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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
  • 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
    % t  k* |" l. ~0 ]. g' w3 T
    : t: ?! h6 G' k9 Q4 S4 i6 S解释的不错
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    $ W0 n+ w! r  u( \5 o  P& T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 i' n; ]( v) j6 z- S' |
    ! N" P% n5 t# z9 V, X4 K" D
    关键要素  E5 I  S/ y2 r
    1. **基线条件(Base Case)**; w$ |! f" X1 `
       - 递归终止的条件,防止无限循环" [3 q+ Z% n5 \# X/ q5 C2 @
       - 例如:计算阶乘时 n == 0 或 n == 1 时返回 14 F# V; P- ~8 m# p& p  i+ A# q

    * T% U. h( j- |/ f2. **递归条件(Recursive Case)**
    * c& c$ n7 z& d& C   - 将原问题分解为更小的子问题" w0 ~2 ?4 W/ H/ }9 I2 L  ^
       - 例如:n! = n × (n-1)!8 P  w7 t0 {: r9 {) Q

    ' ?- r% ?, f! V3 B 经典示例:计算阶乘
    7 A, t, g9 L4 G# @# n! bpython
      x, S  Y  ?5 s8 Qdef factorial(n):0 H" F1 _) t4 c: f
        if n == 0:        # 基线条件9 E# t) `- J6 A3 ?+ ]2 @
            return 1
    ; R! P: b$ p" M3 S/ ~0 O# K) l    else:             # 递归条件* d" y) ^: U% V' |) x; l3 B
            return n * factorial(n-1)
    - Z+ S2 Z5 h/ `( o2 `, o9 A执行过程(以计算 3! 为例):) h6 [6 F3 A' m+ D3 \7 {7 S# d  H. t
    factorial(3)
    2 e$ P" u+ u5 g2 p) f8 n5 d6 z3 * factorial(2)5 Z7 W" M5 v" ]& R" n1 g
    3 * (2 * factorial(1)). @1 f1 s! F6 |1 e9 `
    3 * (2 * (1 * factorial(0)))
    , y$ P7 U# t1 n  J8 r2 T3 * (2 * (1 * 1)) = 6) R- X' F; D) J' K
    1 }4 Y0 `+ U6 R% {1 ?8 K
    递归思维要点3 c% E, w4 p" W, {; x. R
    1. **信任递归**:假设子问题已经解决,专注当前层逻辑
    # ~. W/ b: c! {& y3 k2 s8 n2. **栈结构**:每次调用都会创建新的栈帧(内存空间)! ]' `8 `7 X5 T& d, B' r
    3. **递推过程**:不断向下分解问题(递)0 o0 `% o: m; }& p! e
    4. **回溯过程**:组合子问题结果返回(归)
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    注意事项
    ( f6 S" `' V% m3 W( l7 {必须要有终止条件! B. x" p8 I( Q8 E" e  q8 w
    递归深度过大可能导致栈溢出(Python默认递归深度约1000层)
    - x9 l' Z, L+ ?: G* y某些问题用递归更直观(如树遍历),但效率可能不如迭代) w- U( G$ ^  y% u$ \$ q) J
    尾递归优化可以提升效率(但Python不支持)9 X) H" o  R7 }, q. }' |) F

    : i0 K( p+ R% C5 l5 d 递归 vs 迭代1 w7 X; f. ?/ u, k& }
    |          | 递归                          | 迭代               |! m) s0 d, y3 i8 e% @( K& j
    |----------|-----------------------------|------------------|
    ) ~3 k  P  T- k  @. h& A| 实现方式    | 函数自调用                        | 循环结构            |
    # F7 X6 M3 B0 x0 v% y% w" _( [* @| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
    0 Z& ^, m+ N0 u0 K, o: S& H7 N  n# ?6 x| 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         |
    & i2 V* J! a3 P- f! U" g# T| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |$ d, @! ^% E; {- }0 S

    : M5 O7 p' V  ]8 Y' v- G 经典递归应用场景
    " v' y$ y6 x$ [7 E' N" M1. 文件系统遍历(目录树结构)
    ' a- \! @+ t6 I! T2. 快速排序/归并排序算法
    " v  V' ]6 h' J" M' M* E  I. [/ a3. 汉诺塔问题- a3 O# }2 i' a+ V1 @. F, k4 U  J
    4. 二叉树遍历(前序/中序/后序)# G& R/ [" e. q0 c: L
    5. 生成所有可能的组合(回溯算法)
    8 o% L/ g3 s2 k9 x, P
    $ U; O( ?4 U' X; {' ~% e& X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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  • TA的每日心情

