# w8 h    if n == 0:+ b/ p: x- r2 S0 `' Z
            return 1" C- C, E$ ~- f" n5 J% N
        # Recursive case* n) v8 v2 P5 s  U) o8 j
        else:
    ' b* W  w. F# t$ b6 H# b- A: f        return n * factorial(n - 1)
    - h  a- Q2 |6 Q& n6 }4 |( h4 C% O2 {+ \" K) P2 V
    # Example usage
    # N' y, ?7 c; N0 L, O; Tprint(factorial(5))  # Output: 120/ B; ?/ i; d0 ^' M

    . l0 r7 B! y$ z* _How Recursion Works
    , F) G6 W) i+ V) L$ B- \" H$ y. M% p
        The function keeps calling itself with smaller inputs until it reaches the base case.
    5 |4 J! C+ _: q7 O' C; E- D
    8 ^- _! Y0 O: ~- o( Q, `9 o    Once the base case is reached, the function starts returning values back up the call stack.+ u2 ~( Q3 o' g/ d( y) q6 j
    , Q3 ?+ B6 n/ o5 D
        These returned values are combined to produce the final result.$ Q( W: w+ C; e5 p- s

    ' }/ S9 J2 X) `. Z; XFor factorial(5):/ x. M/ q6 z2 `

    2 _7 e/ E; U+ t* J$ y: {$ @6 x5 g& ]; f% w3 q9 R# w
    factorial(5) = 5 * factorial(4)
    4 `( ~& W5 }# M0 ~3 i2 {+ |factorial(4) = 4 * factorial(3)1 w" e/ b! x! b% f# Z- U
    factorial(3) = 3 * factorial(2)$ K2 w" q8 i; r: J$ J4 W
    factorial(2) = 2 * factorial(1)
    $ O  W8 A- K9 X; y0 r; Qfactorial(1) = 1 * factorial(0)
    1 r$ f' ?  R! d& A4 l; A% y8 lfactorial(0) = 1  # Base case
    " |: M9 \. Y4 U
    ; f! V* {8 [1 w/ @6 F" t% F/ lThen, the results are combined:" [5 ?: l# \  t: ]
    # ?1 n# e" _/ U; L) q0 Q* V+ p
    % ?8 x1 b1 Q& k4 E
    factorial(1) = 1 * 1 = 15 X7 t# ]# b% I2 O, o& L) F. t
    factorial(2) = 2 * 1 = 2% M! N' j* p3 E/ ]
    factorial(3) = 3 * 2 = 6
    ; a$ v! q' @( V' q- n! |' h5 pfactorial(4) = 4 * 6 = 24
    * y+ I# ]: t: y* e! t# Zfactorial(5) = 5 * 24 = 120
    0 N  |* `' a  m3 B; [0 G8 g# N7 }! m$ Q0 x( @- V8 u1 @
    Advantages of Recursion
    ! \! x( `% u0 d! G3 ?
    6 `5 Y% S. ~' {7 o    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).9 g( v# p* e8 w) }% o- q
    * Q( \! j3 S# g8 k- k6 p8 _# c( ~: l
        Readability: Recursive code can be more readable and concise compared to iterative solutions.
    ! z0 Z  @. h" e+ M' H' ?' _* o0 y  D, j' z" [  @1 M' i, ?5 a$ H
    Disadvantages of Recursion0 g6 [% ]# Q; U+ ^8 C% ~! I3 A

    9 }3 s" Y  s' r" K2 x    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( X  K8 j' J3 B0 b6 T

    - L) _8 F; U9 U( p9 b7 C+ p    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).5 `1 b1 a4 _& R" t1 O

    6 e' \5 Y& Q' ]7 |6 Z, oWhen to Use Recursion4 ]3 }5 |5 C0 K0 F' o: t) R) e

    . I- Y5 U$ k) A" a# S/ U$ M    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).& J5 O+ |2 q/ k/ ?
    + X3 S6 f( g/ t. K, P8 [, H
        Problems with a clear base case and recursive case.
    0 Z3 ]. M" C# j; u; b
    7 M& h4 D$ ]; |" {Example: Fibonacci Sequence" H( E/ c8 I! E: U: A

    : y  ]" L# r8 J4 F) e* GThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:7 A( b) Z$ H! O$ k, l5 \
    $ ^4 w- W+ E3 M* ^$ Z2 y2 f
        Base case: fib(0) = 0, fib(1) = 1/ O' M% ^% R; J1 @8 c
      j# T2 F# B! E/ t  R( U, u
        Recursive case: fib(n) = fib(n-1) + fib(n-2)$ P( Y7 m# E- ~7 B' h* {, u
    / q1 `. C# u) o3 d/ U: b1 G+ A
    python& A: F1 q  ]; `+ @% H

    6 c7 h' |8 m; j4 L4 d5 L! l1 E* y2 {6 |% [# q/ p6 M
    def fibonacci(n):% t& |3 X  Q' u
        # Base cases( S  x* h# X4 {: H8 g; B9 h
        if n == 0:& `, ^: l( O8 }: }7 Q) R5 d
            return 0# z" q4 q* F1 |$ }
        elif n == 1:
    - L; X$ u2 R# ^6 c* s/ y        return 1
    1 n5 m3 j* @0 Z) p: u    # Recursive case, t2 a; X9 B% n/ y+ A* J1 c1 n
        else:0 b4 c; h7 M' \/ c' u, y0 V
            return fibonacci(n - 1) + fibonacci(n - 2)
    3 \/ X! u5 E& o8 |; s2 G, `+ P( R: A2 ~9 G* K. M2 M3 H
    # Example usage
    5 E: \/ b/ {& ^- yprint(fibonacci(6))  # Output: 8- k7 i* W4 b5 ]

    " M* C& g& s6 r7 z2 |: ZTail Recursion
    " L4 n, m" n8 {+ `/ @8 }
      t! o7 B1 p2 z$ a4 x' s2 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).
    / |$ ]% d1 H6 n2 S" N! [* U) P* u8 k1 K
    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.
    回复 支持 反对

    使用道具 举报

    该用户从未签到

    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。
    回复 支持 反对

    使用道具 举报

    手机版|小黑屋|Archiver|网站错误报告|爱吱声   

    GMT+8, 2025-12-21 10:47 , Processed in 0.037793 second(s), 19 queries , Gzip On.

    Powered by Discuz! X3.2

    © 2001-2013 Comsenz Inc.

    快速回复 返回顶部 返回列表