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 1 Q* Q. x0 C0 q0 w7 Q& F 0 x. Q, @+ ?( a6 l4 _" g解释的不错1 J4 E/ t8 v1 ~
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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。 ( _6 R/ z3 r% Q4 ^4 L1 h! S' Y, H9 C# p" s9 P x
关键要素 ! |. l3 q0 u; _- r4 {/ \1. **基线条件(Base Case)** 6 @, U( }$ ]( ?* G9 B1 ] ]+ U - 递归终止的条件,防止无限循环 * K+ ?- X% @5 h/ T - 例如:计算阶乘时 n == 0 或 n == 1 时返回 1 . ]- K( H3 k, j0 `3 l, c 6 u) ], V. ]& E5 u2 z, q: f; x1 P2. **递归条件(Recursive Case)** $ Z( s1 x( w+ R9 M, Q - 将原问题分解为更小的子问题 + _0 q/ z( R m, | - 例如:n! = n × (n-1)! 1 X8 j5 b1 @4 P3 I( h. A( M) b' l2 u2 v s9 z9 R0 L5 V! Y% k/ u Z0 k# u
经典示例:计算阶乘; s( k2 `6 i. @6 ` Q# ?' B/ M
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def factorial(n):5 }, ^' \* \6 `- ~. p1 w8 _
if n == 0: # 基线条件 ( s, R( P# Q% U1 K/ K return 1% H" k# `$ B- w+ Q3 W+ g
else: # 递归条件 3 t8 o' Z$ j0 R: i: Z3 k return n * factorial(n-1) ! }; B: V7 D# p# u" O执行过程(以计算 3! 为例):! f- A3 D9 O- G- J6 [9 T
factorial(3) ! M B/ r. A o1 S, R; E4 c3 * factorial(2)# u! p+ l3 @8 G8 L+ f
3 * (2 * factorial(1))% B5 @1 h8 M. R( B( V
3 * (2 * (1 * factorial(0)))7 M/ L3 k3 S! w3 k- a0 y* b
3 * (2 * (1 * 1)) = 6' ^4 H6 c' @/ w8 M( `: H
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递归思维要点 0 k" S& b& K' N1 T" V1 O' p1. **信任递归**:假设子问题已经解决,专注当前层逻辑5 B- O' R& z! }' p2 n; u: u; v- y
2. **栈结构**:每次调用都会创建新的栈帧(内存空间) . a$ R2 w- o. y8 w6 _' R0 ~3. **递推过程**:不断向下分解问题(递)8 ^1 M& H! L" p2 `* a8 i9 l
4. **回溯过程**:组合子问题结果返回(归) , H4 h% ]# k- W2 i4 A% V % O% Y6 P1 r/ k P r0 X" C 注意事项 * E2 d, g/ G# X. x2 h6 {必须要有终止条件 7 X+ K8 u4 @3 v j递归深度过大可能导致栈溢出(Python默认递归深度约1000层) - M8 I0 Q6 c- k* {( }' K某些问题用递归更直观(如树遍历),但效率可能不如迭代 7 F7 C) w# s" {! ^尾递归优化可以提升效率(但Python不支持) 9 N+ Q0 J" \& a% o& E+ K. t, r0 V' G4 j4 S# Y3 S( A# S6 U: }
递归 vs 迭代 - N1 C$ K" S1 ?) [* E| | 递归 | 迭代 |* P# w9 c" g# r
|----------|-----------------------------|------------------|3 u7 P: n* m0 f& `. \2 j
| 实现方式 | 函数自调用 | 循环结构 |8 S- H7 W. T6 \1 F O3 ]; x/ S
| 内存消耗 | 需要维护调用栈(可能溢出) | 通常更节省内存 | 3 g) s# x2 U4 X6 W| 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 |' b$ w3 B% q1 `1 J' Q
| 适用场景 | 树结构、分治算法、回溯问题等 | 简单重复操作 |) y0 w/ Q0 t! K' b/ p6 H2 ?2 D2 Z P
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经典递归应用场景: W# B. k4 s1 u" j2 B; y0 [( f
1. 文件系统遍历(目录树结构) 2 @) g# e2 o: g. K4 {2. 快速排序/归并排序算法 # f0 i: g7 Q7 |# |7 B) }# Y3. 汉诺塔问题 B/ \# E$ y$ L+ U: ?# U4 L* j4. 二叉树遍历(前序/中序/后序) + i6 ~4 ^, s* `1 c* L5. 生成所有可能的组合(回溯算法)$ n& k J O8 Z- `" Q
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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,$ N8 t* X' m/ V v. V# C
我推理机的核心算法应该是二叉树遍历的变种。& F1 [) y+ v% i& n$ h5 C- 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。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: ! c5 t9 A, `! xKey Idea of Recursion ! X3 _% G# a' _" G% I ) Q$ @4 C- s, z/ s6 qA recursive function solves a problem by:: a# u) S) |6 W0 ~5 v7 V