    昨天 12:17
  • 签到天数: 3175 天

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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,
    % W$ c8 Z) m% x$ q; u. _0 w5 e我推理机的核心算法应该是二叉树遍历的变种。
    2 L; R3 x7 F+ `  \+ f2 Y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:
    + q/ H7 b1 O9 L0 GKey Idea of Recursion8 n+ E& l: H- E. R8 F

    6 B  n( p3 ~+ G2 g1 R& [4 d( SA recursive function solves a problem by:
    2 y2 ~7 \  B3 z- @& F1 u
    0 W7 k$ `1 c3 ~( R- ^    Breaking the problem into smaller instances of the same problem.
    # O- g7 Z, w7 L6 p- r& @* G; e
    & ^& A. q/ m7 J$ X5 G/ x& z    Solving the smallest instance directly (base case).
    / [5 q9 w% ~5 q' C! U: `8 y  C
        Combining the results of smaller instances to solve the larger problem.
    ; m1 w" H  x1 f
    % d# Q  u0 l2 CComponents of a Recursive Function
    4 ]! p8 I1 T, m+ N8 Z5 k/ C$ _) H; d5 N; Y3 H/ v
        Base Case:, ]1 j8 l- y% D* i* S

    4 {- P. j2 C/ e4 F( z/ D        This is the simplest, smallest instance of the problem that can be solved directly without further recursion.
    ! i2 h' a4 a8 X  g' h4 _
    2 Y/ T9 ]" k# `  d" W+ \; e3 I        It acts as the stopping condition to prevent infinite recursion.
    . v8 V  O1 E, v  |! x3 n. J! p# T" G1 K9 o0 k" A
            Example: In calculating the factorial of a number, the base case is factorial(0) = 1.4 q5 H0 }1 q" X0 }3 I
    . R( @$ y6 R2 ?+ B% q" ?( c
        Recursive Case:
    9 D3 P1 C9 S3 T6 j
    ; p( x5 T' I1 ^% V* m9 L! Y0 A7 p6 F. v        This is where the function calls itself with a smaller or simpler version of the problem.
    8 y3 b6 g8 |4 \- z. w
    ' @/ N1 @4 }! r0 \* E& N        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
    * M4 P% l6 Z! O5 e
    + w3 Q% Y3 N0 U) P9 M% eExample: Factorial Calculation
    % `7 a) f* H0 d6 D" s. ]+ ?% `6 D
    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:
    ' ]/ i' j2 O1 o9 L. i8 [$ F! K3 ]3 c$ W: o$ ?: n
        Base case: 0! = 1
    ( n6 Z8 Y) I+ ~  g  e0 g9 P, N. R, V+ w7 @, x  @) D. A
        Recursive case: n! = n * (n-1)!
    $ R8 [- I% B' C+ z) C$ V
    5 ^% ~$ e9 j, I0 W( V: L$ @Here’s how it looks in code (Python):) `5 L/ \: t  d+ T9 @
    python
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    ( r  t. B6 o* V6 r9 q% V" g8 X# n& J" Z( j. _3 V% g! r
    def factorial(n):) X1 B& o  m/ B7 f  y8 s7 `. c
        # Base case
    ; x5 r0 r8 S* v+ x  U3 E    if n == 0:
    ) z, z/ [. R5 @- A9 p7 f1 ?* w, k        return 1
    1 T: D" d) w- B$ s) T4 U    # Recursive case
    # Q1 K  `2 ]1 ]: d1 s# m    else:
    7 e- F) W# d# O        return n * factorial(n - 1)
    6 a; ]' z# i# y* T6 F: Y
    / `) f% b  V& H# ~: J5 G9 P8 }# Example usage
    ! M0 x" E2 L5 I. M8 zprint(factorial(5))  # Output: 120
    - M' @& A1 O1 w$ k7 z$ [% u7 Z; [" k* r- G3 Z; ^& u! W5 y+ ?3 `, `7 w
    How Recursion Works# @! E# ?/ `! _+ d3 L6 h$ f. z* x

