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 9 D j% {3 k( [& ~ + P7 Z D( p+ v$ A解释的不错 7 h) O6 X$ i! l! E1 L7 C& l& u& f5 S
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。 ; Z+ k! B) C( w1 X4 a; O: V$ z3 M/ k* t: n9 `5 k; i
关键要素5 a, q; k' V7 s1 Y$ |! F9 E" a
1. **基线条件(Base Case)**# k8 F4 }8 R6 p
- 递归终止的条件,防止无限循环 ) R# q. w: q! y! e+ ]% i ~+ _ - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 0 @4 y/ b! |3 K8 I. ?2 F) a1 o- J( x
2. **递归条件(Recursive Case)**9 E( t9 a1 A" w! Z3 v$ \
- 将原问题分解为更小的子问题 3 w4 d1 h7 W! n9 ~" k9 a$ \3 ] - 例如:n! = n × (n-1)!! e$ z+ B2 A; W# V2 K
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经典示例:计算阶乘( N% ]2 y( X& F- W# p! ?7 f
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def factorial(n): 1 L- x Z8 M1 p if n == 0: # 基线条件( l. }4 V: Y" A" e b. h
return 13 y* i9 J. J2 @* k& |
else: # 递归条件& t2 g( c5 n: O3 l3 ?0 W2 b
return n * factorial(n-1)4 D9 y3 j: ?/ }1 Y8 ?
执行过程(以计算 3! 为例):! L! m% ?8 w# P8 p6 A, z% i
factorial(3) 1 \! `9 l. b9 g6 L3 n3 * factorial(2)' U- o3 \' W- h; c0 p0 i9 q7 A
3 * (2 * factorial(1)) & c G* a: S9 d# }, V: S3 * (2 * (1 * factorial(0)))% `" }9 D! r0 f, G) {
3 * (2 * (1 * 1)) = 6 * [+ r- a0 `+ ?- b4 W1 u; e9 H% e- L
递归思维要点, |' G9 L& J8 ^" }0 [7 u8 c
1. **信任递归**:假设子问题已经解决,专注当前层逻辑 ; L/ x/ R* @/ }7 N3 m+ }4 s1 p4 {( n% G2. **栈结构**:每次调用都会创建新的栈帧(内存空间)" J3 B; Y9 \% |" u9 N) Y+ F
3. **递推过程**:不断向下分解问题(递) ( I \( ]; s) _4. **回溯过程**:组合子问题结果返回(归). G/ h7 R: {: W: _1 a, C* ^
9 Z# R2 i( u- h& I. i' D' `5 l 注意事项 . m/ H# L" Z3 r/ C" ~; B必须要有终止条件 7 t! l' W( w% w2 |& J/ l8 K1 e递归深度过大可能导致栈溢出(Python默认递归深度约1000层) 7 K/ D7 ^4 T& j J8 y某些问题用递归更直观(如树遍历),但效率可能不如迭代 8 |2 s4 i: i( _3 u- G尾递归优化可以提升效率(但Python不支持)8 a+ f y4 `% d. k! w. v" o
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递归 vs 迭代5 u, y+ I. U" K2 t
| | 递归 | 迭代 |8 s; n0 D, d5 ^. q6 D/ j% ~' y; J
|----------|-----------------------------|------------------|# K+ {* F5 `7 m \
| 实现方式 | 函数自调用 | 循环结构 | % N7 s t7 i3 d. w| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | ' U8 B1 U7 ?2 K3 H0 y| 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 |' c8 j% U* f6 _. N: l
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |% h4 m% J" a# @! t" K8 u* F- z/ n
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经典递归应用场景5 h7 l$ M4 O, H4 o
1. 文件系统遍历(目录树结构) + K4 h F' r0 l9 k; C' ^; I2. 快速排序/归并排序算法6 g: B/ }" _. f
3. 汉诺塔问题 + _0 u2 @$ V9 e1 _# x) e* q1 o8 B9 j4. 二叉树遍历(前序/中序/后序) + B6 P$ g% k2 f0 x5. 生成所有可能的组合(回溯算法) - H0 d, V: s& v: x! H# D+ ~5 n" }0 w9 A0 ?8 q5 j D/ p
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,9 P% B3 H: m9 f6 r- N* @# ~
我推理机的核心算法应该是二叉树遍历的变种。. H! w0 t5 b$ W
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: 5 W# {: d# P J0 E. ZKey Idea of Recursion 5 N, I0 b% p2 v; l n; R ; r/ f0 I; P9 S3 a! V9 p- |A recursive function solves a problem by: 1 s6 O. i& C W 8 w$ [& J$ y* _* q/ ^ Breaking the problem into smaller instances of the same problem.: q2 j; P9 t- m/ T
9 R: _* v# |7 j' { Solving the smallest instance directly (base case). 7 B' x) I, n4 P! q+ t7 c' U3 X/ [ ( _4 K! u9 r( U Combining the results of smaller instances to solve the larger problem. : }( w4 r' E& Y8 W . z. u3 f5 B9 `3 D1 I0 E, [6 dComponents of a Recursive Function( m7 k( }3 A2 s( m7 {! D: A
( k" q* p) ]! C- t9 F" g( u! n This is the simplest, smallest instance of the problem that can be solved directly without further recursion. J- n {7 Q% y6 L; k" K8 y( b
! h) w$ t4 y& M; N# k5 W% x It acts as the stopping condition to prevent infinite recursion.! r3 j9 m1 \% [; `# P; ^* ^
0 T9 V" U4 Z- r Example: In calculating the factorial of a number, the base case is factorial(0) = 1. + `" G# N5 e1 j" x9 l. H' W 3 S0 I( @+ t/ y Recursive Case: & c4 ~- X* \! b% q$ T1 ~# @ 4 J3 M& q0 N# S& c) c This is where the function calls itself with a smaller or simpler version of the problem. / b& I" L' ]5 O0 L% {# V# M 1 ~" H' E) J# ^3 U: Y0 L0 B Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).0 t1 R2 N# ]" U; d
