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标题: 突然想到让deepseek来解释一下递归 [打印本页]

作者: 密银    时间: 2025-1-29 14:16
标题: 突然想到让deepseek来解释一下递归
本帖最后由 密银 于 2025-1-29 14:19 编辑 $ r) g; r+ r& k3 a5 M: \

' u3 P) ?7 y+ Z6 |# [解释的不错
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6 T. R& _; @0 h# |* L6 [& r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。
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关键要素, `8 G! J2 b3 s% K5 f4 \
1. **基线条件(Base Case)**
% I0 T0 [! _3 A% \( }3 i& b   - 递归终止的条件,防止无限循环. J9 R& z8 ]7 z4 F/ H. r, p
   - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1
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% I- i, Q- g6 w6 B' z8 @2. **递归条件(Recursive Case)**
" P1 ]. j7 ?8 e   - 将原问题分解为更小的子问题
3 G9 g. ]' s  |1 U+ A  S   - 例如:n! = n × (n-1)!
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3 _; \3 \9 I; U: ?9 f1 x 经典示例:计算阶乘
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def factorial(n):
# P6 B, a6 h) y    if n == 0:        # 基线条件
4 L' G( j) I( h- W# u  z        return 1
- Y3 x0 `. W8 x    else:             # 递归条件
% g; H% l; m; q5 z; M        return n * factorial(n-1), F6 \6 E9 |% ?( {) H2 F
执行过程(以计算 3! 为例):
* h) S* d0 v  ~, i0 g* j' afactorial(3)
  t+ O  s4 u: r& D3 * factorial(2)
: v& V& f) v) \1 V6 V3 * (2 * factorial(1))0 s6 F* ~2 M/ F0 s
3 * (2 * (1 * factorial(0)))
9 ]) R  W- L2 S4 q3 * (2 * (1 * 1)) = 6. U3 O4 h: v9 t9 p5 e

5 O6 W3 |4 Z; N+ x+ Z0 P: v 递归思维要点
. r% \8 l9 J! I2 X7 W! U2 s' f1. **信任递归**:假设子问题已经解决,专注当前层逻辑2 ]% h# t* D# l$ U5 [$ I
2. **栈结构**:每次调用都会创建新的栈帧(内存空间)0 \, K9 a& ?& R# r
3. **递推过程**:不断向下分解问题(递)( J$ c! E1 Z" u7 V' }
4. **回溯过程**:组合子问题结果返回(归)6 p& ^! ]6 D- ^$ d* o" N/ Z1 B0 g
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注意事项
8 C  I& \% O+ Y" b必须要有终止条件
! h/ f- C9 P9 j递归深度过大可能导致栈溢出(Python默认递归深度约1000层)5 v0 m* S) R% L" g6 b; X% X( A
某些问题用递归更直观(如树遍历),但效率可能不如迭代% e' y# y' R$ e' y: U+ O$ g" A4 J) P
尾递归优化可以提升效率(但Python不支持)
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& ~; f( b) F6 E$ p 递归 vs 迭代
1 w6 l- s- I0 f; @5 ~0 {# `|          | 递归                          | 迭代               |
+ D" ?* N9 J" ^% R8 m5 _|----------|-----------------------------|------------------|
# |: z* N4 j) s& b$ i: L4 K| 实现方式    | 函数自调用                        | 循环结构            |/ |' P! l$ J% G7 G- B/ ~
| 内存消耗    | 需要维护调用栈(可能溢出)               | 通常更节省内存         |
! x/ Q8 i2 y' y% R$ d5 @* }$ ?# F| 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         |
. I* w# e" J' j& y6 _| 适用场景    | 树结构、分治算法、回溯问题等               | 简单重复操作          |  l/ j" w# Z; r: G
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经典递归应用场景1 L, q! Q& [2 J* y
1. 文件系统遍历(目录树结构)' `9 z8 g/ F1 H% ^2 C/ a! m0 R% n4 Z
2. 快速排序/归并排序算法
9 k5 d. b1 V, Z8 E$ k0 v4 ?3. 汉诺塔问题/ Y7 _8 z: k1 w7 A' [
4. 二叉树遍历(前序/中序/后序), V" N: Z; r% P; ]
5. 生成所有可能的组合(回溯算法)
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; M& G% P7 {5 @" g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。
作者: testjhy    时间: 2025-1-30 00:07
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/ [7 K% s4 m# Y8 y" i
我推理机的核心算法应该是二叉树遍历的变种。
4 `, d6 Q  K. i  k( b/ e/ o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。
作者: nanimarcus    时间: 2025-2-2 00:45
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:
# J8 q% V* t: s) ?8 s* jKey Idea of Recursion1 `- D4 U/ \+ O9 J' Y

. F1 U' e3 [0 N; Z% m: UA recursive function solves a problem by:
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    Breaking the problem into smaller instances of the same problem.5 y# \* I4 ^. U2 L7 x; C% M0 U

2 b' V& k7 F3 P& D6 D3 o    Solving the smallest instance directly (base case).
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: c1 G4 v" z0 T" N8 j: `    Combining the results of smaller instances to solve the larger problem.
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Components of a Recursive Function( \+ }( F: I$ p( l" ]7 l/ t* a3 b

4 ^% {6 j, n* p6 P; w2 U% c    Base Case:' N8 N& K7 O  @* |; }# |0 R3 P1 Z# |
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        This is the simplest, smallest instance of the problem that can be solved directly without further recursion., T& Q4 [: z* O& y$ D