    4 V/ _- _2 P2 V    The function keeps calling itself with smaller inputs until it reaches the base case.2 w' z5 o& H, r! k2 l
    + c/ w& G4 N6 w
        Once the base case is reached, the function starts returning values back up the call stack.. A3 W5 K5 R6 i2 o# Y# v
    ' f  e/ J* d+ M+ ^
        These returned values are combined to produce the final result.
    7 y) Q) l6 ^- y1 \  w" X' y1 D7 a. I/ m* ^7 S$ O
    For factorial(5):
    7 b( x- w1 o5 u8 u- _+ u. u% e3 w; Z
    2 [9 X% u# S% G
    factorial(5) = 5 * factorial(4)
    1 x0 E! F$ M9 F  n: \; z5 Tfactorial(4) = 4 * factorial(3)
    - U+ U, e$ W/ hfactorial(3) = 3 * factorial(2)
    8 Y) j" T3 h+ m- ?8 N! C, H( F: h4 Wfactorial(2) = 2 * factorial(1)! w! H0 V$ s5 \: B2 |
    factorial(1) = 1 * factorial(0)" w+ B0 C) i4 f! }$ U( `( z$ }
    factorial(0) = 1  # Base case
    ; M4 A/ v9 k* s3 V$ L7 x/ J$ `: M1 Z- Y2 k
    Then, the results are combined:" T. N5 g: v" a, _7 A. U
    $ s7 |: y" a7 N8 g$ K- W4 n
      M; R) B( ]4 O% B7 H
    factorial(1) = 1 * 1 = 1
    3 T" w5 A# _7 m! ^* |$ jfactorial(2) = 2 * 1 = 2
    ' [) v8 E: }" B. ifactorial(3) = 3 * 2 = 63 `! Z) N" d4 `8 j
    factorial(4) = 4 * 6 = 24
    ( `- X  o+ j: Z) Vfactorial(5) = 5 * 24 = 120* K2 x$ p9 V6 v4 K7 D

    . F7 h8 Y8 W  Q( f, m$ @& V5 s$ cAdvantages of Recursion/ x3 s: p' O' \3 Q8 m  W

    0 {' x* ?8 G! A+ L0 n# Y" m5 p3 g$ C    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).
    . q, j. u; b* m/ b& h' g
    + Q% x( v0 j# X! e+ u    Readability: Recursive code can be more readable and concise compared to iterative solutions.( D1 u4 x+ P  ~

    - c( w3 d' y: q1 |9 z! `/ oDisadvantages of Recursion
    . p4 J% n" G% l8 C  S
    , q, E- Q3 F& {+ i/ ^    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.: Y% Y8 V, T  q

    9 k) Q" v/ m6 e0 E  q  ?    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
    + N- `; K* f2 F2 v: @; W  i; p" I# v7 a% d/ ^0 K; s6 F3 b1 @
    When to Use Recursion; T) @- K5 }/ b' s7 U# \7 K4 e* t
    - z& j8 i$ q' ^  Z9 a  {
        Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
    ' m% `7 q$ R) o/ \* o  b" \2 [. [
    ( v* w# z( k8 f- z8 ]7 ~    Problems with a clear base case and recursive case.4 u% X! I+ {1 h+ U! I

    % j0 A6 E0 L3 H: p; p# q, RExample: Fibonacci Sequence
    2 t5 f' H/ [; }! C4 R# F
    1 a! z* P+ [1 _) z8 H( W) LThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:
    " f1 U/ X; h0 c! T. h' B
    3 s5 b6 J) B0 X/ H" o4 `$ X% N" i    Base case: fib(0) = 0, fib(1) = 1* x* `4 C9 j" B( {6 B6 Q* y

      I- G' r2 l% ]4 R# H    Recursive case: fib(n) = fib(n-1) + fib(n-2)
    & l$ Z4 k8 ?  U: }% z
    ) Y$ Z* p( D1 I7 ~% M5 M" H! Apython$ d5 S' B2 Q8 k% H, A
    " ~# I& S6 I/ O' X

      L" v' r* h/ l# u# |( ndef fibonacci(n):3 {# |2 X- b# g+ c' ?: l
        # Base cases
    5 l) L0 I. D- q    if n == 0:
    3 O2 `) i' k, r& P: I& g        return 0: z3 c( a+ }; R1 N$ B$ x9 f
        elif n == 1:
    ' e! t' f  M/ b0 j        return 13 n0 ?! o( |; T: a  a8 ^. M2 ]& E
        # Recursive case
    0 B7 Y% }! \. g* |    else:/ ~+ X/ N1 ]( k( [+ C0 N6 q9 N
            return fibonacci(n - 1) + fibonacci(n - 2)
    . e& t; O& ^- u! M
    * y; l8 j  z: X# Example usage1 ^* S: C- w% I+ H! r5 m- G- h& S
    print(fibonacci(6))  # Output: 8
    ) K% S. P% @0 ?. q* h0 E5 x4 a8 o1 [5 P& M1 z1 P; B
    Tail Recursion2 s0 \- Y+ R7 Q/ o9 i+ i- s
    . V8 o1 h% U0 p  ^3 a! ^( T* K
    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).
    9 b2 q, Y* j) |2 o4 t/ ], n) X) {
    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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