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Example: Factorial Calculation 8 t. }$ Z3 y; h+ I1 `* f ' G6 D# W( a3 f7 _" @' q- G; wThe 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: 2 w# H/ t; |! }+ N % a& n) v( r- o6 L5 `. ` Base case: 0! = 11 Y4 _/ @$ F/ o8 r) R. D8 a( f
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Recursive case: n! = n * (n-1)!* s5 L- ]4 p5 p, L0 b+ g( E
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Here’s how it looks in code (Python):8 x" Q B6 V# |# v) w1 v0 s! [! c2 R
python 9 }$ ?( u8 }5 b: @* G. X e; q+ V# z$ L# J; U& X# I5 T$ N x
# e7 ~) C8 b# f! W6 H) ?def factorial(n): & r9 v; `" p- U }7 ~; } # Base case ! X: d1 ]$ C' J2 W. ` if n == 0: ! `' {5 y7 t) R: t return 1' J" R2 ^ e. l# K
# Recursive case . Y t" Q3 G! K S' U4 P else:* S$ n& |2 l2 V# V `0 v" H
return n * factorial(n - 1) % l" @% a- o- G; z' Z1 V- o1 F+ O7 M ( t! P, Y" g9 e) U& u- p9 Q, y: @# Example usage/ @. a: s/ N0 [' Y, R1 [# p
print(factorial(5)) # Output: 120 ) i- W J/ j1 c i+ p 8 o' u# e, [2 rHow Recursion Works }; ] G0 ]% N$ j& Z' t7 {) e1 T ^6 {+ L) Q
The function keeps calling itself with smaller inputs until it reaches the base case.( i% p/ E* j$ x K4 x
) O: V3 O1 i7 |0 A$ h Once the base case is reached, the function starts returning values back up the call stack. ) V0 ~* B0 w a8 B- @! p, y1 n# z- f* d. P# W4 X
These returned values are combined to produce the final result.) K" t+ b. Y7 A5 Q6 t/ ?2 S
; [$ _; W4 Y- r. g) d9 I* x6 I 2 S4 ~6 K- y+ s( I6 zfactorial(1) = 1 * 1 = 1$ f7 L0 I' s9 K, _1 G/ Q7 {
factorial(2) = 2 * 1 = 26 ]. _5 W/ r7 Z9 O# D
factorial(3) = 3 * 2 = 6 / M5 I# t3 G" k( u# x9 nfactorial(4) = 4 * 6 = 245 x: ?) x7 ]- N( V$ w( b: _/ Y
factorial(5) = 5 * 24 = 120 0 e5 g( Y5 R* [, {( W; b4 Y$ I$ Y4 ~5 V( w6 j
Advantages of Recursion % ~& ?* `" ^( g" R+ E9 @0 T& @2 m
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). 1 J( Y1 D2 E; q# r/ t2 T' _! l. [' e8 l( c* f
Readability: Recursive code can be more readable and concise compared to iterative solutions. 7 F( Q% E! Y: ]# ~" A( _/ C' q. E# u0 ^- Z: L8 Z) u
Disadvantages of Recursion % `: b7 i+ ?& v3 A! b' @! I/ ]4 A/ I5 U0 P; |% b' s
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. \% ^5 V, \) v6 n! q$ B
O; V! }6 j0 @7 B3 J/ e; Q. Q& o1 P Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). : q& V& I$ Z# i" t' n. p( @3 o a7 B
When to Use Recursion ; }& V" P/ c' A- a' j( F* ~7 E8 m2 [# e8 z
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort).0 K" o" Z9 V- @6 e# x
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Problems with a clear base case and recursive case. $ L. o2 f7 L8 l$ y$ `# C, n- {# j- R. t0 ?
Example: Fibonacci Sequence 6 v9 z# e9 A. h7 k/ j: \, _2 f% n8 n. {8 N; c+ j& _ H. p" b
The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: + O: o- S6 h% l O : M- H! ^' A' o0 M4 f4 D& n Base case: fib(0) = 0, fib(1) = 1 ! v0 R/ j1 V$ ]9 n( W+ \1 H! `) _ / v3 c5 r9 }2 t7 G+ ~ Recursive case: fib(n) = fib(n-1) + fib(n-2) ! b0 ^2 x6 l/ S 9 R; A" T0 a lpython" n3 g. X" @. `3 r
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% `0 z* `: C* g& Zdef fibonacci(n): - u: Y8 i$ r! C% M # Base cases 1 m2 }- d3 Q# a- d+ [+ [0 b; x' ^ if n == 0: H% j5 X/ i4 I5 T! X3 s) V
return 0 ; W; c; E- d4 p elif n == 1: X9 Z+ F! h) z# S return 1* X- G5 o4 A0 T9 x0 t' F. p1 b" R
# Recursive case# M9 c4 c* L8 q
else: ; L( G( p& _! p9 Y return fibonacci(n - 1) + fibonacci(n - 2) p* p8 Q# [ u* e0 m' d2 J+ \& A+ m
# Example usage+ X8 I9 C( i7 T
print(fibonacci(6)) # Output: 8: F1 a; k9 P& ^3 L
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Tail Recursion8 A# P3 `7 j! K- E5 G' J! k
6 ^' A2 Q0 X. a& 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). + @: r2 F! W7 d" X, W " b6 x; |6 i* b2 c9 JIn 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。