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 `7 j5 T, u; C! K
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解释的不错+ |+ Y; e8 D$ I) H
: Z9 M! a0 o4 r% y7 L1 t- F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。 1 W* U% ~' I; t8 j7 g3 C' q . E8 ]; e# Y% |+ L5 Y% g F- v4 D 关键要素+ |) L5 Z3 U$ ?: H3 ~$ W7 M9 j5 o2 @
1. **基线条件(Base Case)**) F! G+ o1 p9 q
- 递归终止的条件,防止无限循环 ) e9 b/ f0 T- S/ N. { - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1, U8 l A- c4 v% H* J0 z
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2. **递归条件(Recursive Case)** 0 K+ ~9 u d: T4 D; F - 将原问题分解为更小的子问题; w% F9 t. w8 a& i% j( J. M
- 例如:n! = n × (n-1)! ( Y5 w$ T2 u4 R, U* N ) T! \) e5 H6 R& L7 p4 J7 F- d 经典示例:计算阶乘 6 p# @6 u. ]: X, h! Q" ]* v( ipython # _: s6 A& O6 P% H6 xdef factorial(n): - s3 Z! @ a7 p$ g8 e0 S if n == 0: # 基线条件* U" o; y+ T1 u
return 1 ( b% F4 o2 l& x; N* ]$ o. F5 B, a else: # 递归条件 6 g9 A- b1 }; Z return n * factorial(n-1)$ s+ s. D( C, q' F# y- H, b1 U9 T
执行过程(以计算 3! 为例):0 `- B6 _9 S& T" u' Z' X
factorial(3) - L% v- Z; v/ E- N) P. R9 `- `3 * factorial(2)2 p" f8 _9 j) }# L
3 * (2 * factorial(1))1 O; W5 T& G U: H3 W
3 * (2 * (1 * factorial(0)))* B) z1 e* [2 |! C
3 * (2 * (1 * 1)) = 6* b: T1 m9 n/ T: i6 S) y: ]4 s
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递归思维要点 3 T( ?' |7 ?- Z6 r4 I0 j- c$ x1. **信任递归**:假设子问题已经解决,专注当前层逻辑 - J+ T. G( o b2. **栈结构**:每次调用都会创建新的栈帧(内存空间) W f- j- U4 n' }3. **递推过程**:不断向下分解问题(递) 4 s( j+ e \. k! a' i U* F9 m4. **回溯过程**:组合子问题结果返回(归) & t& [% G, u; w Q5 H8 D $ g0 W* D8 p! ^ 注意事项2 V: U: I, o7 ]3 J* L' F
必须要有终止条件 9 p u, s/ h$ l3 F& Q/ X递归深度过大可能导致栈溢出(Python默认递归深度约1000层)" S* g' i1 o9 g0 ], C/ Y1 M
某些问题用递归更直观(如树遍历),但效率可能不如迭代5 E- y. D3 j9 d0 b8 b/ s0 Z
尾递归优化可以提升效率(但Python不支持) % v6 \2 G) d: B- n `# u S# M: ?7 e) m
递归 vs 迭代8 a8 E9 d2 l. L
| | 递归 | 迭代 |0 T- }7 p/ F: M
|----------|-----------------------------|------------------| 4 {! o% n& r) D! l, C| 实现方式 | 函数自调用 | 循环结构 | : c) v9 r5 R: o* x) d" F% q7 W| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | / u' }- C0 g" h" u- I| 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 |8 l5 E$ U: f+ p+ H9 r3 z
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |/ v4 @3 z! v5 T: m
% m* z* F% L- u8 G M4 i 经典递归应用场景$ ?1 E$ h9 Q" c2 C
1. 文件系统遍历(目录树结构)$ d% {& K$ D+ r: D
2. 快速排序/归并排序算法0 |6 E [$ t5 h
3. 汉诺塔问题8 U$ D$ }8 y2 g* p
4. 二叉树遍历(前序/中序/后序) & P- w& _/ y& o$ V5 D5. 生成所有可能的组合(回溯算法)( ]) `4 O f1 V% l9 Y1 W- ^2 n
' f' D' _. V6 o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, 7 k2 F( i( ]' ?' D8 ^9 Y) y. f# A% s我推理机的核心算法应该是二叉树遍历的变种。 5 X. ]# P/ B6 \; 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。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: ' {# H$ v' Y: D! v0 dKey Idea of Recursion' l3 f7 o4 P9 e& Z$ i/ w- b2 u