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Breaking the problem into smaller instances of the same problem. + b: @0 ~2 b% L: g) E$ Y; b! e* r1 d" Y
Solving the smallest instance directly (base case). ! G5 T$ l5 w& `$ Y: k9 ^! d 6 A' t+ B) V \" P7 x Combining the results of smaller instances to solve the larger problem., t4 B/ }- a2 p* u$ }
& K, }4 k/ x/ K9 ZComponents of a Recursive Function 5 V* g6 P9 t, ~) d; |# B1 \5 c) |# S* \2 M
Base Case: 4 g/ A7 g# Y0 G' D9 ?' u' s; y/ E# h7 i) F( S5 X7 I
This is the simplest, smallest instance of the problem that can be solved directly without further recursion. B8 o* `" X+ e! M1 E ( Q$ k F# ?% t It acts as the stopping condition to prevent infinite recursion.; V% W q2 @- V' d/ |
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Example: In calculating the factorial of a number, the base case is factorial(0) = 1.; u7 p* h* ]6 L
( A* q9 x& _ V, M: w- d) t8 f Recursive Case:4 o/ {4 K1 m( j; c! ~/ I
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This is where the function calls itself with a smaller or simpler version of the problem.% V1 g$ B6 b, u5 s1 P6 P" B
- i. y y* _6 }' V8 p" H# I Example: For factorial, the recursive case is factorial(n) = n * factorial(n-1).( e/ N0 b, w; e' z0 a
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Example: Factorial Calculation 1 [2 x6 D8 y9 ^! C' G0 |0 m$ l" h2 K
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:* N2 x/ Y/ _- l7 M9 n, |+ U& ^
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Base case: 0! = 1 , b$ {; N3 B g2 ?. V; L2 v8 | : x2 j: [! `2 s$ y" {* n0 { Recursive case: n! = n * (n-1)! , h" d! p1 b( h1 @7 [' b- K0 V" i8 b Q2 g0 v, a; F- s/ L
Here’s how it looks in code (Python): & r& ]4 ]! f; z5 u+ jpython2 t6 i( x8 {% j4 v k' I$ M8 m
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def factorial(n):* O' {. p6 n3 n- ?* j6 m
# Base case . S3 N+ s3 O- G" w' O( g- R2 | if n == 0: " Q! H+ `* l( l* _( j! M( x! c4 I return 1 0 W: c' X; Z, J. c$ C # Recursive case 7 d+ A$ L: z" Q else: 6 B+ G7 A1 B- D return n * factorial(n - 1)7 q; c& C7 O1 i3 q
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# Example usage1 C: `* M1 T' n# O
print(factorial(5)) # Output: 120 % D- F/ p; u5 z! a; O, }8 {2 O , L; Z- d1 J. x+ R1 W. THow Recursion Works 5 A5 t4 G; H- ^8 v) h3 N( | b6 r6 h; {3 u8 p
The function keeps calling itself with smaller inputs until it reaches the base case. & e0 J7 n5 d2 P1 n) n- W4 C" q 3 e7 r! k' R5 D) l Once the base case is reached, the function starts returning values back up the call stack. 1 U5 S% }! `3 n7 @2 R3 { ( r/ O7 ?1 j( C# ]& q$ w# j$ z, z These returned values are combined to produce the final result." x P. l4 J! D. Q2 T' i4 m: c5 s# U& I
) a1 z7 K1 B# Y7 L1 `For factorial(5): ]0 n, `0 \, S$ k: x% Z- L- G