6 }/ C: Q3 K, w* w        It acts as the stopping condition to prevent infinite recursion.. U* m" X) C8 u( l
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        Example: In calculating the factorial of a number, the base case is factorial(0) = 1.
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5 z, v9 T5 T0 F3 E* u* q7 i    Recursive Case:
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3 F' W* Z) u9 {) d) N0 @        This is where the function calls itself with a smaller or simpler version of the problem.6 `$ r& L0 }* I5 J3 ]6 |7 x' j
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        Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).
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Example: Factorial Calculation3 ?) F( O5 @* K9 g; G8 K

* a7 v  P. v, m' ?7 aThe 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:
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9 ?! c' x5 @# l' n0 r% A; ?    Base case: 0! = 1
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' a8 f& |+ J$ V! z& ?* R! W. q    Recursive case: n! = n * (n-1)!
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Here’s how it looks in code (Python):4 i# J" m8 \& ~6 j; N0 U& Q4 E" _
python6 h/ ^' e5 S+ U% y9 V% }8 @5 m0 j0 F
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5 u9 |) D0 ]* Udef factorial(n):
2 K" {$ [4 J, D3 ~    # Base case
" Z+ e( L9 R" u$ d6 ^- y    if n == 0:" ^% J. N9 Y0 }& H+ n1 c
        return 1
7 i6 p* ?4 W6 o$ U9 r    # Recursive case! Q& V  H" w( ?# T* y' _4 K+ {
    else:4 B( l7 a5 t- {- c4 u1 T9 X
        return n * factorial(n - 1), M( a6 s, t5 @5 x8 q& k% Z
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# Example usage
& O, l! h: w! r: t2 b  @  g" gprint(factorial(5))  # Output: 120" M( {% c8 `; ?3 E9 u

% \! w+ n6 w' s3 iHow Recursion Works
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    The function keeps calling itself with smaller inputs until it reaches the base case./ P- ]# r! p2 F+ c+ d: \  x0 W
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    Once the base case is reached, the function starts returning values back up the call stack.# F3 }6 v7 K8 J, U- K3 A# Y# J
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    These returned values are combined to produce the final result.9 W$ c0 o" ], A$ I) j; q% n
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For factorial(5):
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factorial(5) = 5 * factorial(4)1 R, h* C6 T, m3 g; F
factorial(4) = 4 * factorial(3)
" F. F+ A+ l( p# D& Vfactorial(3) = 3 * factorial(2)$ V( E0 G* l7 m7 i9 ]
factorial(2) = 2 * factorial(1)* j- n  m1 O. c* Y
factorial(1) = 1 * factorial(0)2 S5 i  H/ R, y  @5 D
factorial(0) = 1  # Base case
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, h: v9 X# R5 D/ OThen, the results are combined:
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. n, O4 L6 y( k+ n* D* O" P% Sfactorial(1) = 1 * 1 = 1
% q# u$ Z( K) C, q. \factorial(2) = 2 * 1 = 2
) _: q& `; H& X, c0 }5 ]factorial(3) = 3 * 2 = 6
% i/ c" E1 n2 j& h4 J7 @0 `factorial(4) = 4 * 6 = 24: g2 t* q' z% Q. Z- n( p! J% _
factorial(5) = 5 * 24 = 120
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: k. U% c, q& v" @Advantages of Recursion
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    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).2 I" ~  K) s2 L, h& G3 W* \& }

& y: J- x( [, @# o: |0 T5 L    Readability: Recursive code can be more readable and concise compared to iterative solutions.
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Disadvantages of Recursion% O$ ~6 k$ T' z) V7 }  @

0 L" a( V9 u0 r" W2 h/ b    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.! y1 v& q$ W, N) _( o! A
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    Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).
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4 G! m3 P6 x+ z8 t" t4 F1 M9 kWhen to Use Recursion$ [$ A3 B  ]$ R1 R6 g! V% V

7 y* P. k8 m8 Q4 }. _    Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).
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0 c) N: s' E3 L+ ]# c! \7 x. |4 D    Problems with a clear base case and recursive case.7 q+ z; p1 z( p! g

0 F0 j7 W) O/ l+ G8 e2 v) ~Example: Fibonacci Sequence
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2 K& I7 W! w+ n" O* F1 rThe Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:8 f9 {& E8 X% s& U4 r( t& P
+ `$ p; [, P  ~/ i" R8 L7 p3 t8 p' t
    Base case: fib(0) = 0, fib(1) = 1
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    Recursive case: fib(n) = fib(n-1) + fib(n-2)
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# Z* ?3 {% ~6 J: Z* _3 gpython: T) y: J9 ^7 M1 G

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def fibonacci(n):7 W0 W/ V6 @3 F- z/ B+ n) }0 ]6 }: n
    # Base cases
! m9 \4 o1 b' X" C$ Y    if n == 0:+ r7 M9 S$ I5 o* `) _8 P; n( t- O
        return 0
; |' U* {0 X! ^1 y% h7 b    elif n == 1:
1 Q& `6 U! Q" B3 M/ D1 y" m' a        return 1
2 z" L0 }4 m2 X* v* w3 U, ]. K+ ?    # Recursive case
$ {3 b1 f6 `& l5 F7 h    else:* E# [1 P- E: k9 f$ h
        return fibonacci(n - 1) + fibonacci(n - 2)
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! L# d8 I" H1 G# Example usage+ U; ~8 z" i* I' A
print(fibonacci(6))  # Output: 87 B# j3 C2 M* e) x1 a9 h2 `  p! y0 \

( [: t% _+ d# j4 M  b8 E, D& ^; ~Tail Recursion
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, s% u) {% `; ?& eTail 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).
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# j: Y6 d; B" ]! \/ n8 k! d. p: ^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.
作者: nanimarcus    时间: 2025-2-2 00:47
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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