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A recursive function solves a problem by: 2 N7 E; I; Z0 W$ y ! ~8 I+ h6 a- m Breaking the problem into smaller instances of the same problem.5 w# K/ i2 b- v7 i) F5 f
! E' H: j$ K L, c Solving the smallest instance directly (base case). ' ^8 M- |' E( w / S0 U$ Z0 Y7 t! |+ |8 C Combining the results of smaller instances to solve the larger problem. # T5 ^ K0 L3 p! W- e& \ . a* M2 T1 o/ u% {# i: b: m6 BComponents of a Recursive Function9 N1 T. `& Z8 N0 R( f, b5 [
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Base Case: . K$ h# I0 R v0 c$ h$ a$ z8 F % F6 |' B! w( h' A This is the simplest, smallest instance of the problem that can be solved directly without further recursion.5 R" S, L4 n# S2 a' M+ \: v
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It acts as the stopping condition to prevent infinite recursion.6 T" q, y7 x& n. n' p
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Example: In calculating the factorial of a number, the base case is factorial(0) = 1.0 y( k6 o( h E8 X0 G
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Recursive Case: 7 u# v5 [/ i+ ^* W: {9 w 2 w/ r( G4 r9 I4 f This is where the function calls itself with a smaller or simpler version of the problem.- b' F6 j; n- O
1 T6 O; Y2 N/ d4 m4 d, w Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).3 F! H9 i) s' ~1 M
% B2 ~; L& q! d# fExample: Factorial Calculation 9 P8 O; Y2 d' |! Y$ C" n F* [9 |! I. \/ _: r x0 ~( B' X# E3 y
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: - w$ R) V, [, ~) i0 N" r+ K 1 X8 m& F6 e% g/ C Base case: 0! = 1# X8 h8 A) P: y9 ~
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Recursive case: n! = n * (n-1)!1 l5 r$ M7 E q! j3 X$ J8 [
' t5 r. f( D" t2 }0 O* v, [( w9 T# x. N9 V oHere’s how it looks in code (Python): * ^5 N) e0 E8 S$ b3 E4 x# D! Gpython 5 ^0 [" o. Q a. N" \: J& T * S. H; q9 J3 }) _ ( O3 o3 |0 ]/ D" Fdef factorial(n):" V+ S; q! G. `9 u
# Base case7 { @5 }; `$ b1 n3 r& T
if n == 0: : k2 v! d! u0 P; S return 1 # a/ K, a/ M7 c # Recursive case, ~/ u7 P9 o* O+ v# v. H+ ~8 ]8 h& i
else:7 o' I8 o; L6 s& x
return n * factorial(n - 1) 0 G+ z" _# o5 P4 @$ Q# f" l5 E+ c; |" b7 C# a, \- o
# Example usage! T1 T4 |, \ Z3 p* `, i/ l. {
print(factorial(5)) # Output: 120 0 B, X c" h% g ; [3 c: F! T' ~2 @1 O$ WHow Recursion Works 5 g! i- T# N3 u( b0 ^6 Q+ ^2 U' k
The function keeps calling itself with smaller inputs until it reaches the base case.. s- P# A1 }6 P) F
% l8 H! n; M3 ^) ]1 k A; Q( S Once the base case is reached, the function starts returning values back up the call stack. ' l2 L8 N/ ^' G' T 7 d! g7 C' S- U0 i, l9 \- D These returned values are combined to produce the final result.% D9 n: [0 }: K6 r