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factorial(5) = 5 * factorial(4)) A6 U6 w- P7 H; x: C: ~6 [/ H$ s
factorial(4) = 4 * factorial(3)2 \3 j6 l4 k9 r; n$ c0 Y5 N
factorial(3) = 3 * factorial(2)8 |! B5 ?% q" c
factorial(2) = 2 * factorial(1) ) p* X p& P! n6 t% `6 A2 X) Afactorial(1) = 1 * factorial(0) 7 x* I) G! D* Z- f/ N( d) ufactorial(0) = 1 # Base case ! i- `5 X/ z! U: v r* l0 Y* n+ U" X
Then, the results are combined:( k* R1 k. O2 j) S" G3 i' z* h
( s& q' w8 Q* }9 Q' ? 9 N. J- c. v" ?; { Z g$ B% efactorial(1) = 1 * 1 = 1 , L+ \% a0 [6 ]0 x) x! Yfactorial(2) = 2 * 1 = 2 9 p, r9 H1 D6 ] Efactorial(3) = 3 * 2 = 6/ L L3 H1 [5 a5 U3 j* B
factorial(4) = 4 * 6 = 24! v/ L+ W8 `5 \+ A+ ?
factorial(5) = 5 * 24 = 120; l3 S' X8 @0 `* E& R% B
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Advantages of Recursion5 @( U* ]) u" f) f
: r; A! Y: T: D5 a, J2 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).' Z7 X& m% E' Z5 s; L0 e
8 {1 _; | q7 ]5 y Readability: Recursive code can be more readable and concise compared to iterative solutions.! d4 w: V) A }9 T0 |' y8 l3 a
: a4 ^. [% J1 U$ }9 R6 S4 wDisadvantages of Recursion! u8 @& Z1 }& g) u
. _/ s' L! o* ^5 b" Y; p 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.( B4 }4 O# Q& s R" A
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Inefficiency: Some problems can be solved more efficiently using iteration (e.g., Fibonacci sequence without memoization). 3 E6 B. S) _- _( [' A+ z* }- h8 h7 C( r' p
When to Use Recursion 4 G+ g" Y5 z1 ]$ m( \* o$ y, m, K. m' x3 h: \7 s5 `1 o; B$ I0 n
Problems that can be broken down into smaller, similar subproblems (e.g., tree traversals, sorting algorithms like quicksort and mergesort). 1 |1 U- o: H0 d) K0 a2 H- Q: W1 }# Q; E/ b
Problems with a clear base case and recursive case. 2 T k1 S' F6 o' x7 D+ x: c5 p/ v& `! r, T
Example: Fibonacci Sequence1 Z; m1 N1 G ]; {; Q; U
) h2 }: |) i% I4 ?& N! \The Fibonacci sequence is another classic example of recursion. Each number is the sum of the two preceding ones:. p$ o& j; {0 t1 _" v# t
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def fibonacci(n):: x4 @. M* i& f$ W9 ~2 ~
# Base cases. B" s: p! S* B* `! b# _9 H
if n == 0:* _% L! j9 P; _" O
return 0 2 ]0 O. Q) ~, T/ P2 N elif n == 1: ) Y* B! R9 m0 I- X9 c$ O return 1 9 M- |; D# @1 [2 d # Recursive case 5 o- i8 K7 U& l( H0 D else:) I( i9 U8 a% ^& P! q# \+ Q7 p
return fibonacci(n - 1) + fibonacci(n - 2) ! b b4 |* i' d) s+ T6 d, U A" I( A) V, d0 w# W( x$ O
# Example usage& U( h; m% i7 S0 l$ O
print(fibonacci(6)) # Output: 80 T+ \( O1 b; e0 S+ M3 T
% R; x s) s3 i) w. [Tail Recursion! ^- M0 ?* Q6 y$ v; \2 h8 o- x
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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). % o* H! v" V# C) y+ O% ]$ ? + V6 v1 V8 L# M# h8 c# v- N7 AIn 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。