X+ [# z! i$ z: s" ]2 r& |+ Nfactorial(5) = 5 * factorial(4) " Y5 p# q. y. l* efactorial(4) = 4 * factorial(3) & B+ V, T- ~# x8 Lfactorial(3) = 3 * factorial(2) 9 B. B x; {* i; C0 Y* n. }! Xfactorial(2) = 2 * factorial(1); Q( s7 v+ x7 X$ y
factorial(1) = 1 * factorial(0), p% u4 g' C0 G2 e
factorial(0) = 1 # Base case9 g1 Q$ T; f/ ~( d8 p
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Then, the results are combined: , k1 \8 W+ r8 l+ l* h$ `4 u8 C5 f6 U5 L
4 C' C5 Z5 f7 c3 g. yfactorial(1) = 1 * 1 = 1 / m3 y; ` L; B' Cfactorial(2) = 2 * 1 = 2 - x8 \7 i4 p5 l' Ufactorial(3) = 3 * 2 = 6 3 s/ E- O. E- g. b' L M- _$ [7 U" Wfactorial(4) = 4 * 6 = 24 , v- Y W5 n, r0 z) \ Ufactorial(5) = 5 * 24 = 120 0 N9 \1 r4 Z3 G ! n* `; \4 b; ?5 ZAdvantages of Recursion5 x, y' Y9 ?! t( d
, d$ z3 p. u8 k 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).( d7 A8 J8 z4 V3 c' x
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Readability: Recursive code can be more readable and concise compared to iterative solutions. $ q4 S$ Q# x( ]- L e, P I) ?0 _; t3 v4 X8 rDisadvantages of Recursion $ q( l8 t# B2 Q1 z8 R 5 }& Q% Y' ?- L3 i1 z6 k' | 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. 2 j: [- A" B" g7 g m! Z- v- T' D% Y ! O I0 i( t7 T. C" ~ Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization).# a! V$ ]1 v! q0 S7 M; F3 L
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When to Use Recursion6 w% \+ G# K E& v6 T* C# {3 F
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Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). + {$ ^8 v+ }$ R" `5 \# ], Q5 M3 S2 f" {6 Y2 B! l. O- g! ?
Problems with a clear base case and recursive case. - R: S4 _. F% d; x3 M5 }7 i4 Z" Q; N7 K& |6 ?
Example: Fibonacci Sequence; s% w5 k4 |% F d9 r8 [
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The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones: 0 A- e. F; ?( ^7 K( }" E( u. ~5 H/ ^1 x% K
Base case: fib(0) = 0, fib(1) = 1 1 m" K" }: Z! C8 z) g* E3 d' U - e9 o5 [8 V* o" k" Y Recursive case: fib(n) = fib(n-1) + fib(n-2) - Z- R) z/ n8 y. O, f) \2 O ( X, ^# H$ w/ |5 Wpython $ Y" O3 d1 f. i2 M % L( F9 o' s* w E& h1 `- Q- a 6 B- q8 Q: v" p$ s9 {def fibonacci(n):$ \ h; ^ i7 e; T
# Base cases - ?- [6 X- k; s$ x; m if n == 0:2 h* h* e: g1 e1 l- S
return 0 , d. ~1 \$ ]; j% g! S( A elif n == 1: 4 h1 M' u1 J* E; B1 [5 X return 1 $ z: u$ n+ v$ ^+ z4 Q1 s7 T # Recursive case : ?) z; Y, p1 ^* g0 s" \ else: 0 s, F5 a" S9 Z9 y$ q return fibonacci(n - 1) + fibonacci(n - 2)1 d$ o U' s5 f
: U( M M5 w% A4 Q9 L# Example usage9 u- P4 p0 `% m: P
print(fibonacci(6)) # Output: 8 # L z7 @ o2 C8 B7 J3 Z6 _) R 7 r( b3 ^( I& \1 M7 x' o3 A+ U' ~Tail Recursion / z1 N; z$ t1 v& _! ^9 B5 Z! a0 u! z) ~2 T# K% m
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). , h! @# L6 y: H' u & k5 `6 A% N8 }4 fIn 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。